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Contents
1 Welcome
2 LSAT Basics
3 Game Time
4 Madness and Method
5 The Big Two
6 Game Plan
7 The Questions
8 The Blueprint
9 World of Ordering
10 Basic Ordering
11 Games Logic
12 1:1 Ordering
13 Under/Overbooked Ordering
14 Adding Tiers
15 Hello Grouping
16 The Big Four
17 In and Out
18 Two Groups
19 More Groups
20 Profiling
21 Combo Games
22 Neither?!?
23 Play the Numbers
24 Scenarios
26 Timing
27 Next...
1/WELCOME
THE BLUEPRINT
Welcome to The Blueprint for LSAT Logic Games, the best guide to Logic
Games ever written (if you don’t trust us yet, just ask Juan, the friendly
fellow who sells tamales across the street from our office). We are very
pleased that you have selected Blueprint, and we are confident we can help
you substantially improve your performance on this challenging section.
Blueprint was founded in 2005 by a group of people who believe that it’s
possible to learn difficult concepts and have fun at the same time. In general,
LSAT prep materials are about as exciting as listening to hold music while
waiting for tech support. At Blueprint, we don’t think you have to be bored to
achieve a high LSAT score. In fact, we believe students learn better when
they’re having fun. To that end, we’ve designed our live classes, online
courses, and books to be entertaining as well as informative.
Now, don’t get us wrong. At Blueprint, our first priority is the LSAT. And
we definitely know the LSAT. We actually know far more than any group of
people really should know about the LSAT (at least if they’d like to maintain
healthy relationships and functioning livers). But this LSAT knowledge has
enabled us to help thousands of students improve their scores significantly.
We also have Matt Riley, Logic Games assassin and the author of the
Blueprint Logic Games curriculum. Matt excels in three things in life:
Connect Four, spinning a basketball on his unnaturally flat index finger, and
solving games. Matt scored a 179 on his own exam and has never missed a
question on the games section of a real LSAT. But it’s not just that Matt is
able to solve logic games himself; he is also able to articulate our method and
teach it to others. Thank goodness he turned down law school to found
Blueprint, even if his mom still disapproves of the decision.
We’re happy that you’re here. Strap in, kick back, and
grab a tamale. It’s going to be a fun and educational
ride.
2/LSATbasics
LSAT OVERVIEW
The LSAT, for those of you who accidentally opened this book instead of US
Weekly, is a standardized test that every student who wants to attend an
ABA-approved law school1 must take. It consists of five timed sections of
35 minutes each and a writing sample, which is also a 35-minute section.
The test is scored on a scale of 120 (lowest) to 180 (highest).
Sections of the LSAT
The scored sections of the LSAT include two sections of Logical Reasoning,
one section of Reading Comprehension, and one section of Logic Games, not
necessarily in that order. In addition to the four scored sections, there is an
“experimental” section. If that sounds like it involves questionable decisions
made at college parties, not to worry: The experimental section is not scored.
The section can vary between Logical Reasoning, Logic Games, and Reading
Comprehension. The Law School Admission Council (LSAC), which
administers the LSAT, uses this section to test future LSAT questions. In
addition, the experimental section is not identified during the exam, so you
will have the pleasure of expending energy on a portion of the test that in no
way affects your score.
There is also a writing exercise at the end of the exam, which, as far as we
can tell, tests your ability to write an essay long after your brain has died.
Like the experimental section, the writing sample is not scored. However, it
is sent to law schools along with your LSAT score. The writing sample
challenges you to write with a pencil on paper for 35 minutes at the end of
about three hours of testing, so your chances of suffering a major hand cramp
are quite good.
Here’s a table that illustrates the six different sections
of the LSAT:
Section
Time
# of Questions
Logical Reasoning 1
35 Minutes
24 - 26
Logical Reasoning 2
35 Minutes
24 - 26
Logic Games
35 Minutes
22 - 24
Reading Comprehension
35 Minutes
26 - 28
Experimental (Not Scored)
35 Minutes
22 - 28
Writing Sample (Not Scored)
35 Minutes
N/A
LOGICAL REASONING
There are two scored sections of Logical Reasoning on the LSAT. Each one
contains between 24 and 26 questions, so you’ll answer approximately 50
Logical Reasoning questions, which means that Logical Reasoning makes
up approximately half of the test.
1. Everyone in Biba’s neighborhood is permitted to swim at Barton Pool
at some time during each day that it is open. No children under the
age of 6 are permitted to swim at Barton Pool between noon and 5
P.M. From 5 P.M. until closing, Barton Pool is reserved for adults
only.
If all the sentences above are true, then which one of the following
must be true?
(A) Few children under the age of 6 live in Biba’s neighborhood.
(B) If Biba’s next-door neighbor has a child under the age of 6, then
Barton Pool is open before noon.
(C) If most children who swim in Barton Pool swim in the
afternoon, then the pool is generally less crowded after 5 P.M.
(D) On days when Barton Pool is open, at least some children swim
there in the afternoon.
(E) Any child swimming in Barton Pool before 5 P.M. must be
breaking Barton Pool rules.
In this section, you will read a short passage followed by a question. To the
right is a question about a nice guy named Biba. You’ll have the joy of doing
this 24 to 26 times in 35 minutes. Sometimes the questions ask what must be
true or false based on the information; sometimes you’ll be asked to weaken
or strengthen an argument. Still other times, you’ll be required to identify
how an argument is flawed.
What LSAC is testing with Logical Reasoning is your ability to break down
and assess information quickly. This section crucially depends on your ability
to judge the validity of arguments without focusing on whether the statements
are true or false. In this way, logical reasoning prepares you quite well for
being a lawyer.
READING COMPREHENSION
Reading Comprehension consists of three long passages and a comparative
reading passage (itself made up of two, shorter passages). The subject matter
is invariably riveting, encompassing everything from the oral traditions of
Native American communities to various theories on the extinction of the
dinosaurs.
Each passage is 50 to 65 lines long and is followed by five to eight questions.
All told, Reading Comprehension is typically the longest section on the
LSAT and totals between 26 and 28 questions.
LOGIC GAMES
The third scored section on the LSAT is known in fuddy-duddy parlance as
Analytical Reasoning, but we, and most of the known world, affectionately
refer to it as Logic Games. In this section, you will be presented with an
introductory paragraph and rules, then asked questions about the way in
which the variable sets can be organized. Logic Games might involve clowns
getting out of cars, cereals displayed at grocery stores, or monkeys being
placed in cages. Despite the wide-ranging and sometimes absurd nature of
games, it turns out that the number of game types is actually quite small.
THE MOMENT OF TRUTH: SCORING
In order to understand scoring on the LSAT, we’ll need to discuss the
difference between a raw score and a scaled LSAT score.
Let’s use an example to show you how this works. Meet Sue. Sue took the
December 2011 LSAT. Sue eschewed a vacation to Panama City with a man
named Ricardo who made her feel both safe and dangerous to study for the
test and was rewarded by answering 78 questions correctly out of a possible
101. After LSAC ran all of the tests through their fancy, 1970s-era Scantron
machine, they determined that Sue and her 78 correct answers placed her in
the 90th percentile. This means Sue beat out 90 percent of other test takers.
LSAC then references her score against a percentile table and emails Sue her
final score about three weeks later—in this case, a 163.
Raw Score
Scaled Score
98 - 101
180
93
175
87
170
80 - 81
165
73
160
64 - 65
155
56
150
Here is a truncated version of the score conversion chart from the December
2011 LSAT. The score conversion chart changes slightly for different LSATs
depending on how the test- taking population fares, but it has stayed
relatively consistent over the years.
Remember, your score is determined by how you fare in comparison to
thousands of other test takers.
What does the LSAT test?
Now that you have an understanding of how scoring on the LSAT works, it’s
quite natural to wonder what it tests. In a nutshell, the LSAT tests the skills
you’ll need as a law student and as a lawyer. This includes constructing and
analyzing arguments (Logical Reasoning), understanding dense passages of
material (Reading Comprehension), organizing information intelligibly
(Logic Games), and how to bend a spoon with your mind.2
What the LSAT does not test, however, is rote memorization. Rather than
asking you for the capital of North Dakota (Bismarck) or the German
Chancellor during World War I (Bismarck), the test challenges you to think
in a certain way. This could involve weakening an argument, identifying the
main idea of a passage on Thurgood Marshall, or deciding whether or not the
third mannequin wears a blue hat.
The downside of this method of testing is that you
cannot “cram” for the LSAT; you must acquire the
skills for it through diligence and repetition. While
this is unfortunate news for those of you who
scraped by in college writing your papers hours
before they were due, it’s good news for those of
you who enjoy reading and thinking critically. Like
all skills in life, some good practice can really enhance your abilities in
these areas.
LSAT ADMINISTRATIONS
The LSAT is administered four times a year: February, June,
September/October, and December.
As you can see, the Sept/Oct LSAT has historically been the most popular
administration, followed by the December exam. The June and February
LSATs have fewer test takers, but each still sees around 25,000 students.
The LSAT is typically given early on a Saturday morning, due to LSAC’s
insidious plot to ruin your entire day. The June exam is the exception, as it is
given on a Monday afternoon.3
For exact dates and times, check the LSAC website at
http://www.lsac.org.
Is there a “best” time to take the LSAT?
There is a “best” time to take the LSAT, but probably not in the way you’re
thinking. Unfortunately, it is not the case that one LSAT administration is
easier than the others. If you ponder this for a second, you’ll probably see
why. If the February test, for example, were always easier than the December
test, this would be spectacularly unfair for the December test takers. More to
the point: No one would take the December test, and everyone would take the
February exam. To guard against this, LSAC institutes elaborate procedures
to ensure that all the exams are fairly equal.
However, the simple ease of the test isn’t the only reason to take a particular
administration. Most law schools work on a rolling admissions basis in which
students are admitted as applications come in. It’s generally better to apply
earlier in the process because there are typically more seats available.
Law schools typically begin accepting applications in early fall, so taking the
February, June, or September/October LSAT will ensure your score arrives
early in the process. All law schools accept the December LSAT, but you’ll
want to be absolutely sure your applications are ready to go by the time you
get your LSAT score. Some schools will accept the February LSAT for the
same year in which you’re seeking to enroll, but many won’t. Taking the
February exam to matriculate the same year will generally ensure that you’re
at the bottom of a squirming, anxiety-ridden mass of applicants.
But whether you apply early or late, the biggest component of your
application is the LSAT score itself. It is such an important piece, in fact,
that applying later with a better LSAT score usually outweighs applying
earlier with a lower score. To that end, the “best” time to take the LSAT is
when you have the most time to prepare for it. That is why so many people
take the September/October test, when the principal part of their study time
falls during the summer months when school is out.
IS IT REALLY WORTH THE WORK?
The LSAT is the single most important piece of your law school application
(apart from substantial donations to university library foundations from a
wealthy family member). Conventional wisdom holds that the LSAT
carries the weight of about 60% of your application. Because of its
importance in the application process, the better your LSAT score, the better
the law school you can typically attend. As law firms become increasingly
selective in the hiring process, a better law school can mean a better job. A
better job often equates to a better-looking spouse, which in turn can
influence the attractiveness of your offspring.4 In other words, a higher LSAT
score can equate to an evolutionary advantage for passing on your genetic
traits. That’s just basic Darwinism.
Not only can a higher LSAT score get you into a better law school, it can also
result in scholarship money once you’re there. Why not save on tuition so
you can use your money for better things, like purchasing a taco truck or an
exotic pet farm?
Finally, more so than any other standardized test, better performance on the
LSAT correlates to better performance in law school. So in studying for the
LSAT, you have the satisfaction of knowing it can pay dividends in law
school.
Why devote a whole book to Logic Games?
Of the different sections on the LSAT, we find that our students are initially
most frightened of Logic Games. (Logical Reasoning generally causes
amused resignation, and Reading Comprehension mostly leads to anger.)
This is probably because Logic Games look nothing like anything you’ve
seen before. Most people are familiar with long passages (Reading
Comprehension) or with paragraphs of information (Logical Reasoning). But
trying to decide whether the third dinosaur is green or mauve can be a foreign
experience (unless you’re a paleontologist throwing a Mardi Gras party).
There is good news, however! As odd as logic games initially appear, they
are eminently learnable because they don’t actually test that many different
skills. In fact, we find that most of our students see their biggest increases in
Logic Games.
More to the point, learning to master Logic Games will reward you on other
sections of the LSAT. For example, the diagramming skills that enable you to
see that the veal being served on Wednesday forces the tilefish to be served
on Friday will also have a big payoff in Logical Reasoning.
THE BLUEPRINT ADVANTAGE
Now that you’re convinced you need to hone your games skills, you might
question why our Logic Games method is better than other companies’, or
why you can’t just tackle logic games with good old-fashioned know-how.
Blueprint has devised the most effective approach to
Logic Games. We break down all the games into just a
few categories, then provide you with the tools to 1)
recognize which category the game falls into and 2)
solve it.
As you will soon see, our approach is simple, yet powerful. Rather than
throwing random games at you, we have isolated the basic processes driving
all games. By helping you identify these salient features on your own, our
methods will help you simplify any game that is thrown in your direction.
But there’s something else that sets us apart. Other Logic Games books will
give you a game, ask you to do it, then explain how to solve it. This means
that readers have to reconstruct, as best they can, when they were supposed to
diagram something, or how an important deduction was made. In The
Blueprint for LSAT Logic Games, however, we’ve broken down all the games
and the questions into their component parts. We tell you what to do the
minute you receive new information. As soon as a rule indicates how to
construct a setup or leads to a deduction, we’ll walk you through the process
of how to recognize it and what to do with it.
We call this step-by-step process Blueprint Building BlocksTM, and it walks
you through every moment of a game.5 It’s as close as you can get to having
an instructor with you while you’re reading (which, depending on your
location while reading, could be a little creepy). It also requires a lot of work
on our end, which is our best guess as to why we haven’t seen it anywhere
else. Whereas other companies explain how to solve games after the fact,
we’re able to work through games with you, step by step, to show you the
crucial maneuvers that occur at each stage.
After reading every single logic game available to man, woman, or armadillo
(one of nature’s more under-appreciated animals), we’ve also figured out that
there’s a whole bunch of stuff the test makers ask about over and over again,
so we can give you tips on how to look out for common patterns as well. In
the end, we collapsed all of this knowledge into this book so you’ll absolutely
kill the Logic Games section on your LSAT. And by “kill,” we mean achieve
a score that will get you a nice glass office and an array of expensive vacation
homes.
YOUR STUDY GROUP
Now that we’ve covered the basics of the LSAT, it’s time to meet your study
group. No one likes to study alone. You’re probably reading this by yourself,
possibly somewhere in the badlands of South Dakota. We all need friends,
particularly for something as traumatic as LSAT study. Sit back and say hello
to your new comrades.
The Ninja:
The Ninja is very powerful, as you can see by his extremely
long sword. Your goal is to become the Ninja, for he knows
games like the back of his karate- chopping hand. At various
points, he will contribute Ninja Notes to which you should
pay close attention. These will be advanced comments about
specific games or techniques. Listen to the Ninja, and one
day the Ninja you will become.
Cleetus:
Poor Cleetus. Unfortunately, not all of us can be born with
the correct number of chromosomes. Or teeth. You see,
Cleetus is not very skilled at Logic Games. You definitely do
not want to be Cleetus. But Cleetus will commit some of the
common mistakes that students make, and those are the same
mistakes you need to learn to avoid.
Ditz McGee:
Say hello to Ditz. She doesn’t really want to be here. Truth be
told, Ditz wants to become a pop star, but the singing career
has not quite taken off. So Ditz is thinking about law school
as a backup plan. Since she is lazy and isn’t committed to
studying, Ditz will try to skate through and find ways around
actually learning. Like Cleetus, Ms. McGee is the voice of
mistakes you’ll want to avoid. Remember, there is no quick fix for Logic
Games, just like there is no quick fix for poor career choices or tattoos of an
ex-boyfriend’s name.
BP Minotaur:
Finally, we come to the Minotaur. He’s a master of
erudition, and his study smells of rich mahogany. In case
you were not aware, he’s a figure from Greek mythology
who dwelt in a maze and ate Athenian youths sent to Crete
as tribute. Now, he’s the perfect figure to lead you through
the labyrinth of Logic Games. The Minotaur will be the
voice of Blueprint and lend helpful tactical hints along the way. Plus, we like
his horns and his monocle.
So that’s it for the introduction, friends. As we stated earlier, the world of the
LSAT and law school admissions is a competitive one. There is a pot of gold
at the end of the rainbow, but you have to beat out all the other leprechauns.
If other students are studying harder or better than you, this will hurt your
chances on test day. You can, however, rest easily knowing this book will
help you ensure that is not the case.
You’ve come to the right place to become an expert in Logic Games, and
this is the right guide to take you through the process. Someday, when
you’re an elder with a grandchild on your knee for whom you’ve set up a
trust fund (provided he or she doesn’t squander it on drugs and orgies),
you’ll look back on what you did here and say: “Man, Blueprint *+%*ing
rocked.”
1 What is an ABA-approved law school, you ask? It stands for the American Bar
Association, whose job it is (besides making your life relevant with a little thing
called the Bar Exam) to make sure that law schools are reputable. If your law school
is not ABA accredited, it means you may not be able to take the Bar Exam in any
other state, thus potentially limiting your options for working as a lawyer and
purchasing very expensive automobiles.
2 This is not actually tested on the LSAT. But if you can do it, you should already be
aware that none of this is real.
3 If you need six cups of coffee and a defibrillator to wake up before 8 AM, the June
exam might be for you.
4 While we are very confident in our ability to improve your performance in Logic
Games, we cannot guarantee anything about the attractiveness of your significant
other. You have to work a little bit, too.
5 At Blueprint, we enjoy gratuitously trademarking terms. Particularly assertions
abounding in alliteration.
3/GAMEtime
LOGIC GAMES?
At some point, someone decided to call the Analytical Reasoning section of
the LSAT “Logic Games.” That person deserves to have a celebrity starlet
appointed as his or her permanent designated driver.
For most students, games will feel very foreign at first. Unlike other sections
of the LSAT, your education has provided very little training in how to
arrange six different dresses on consecutively-numbered hangers (something
you get to attempt in a few chapters). Though called “games,” when you first
encounter these little logic puzzles, they will have very little in common with
Scrabble or Taboo. They will feel more like an organic chemistry final. Or a
root canal. Or a cage fight against a UFC champion. These so-called “games”
will chew you up and spit you out. But you will get your revenge. With
rigorous tutelage and lots of practice, this book will give you the tools to
master Logic Games.
Logic Games is the most
abstract section of the LSAT. While
you may have some experience reading
academic passages or analyzing
arguments, you simply have not honed
your ability to assign jugglers to circus
rings yet.
BP Minotaur:
Can I get better?
That’s a fair question. Logic games are a new skill you have to develop.
Think about it this way: Could you improve your skills on a pogo stick if you
really wanted to? Of course. Many students perform horrifically on the games
section initially. But there is hope.
Thankfully, the people who write this wonderful exam appear to be stuck in a
rut. We can only assume they’re the kind of people who watch nothing but
M*A*S*H reruns, eat chicken and rice for dinner every night, and paint every
wall in their house a different shade of beige. They also make the same types
of games over and over and over again. When one game challenges you to
schedule game show contestants and another game involves secret computer
codes, it can be difficult to recognize any similarities between the two.
However, this book will give you the tools to recognize the repetitive features
in any game you encounter. Once you realize you are doing the same type
of games over and over again, your performance will improve
dramatically.
GAMES, UNDRESSED...
We at Blueprint feel the best way to get to know something is to strip it
down. In the Logic Games section, there are always four distinct games, and
each game has three basic parts: the introduction, the rules, and the
questions. Say hello to a game:
Introduction
This paragraph lays out the
situation for the game. It will
identify what the game is asking
you to do (put tigers into cages,
dress up mannequins, etc.) and the
players in the game (soft drinks,
cookies, dinosaurs, etc.).1
Questions 7-12
A soft drink manufacturer
surveyed consumer preferences
for exactly seven proposed names
for its new soda: Jazz, Kola,
Luck, Mist, Nipi, Oboy, and
Ping. The manufacturer ranked
the seven names according to the
number of votes they received.
The name that received the most
votes was ranked first. Every
name received a different number
of votes. Some of the survey
results are as follows:
Rules
The indented rules establish the conditions that must be followed for the
duration of the game. They are
often vague and ambiguous and
must be interpreted to figure out
what they actually mean. In this
regard, reading the rules may
remind you of talking to your
significant other.
Jazz received more votes than
Oboy.
Oboy received more votes than
Kola.
Kola received more votes than
Mist.
Nipi did not receive the fewest
votes.
Ping received fewer votes than
Luck but more votes than
Nipi and more votes than Oboy.
Questions
Finally, the payoff. Maybe. Each
game has between five and seven
7. Which one of the following could
questions. They are designed to test
be an accurate list of the seven
how well you understand the
names in rank order from first
introduction to the game and the
through seventh?
rules. However, if you’ve done
what you’re supposed to do with
(A) Jazz, Luck, Ping, Nipi,
the introduction and the rules, most
Kola, Oboy, Mist
of your work should be done by the
(B) Jazz, Luck, Ping, Oboy,
time you get to the questions.
Kola, Mist, Nipi
(C) Luck, Ping, Jazz, Nipi,
Oboy, Kola, Mist
(D) Luck, Ping, Nipi, Oboy,
Jazz, Kola, Mist
(E) Ping, Luck, Jazz, Oboy,
Nipi, Kola, Mist
8. Which one of the following
statements must be true?
(A) Jazz received more votes
than Nipi.
(B) Kola received more votes
than Nipi.
(C) Luck received more votes
than Jazz.
(D) Nipi received more votes
than Oboy.
(E) Ping received more votes
than Kola.
And when you are done... repeat three times. Oh, and
remember, you have 35 minutes to complete all four
games. Sound like fun?
1 If you have a group of people, the LSAT will generally use a very racially diverse
set of names. For example, a game might tell you that six students—Akhbar, Bing,
Chowdury, David, Ephraim, and Faquishah—attend a musical performance. The
LSAC: Always bridging the cultural divide.
4/MADNESSandMETHOD
THE MADNESS
So how should we proceed to learn about games? The first step might
surprise you. Most people like to learn how to do games in the same way they
learn how to play a new game of cards, namely by just dealing a few hands
and seeing how it goes. Students generally like to jump right in and start
doing games. If you’re playing gin rummy in your apartment, this might be a
fine strategy. If you are at a high limit table poker table in the Bellagio, you
could end up performing unspeakable acts just to scrounge up the money for
a bus ticket home. The stakes on LSAT games are every bit as high, and you
should be equally circumspect about your method of preparing.
The seductive mistake is that just doing a bunch of games without a method
gives the illusion of learning. You will indeed become more familiar with
games this way, which might help to remove some of the nervousness that
makes games unpleasant for you right now. But without a more theoretical
approach, you will never be able to reach your true potential.
We believe that the makers of the LSAT are actually counting on this.
Knowing most students blindly practice game after game, the test will
include games that are slightly different than earlier games. Though
superficial, these differences are often enough to send students who lack
theoretical training into a sobbing panic.
At Blueprint, we have identified the general structures that underlie all games
in the history of the LSAT. As we will soon show you, there aren’t very
many. Once we provide you with these structures, they will allow you to
understand games on a theoretical level, and you won’t be intimidated if they
insert a superficial twist on the actual test. But we can talk about this all day
and you still won’t be convinced. As they say, seeing is believing but tasting
is proof (that actually has nothing to do with our current discussion).
The Challenge
So here is what we’re going to do: You’re going to do a game. Yep, you’re
going to attempt a game with whatever unsystematic, cobbled together
method you are currently using.
If you ace this game in 5 minutes, put down this book and start packing for
Harvard Law School. (Remember, Boston is very cold in the winter.) But we
think that’s extremely unlikely. If you are like almost everyone in the world,
this is going to hurt.
When you’re done, we are going to show you the correct way to solve the
game, Blueprint style. It’s always good to have a clear view of the goal.
When you’re ready, turn the page.
DECEMBER 1997: GAME 4 (18-23)
This is the fourth and final game from the December 1997
LSAT. On average, you have eight minutes and 45 seconds to
complete each game on test day. So grab a watch, egg timer,
or sun dial and set the clock. Your challenge, if you choose to
accept it, is to complete this game in the time allotted. Once
you finish (or when time is up), flip the page and we will
discuss how Blueprint can help this process become more
enjoyable.
Questions 18-23
Nine different treatments are available for a certain illness: three antibiotics—
F, G, and H—three dietary regimens—M, N, and O—and three physical
therapies—U, V, and W. For each case of the illness, a doctor will prescribe
exactly five of the treatments, in accordance with the following conditions:
If two of the antibiotics are prescribed, the remaining antibiotic cannot be
prescribed.
There must be exactly one dietary regimen prescribed. If O is not
prescribed, F cannot be prescribed.
If W is prescribed, F cannot be prescribed.
G cannot be prescribed if both N and U are prescribed. V cannot be
prescribed unless both H and M are prescribed.
18. Which one of the following could be the five treatments
prescribed for a given case?
(A) F, G, H, M, V
(B)
(C)
(D)
(E)
F, G, M, O, V
F, H, M, O, W
G, H, N, U, W
G, H, O, U, W
19. Which one of the following could be the antibiotics and physical
therapies prescribed for a given case?
(A)
(B)
(C)
(D)
(E)
F, G, H, W
F, G, U, V
F, U, V, W
G, U, V, W
H, U, V, W
20. If O is prescribed for a given case, which one of the following is a
pair of treatments both of which must also be prescribed for that
case?
(A)
(B)
(C)
(D)
(E)
F, M
G, V
N, U
U, V
U, W
21. If G is prescribed for a given case, which one of the following is a
pair of treatments both of which could also be prescribed for that
case?
(A)
(B)
(C)
(D)
(E)
F, M
F, N
N, V
O, V
V, W
22. Which one of the following is a list of three treatments that could
be prescribed together for a given case?
(A)
(B)
(C)
(D)
(E)
F, M, U
F, O, W
G, N, V
G, V, W
H, N, V
23. Which one of the following treatments CANNOT be prescribed
for any case?
(A)
(B)
(C)
(D)
(E)
G
M
N
U
W
Ditz McGee:
I feel like I just got hit by a
truck.
That’s pretty normal. But don’t freak out. You are new to this. Before you
choose a new career path, it’s time to discuss how we can help.
THE METHOD
Next, we are going to walk you through the Blueprint way to solve this game.
Here’s a quick warning: This will be a bit confusing at this point. But that’s
natural because you are new to this. The following is not intended to be a
complete explanation of the game, but just some of the highlights. In about
500 pages, you will be able to do this on your own.
Note: There is a complete explanation for this game
online in your MyBlueprint account. We will
more thoroughly discuss these videos in a couple chapters.
In this game, you get to play doctor. There’s a nasty illness, and you have
to select which treatments to prescribe. Time for the big reveal.
The basic task in this game is to select five of
nine treatments.
Based on the first two rules, only two
selections are possible:
One option is to select one antibiotic, one
dietary regimen, and all three physical
therapies.
The second option is to select two antibiotics, one dietary regimen,
and two physical therapies.
In the first situation, all three
physical therapies (U, V, and W)
are selected.
According to the last rule, H
must be hmuvw the antibiotic
and M must be the dietary regimen. That option is completed nicely.
In the second situation, two of the
three physical therapies are
selected.
V could still be selected, with either
U or W.
If V is selected, H must be one antibiotic and M must be the dietary
regimen.
F can’t be prescribed since O isn’t prescribed. Thus, G is the second
antiobiotic.
V might not be prescribed in the
second situation.
If V is not selected, then U and
W both are.
W and F can’t both be selected,
so G and H are the two antiobiotics.
Since G and U are both prescribed, N can’t be prescribed.
The dietary regimen must be either M or O.
At this point, there are a few very important things to note about the game:
For the antibiotics, F is never prescribed and H is always prescribed.
N can’t be the dietary regimen that is prescribed.
As you can see, this game is
simplified greatly before any questions
are attempted. You now know that F
and N can never be prescribed, so you
can quickly eliminate any answer
choice that prescribes either of those
treatments.
BP Minotaur:
The three situations above illustrate that there are way less possibilities in this
game than you might have initially realized. We worked through that process
very quickly and without much fanfare, but that will come later in the book.
The goal of this exercise is simply to show you what can result with the
proper approach.
You can probably imagine that the questions are going to go very smoothly at
this point. But let’s run through them so you can see how fulfilling this
process can be.
Question #18
(A), (B), and (C) each include F. (D) throws N into the mix. Those four
answers are eliminated, so (E) is the correct answer.
Question #19
There’s that pesky F again. Get rid of (A), (B), and (C).
As you saw above in our first situation, if all three physical therapies (U,
V, and W) are prescribed, the other two treatments are H and M. (E) is
the answer again.
Question #20
O can only be prescribed in the third situation above.
If O is prescribed, the two physical therapies are U and W. (E) is the
winner yet again.
Question #21
(A), (B), and (C) are each thrown out because they include F or N, the
evil letters.
O and V are never prescribed together, but V and W are a possibility.
(E) again. Crazy.
Question #22
(D) is the only answer that doesn’t include F or N, so this one is easy.
Question #23
F or N would be great answers on this one. Look, there’s N in answer
choice (C). Nice.
How was that? Much better, hopefully. To be fair, this is a game that has two
great characteristics: (1) It’s very difficult for students and (2) it breaks down
very nicely when using the Blueprint methods. You might not have grasped
every step of our discussion at this early stage, but hopefully you can
appreciate the importance of a good approach. This exercise is designed to
contrast your current, haphazard approach with the most effective way to
attack this game.
So how do I get there?
That’s a great question. As we stated earlier, the difficult part for students is
that they just want to start doing games. It seems logical that the best way to
improve your performance on games is by working through a bunch of
games. But that’s not the case. That’s like saying the best way to improve at
karate is to get in a lot of fights. Ouch.
On the last couple pages, you saw the proper way to attack a difficult game.
But there is no quick and easy guide for mastering that strategy (as evidenced
by the size of this book). You probably won’t see a game about antibiotics
and physical therapies when you take the LSAT. And it’s not as if the letters
F and N are never going to be selected. If there were 10 Easy Steps to
conquer games, students would all ace the section, and the test would have to
change (not to mention the fact that Blueprint would not be able to drum up
much business). But the challenge is more complicated than that.
Games are difficult because they test a method of critical thinking. You have
to be able to analyze a game in a powerful and expeditious manner. This is
the theoretical understanding we mentioned earlier. We will teach you to do
just that. But you have to be patient. We will do plenty of games throughout
this book, but not right up front. There will be a number of chapters that
discuss general strategy and focused drills before we set you free to battle
games about alphabet soup or criminal suspects (two more exciting
challenges that are in front of you).
So let’s start at the beginning...
5/theBIGtwo
CAN IT REALLY BE THAT SIMPLE?
It’s time to talk strategy. When you are first confronted with games, they
might seem overwhelming. This is largely because there appears to be
hundreds of different types of games. One minute, you’re helping friends
move furniture. The next, you’re scheduling workers at a convenience store.
Then, you find yourself choosing cookies for a cookie jar.
I ain’t got no shot at
learning to do all those things.
Cleetus Comment:
Cleetus, you might be right. However, here’s the great news: you don’t have
to!1 It turns out that, in Logic Games, you don’t have to learn to complete
hundreds of different tasks. The masterminds who make this test just want
you to think you do. While the subject matter may vary widely from game to
game (from monkeys to swim teams to toothbrush colors), this is completely
irrelevant to the task at hand. The subject matter of a game may be interesting
or even entertaining, but the topic never determines how you approach a
game. Rather, the basic process you are being asked to perform is what really
defines a game.
Great news - Blueprint has
cracked the code and discovered there
are only a small number of game types
that appear on the LSAT.
BP Minotaur:
You might find this hard to believe, but there aren’t hundreds of types of
games. There aren’t even twenty, or ten, or five. Hold on to your seat...
At Blueprint, our revolutionary approach classifies every single game in
the history of the LSAT according to which of two basic elements it
contains: ordering or grouping.
Yep, just two. All of the hundreds of games in the history of the LSAT can be
boiled down to ordering or grouping. Put simply, you are either lining people
up (ordering) or making teams (grouping). Once you recognize this fact,
Logic Games becomes much more manageable. Perhaps (dare we say it?)
even pleasurable.
ORDERING
Let’s take a look at ordering, the first task that you must master. Ordering
games are all about spatial relationships and discerning the relative positions
of the participants.
Which show is ranked first? How many people finish
before Timmy? Who could have an appointment on
Thursday? Which turtle must place second?
You are guaranteed to see at
least one ordering game on your
LSAT, and there is normally more than
one.
Ninja Note:
Here are some examples of ordering games:
Seven third-graders must be ranked
according to their speed on a tricycle, from
fastest (first) to slowest (seventh).
The doctor must schedule appointments
with six different patients this week,
between Monday and Friday.
The animals are assigned to a row of
consecutively numbered cages, from 1
through 7.
At the talent show, each child must fill one
time slot. The time slots begin at 1 P.M.
and end at 8 P.M.
At first glance, these games might seem to be quite distinct from each other.
However, the basic process in each game is the same: You are determining
the order for a set of players. You should always build a setup similar to the
following to work through ordering games:
The rules for ordering games place spatial restrictions on the players. Some
rules concern only one player, and other rules will place two players in
relation to each other. For example:
Azadeh is faster than Benjamin.
Christi cannot perform at 3 P.M.
Bartholomew cannot visit the doctor
before Dirk.
The ferret is in a lower-numbered cage
than either the lion or the koala.
Ordering games will be the first challenge we tackle in this book.They come
in many different shapes and sizes,
but the basic process is always the
same.
GROUPING
The second task for you to conquer is grouping. You are very likely to see at
least one grouping game on the fateful day you take the LSAT. Grouping
games are not concerned with spatial relationships, but rather focus on
assigning people to different groups.
Who is on the blue team? Which friends are coming to
the party? Who cannot be selected with Harry? How
many students can be in classroom 4?
Here are some examples of grouping games:
Of the eight former child stars who
auditioned, exactly five were chosen for
Celebrity Rehab.
Each of the seven parents must serve on
either the recycling committee or the
academics committee.
The volunteers must be separated into
three different work teams.
Each child receives at least one of three
different types of gifts for
Christmas/Hanukkah/Kwanzaa.
These games are clearly different from the previous ordering games.
However, there is a x: common element to them: Each of these situations
requires you to assign the players
to different groups. As with
ordering games, we will also use a
consistent setup for grouping
games.
The rules in grouping games will focus on the relationship between the
players - restricting the players that must, can, or can’t be assigned to the
same group. There are some examples on the next page.
If Azadeh is chosen, then Benjamin is also
chosen.
Bartholomew and Dirk cannot serve on the
same committee.
Each work team must have at least two
volunteers.
Ebenezer receives more gifts than
Gertrude.
Some grouping games only select one group,
others assign players to two groups, and
crazy games might have up to four or five
groups. But they all have groups, and your
job is to assign players to them.
BP Minotaur:
That covers the two basic processes that drive
games. Once in a while, the makers of
the exam want to throw a new
challenge at you. But it’s not really
new at all - it’s just ordering and
grouping together.
COMBO
You knew there had to be some twist coming. There are tons of ordering
games, and you are going to get sick of seeing grouping games. But what
happens when the LSAT wants to add some complexity? It makes a game
that combines the two. We call these combo games (creative, we know).
While not as common as either ordering or grouping games, they do rear their
ugly heads on occasion.
Here are some examples of combo games:
At the family picnic, the eight family
members are separated into two fourperson teams for the relay race. Each team
will complete four consecutive legs,
numbered 1 through 4.
A teacher must select six of eight students
to compete in the math competition. The
six competitors are ranked from first to
sixth.
Note that there are both ordering and grouping elements to these games.
Later in the book, we will teach you the techniques to build effective setups
for combo games. Since there is a lot of information to juggle, it is crucial to
keep everything organized. The good news is that the rules and deductions
you will find in combo games are the same as ordering and grouping games;
you just have to deal with both at the same time.
The dreaded “combo” games tend to appear
toward the end of the section, undoubtedly another tactic
BP Minotaur:
by the LSAT to test your fortitude.
Neither Games
The vast majority of our time will be spent on ordering and grouping games,
with a few combo games thrown in for fun. While it is overwhelmingly likely
that these categories will cover all four of the games you will see on your
LSAT, there are a few outliers. Once in a while, the LSAT throws a real
curve ball. Instead of asking you to rank people or make teams, you will be
required to draw a map, or discern a pattern, or choose a costume for a clown,
or try to pat yourself on the top of your head while rubbing your belly (okay,
that last one is not real). These games include neither ordering nor
grouping. They are very rare, but we will take a brief look at them later in
the book.
This chapter provides an outline for the entire
Blueprint approach to Logic Games. Since
ordering and grouping are the basic tasks that
drive all games, most of the later chapters will
focus on ways to master these processes. This
elegantly simple structure provides you with a
huge advantage over other test takers.
Up next, we are going to discuss the general approach you should use to
attack all games, regardless of type.
1 If we were talking to you in person, we would try to recreate the scene from The
Matrix in which Neo asks Morpheus if, one day, he will be able to dodge bullets.
Morpheus famously responds, “Some day, you won’t have to.” This is an equally
dramatic moment in your LSAT metamorphosis.
6/gamePLAN
THE FOUR STEP PROGRAM
Games can be intimidating because it’s not always clear when to do what.
The process does not seem as linear in nature as other parts of the exam, and
this can lead to frustrating indecision.
When should I build my setup? Should I spend more
time on the rules? Did I find all of the deductions?
Should I move to the next question? Should I just fold
my test into a paper airplane and weep quietly?
These questions can haunt students and hurt their performance.
Fortunately, we have the best strategy in the known universe for solving
games. We’ll show you how to tackle any logic game with just four steps.
By approaching games the same way every time, not only will you be
armed with an effective methodology, but you’ll also be able to solve the
games within the time allotted.
Every great strategist has a game plan. Up next is Blueprint’s four step
approach for solving any logic game.
Solving a game is dependent
on successfully completing each of
these four steps. Neglecting any one of
the steps will spell disaster.
BP Minotaur:
STEP 1: READ THE INTRODUCTION
The first step in each game is to read the introductory paragraph. This is
when you must identify the situation, the players, and your task.
Understanding these aspects of a game will largely determine the next eight
to ten minutes of your life.1
You will have a natural urge
to get to the rules and questions as
quickly as possible, but read the intro
to each game very slowly and
carefully.
Ninja Note:
During the first stage of a game, there
are two main objectives. First, you
need to classify the type of game that
you are facing (Ordering? Grouping?
Combo?) in order to build the
appropriate setup. The specifics of
constructing setups will be covered
extensively later in this book. For
now, just know that you will need to create an effective setup as part of the
first step in each game. It is vital to identify the type of game you are tackling
so you can construct the appropriate setup. Otherwise, you might build a
beautiful ordering setup... for a grouping game. Not a good day.
The second objective is to identify the variable sets in the introductory
paragraph. You need to account for people, places, gerbils, airplanes, colors,
jock straps, lipstick brands, turtles, or anything else that might show up.
Simplify all of these variables near your setup (normally, the first letter of
each will do).
Objective 1: Classify the type of game and build a
setup.
Objective 2: Identify all of the variable sets that are
present.
In very rare cases, you will not
be able to identify the type of game
from the intro paragraph. The ninja
will teach you how to attack such
games in a later chapter. The ninja
commonly talks in the third person.
Ninja Note:
STEP 2: SYMBOLIZE THE RULES
Following the introductory paragraph, there are always a number of indented
rules. These rules lay out the conditions that must be followed for the
duration of the game.
There are a number of objectives to keep in mind while you work through the
rules.2
1. Always represent rules uniformly.
You will be presented with the same
types of rules again and again, and
consistently representing them
accurately will help you recognize
familiar patterns. Don’t worry, we’ll
show you how to do this soon.
2. Represent as many rules as possible directly on your setup. Your focus
will be largely on your setup as you work through the questions, so
inserting rules directly into your setup will be helpful.
3. Combine rules whenever possible. If Larry arrives before Mo and Mo
arrives before Curly, these rules should be combined. It is always favorable
to deal with fewer rules as you move forward in a game.
Always pay close attention
when one player is mentioned in more
than one rule. If Javaris is mentioned in
the second rule and the fourth rule, a
deduction is sure to follow.
BP Minotaur:
4. The fourth and final objective sounds simple: read each of the rules
twice. Horror stories abound of students who misread or misinterpreted a
rule. If a rule states that Pedro is taller than Oscar, and you symbolize that
Oscar is taller than Pedro, you will be attending the Puerto Rico School of
Legal Stuff quicker than you can say Puerto Rico School of Legal Stuff.
After you symbolize a rule, you should go back and read the rule again to
ensure you understood and symbolized it correctly.
Here is a quick summary of the important points to remember about the rules:
1. Symbolize rules consistently.
2. Represent rules directly on your setup whenever
possible.
3. Combine rules whenever possible.
4. Don’t f#$% it up (read each rule at least twice).
STEP 3: MAKE DEDUCTIONS
Remember how Sherlock Holmes could solve a murder by discovering a
piece of hair in the carpet and combining that with the direction of the breeze
to deduce that the butler used the candlestick in the library?3 Now you can
take a shot at being just like Sherlock. Step Three is the all-important
deduction phase. Despite the fact that this is the most crucial phase in Logic
Games, it is also the most commonly overlooked by students.4
If you make the proper deductions, the game
will feel like a vacation in Hawaii with a fruity
umbrella drink. If you don’t, the questions will
feel like shower time at San Quentin.
In later chapters, we will cover a variety of
common deductions and bug you incessantly to
spend an extended amount of time searching for
deductions. Students always have the inherent
urge to jump into the questions because, after all, that’s where you get the
points. But that is a dangerous mistake. In games, the true challenge is to
process the setup and rules, to get a good grasp on a game before you ever
look at the questions. Here’s a motto to live by:
You win or lose every game before you look at a single
question.
As you improve in games, you will spend more and more time making
deductions. Additionally, as games increase in complexity, they require more
time up front. Here is how a poor student (you, at the beginning of this book)
approaches a game on the LSAT.
Setup, Rules, + Deductions
Questions
3 Minutes
10 Minutes
This 13-minute plan is exactly what you need to avoid. It will lead to
frustration, random guessing, wrong answers, and nightmares in the weeks
following the test. Here is how an advanced student (you, at the completion
of this book) destroys a game.
Setup, Rules, + Deductions
Questions
6 Minutes
2 Minutes
As you work through games,
do not judge your success solely by
right and wrong answers. Your ability
to spot key deductions is equally
important.
Ninja Note:
As you will see, spending more time in the deduction stage trims valuable
minutes off the total amount of time you spend on a game. Okay, enough
nagging about deductions... for now.
STEP 4: MOVE TO THE QUESTIONS
Now, and only now, are you ready to jump into the questions. Armed with a
good setup, a deep understanding of the rules, and some powerful deductions,
it’s time to show this game who’s boss. The questions will present you with a
variety of challenges to test how well you understand the game.5
In the next chapter, we will discuss
specifics regarding the questions you
are going to face. Questions require
different tactics, but we will teach you
the best strategies for approaching any
question you encounter. As long as you
have built the proper setup, symbolized the rules correctly, and made the
necessary deductions, the questions should be smooth sailing.
THE MASTERPIECE
There is an additional obstacle in the games section that we should discuss. In
an apparent attempt to cater to oompa loompas, the makers of the LSAT
don’t provide you with a huge amount of space to work. No scratch paper is
allowed, so you need a good plan.
Duh... I can just erase when I
need more space. I already have my
sparkly pink eraser.
Ditz McGee:
Ditz does raise an interesting issue. That might sound like a reasonable
solution, and we definitely do not want that fancy new eraser to go to
waste. However, despite the $150 or more you spend to take this
wonderful exam, the LSAC does not believe you deserve to take the test on
actual paper. The LSAT is printed on newsprint (comparable to single-ply
toilet paper). If you attempt to erase, you will very likely tear your test,
have a panic attack, stab the person next to you so that you can use their
test packet, get kicked out of the test, and quite possibly end up in prison.
This could easily hurt your chances of attending a good law school
(although it would have the makings for a great personal statement).
You have to learn to work in the space provided.6 It’s very important to
separate your original work on the game (setup, rules, and deductions) from
your work on the questions. Otherwise, you could easily mistake an original
condition in the game with one that is only provided for one question.
If you attempt to erase work
as you move through a game, you run
the additional risk of erasing a
deduction. That would truly be a
shame.
BP Minotaur:
For questions that add new information, sketch a brief hypothetical next to
the question rather than filling in your original setup. It’s very important to be
able to refer back to both the original rules in a game and any work you
completed for a specific question.
When you work through the
questions, do not touch your original
setup and rules. Put down the #2
pencil and step away. You don’t
really have to ditch the pencil, but you get the point.
Below, you can see an example of what your test should look like after you
have completed a game (with more questions on the second page). Note that
everything is nicely organized, and there is a clear distinction between the
original setup and the questions.7
There’s another advantage of
not erasing your work. The
hypotheticals that you build for
specific questions can help you on later
questions by proving whether certain
answer choices can be true or false.
Ninja Note:
As we work through games, we will use the same process every time.
Repetition is key. Logic games are a linear process, and you have to practice
the correct steps every time.
Moving right along, we are going to zoom in and discuss the questions.
1 To help understand the four steps, we are going to make an extended analogy.
Getting through a game is analogous to a successful night at a party. In our analogy,
this first step is assessing the room - seeing who’s there and what they are doing. You
should also check out what everyone looks like (for reference later). Warning: An
inaccurate evaluation (say, thinking a 6 is really a 10) might not hurt you initially, but
it will come back to burn you in the morning.
2 Back to our party analogy: The rules in a game are analogous to the information you
get from the other party- goers. These are the social cues that will guide your evening.
Tamara is recently single. Brad is in a relationship, but does not value monogamy.
The creepy guy over there will trap you in a corner talking about ferrets if you let him.
Now you know the rules.
3 Apologies for mixing pop culture references.
4 Returning to our party, the deductions will determine the outcome of your night.
Sure, you have all the information at your fingertips, but how are you going to use it?
Maybe you learned Veronica is single and she thinks that both you and Steve are cute.
You might also know that Steve is a lightweight, and he begins to drool
uncontrollably when he drinks too much. But will you be able to properly act on this
information? “Steve, let’s go take another tequila shot!” That’s the key.
5 Back to our party analogy: You’ve now met all the other guests. You picked up on
the appropriate social cues, and when the time was right, you made your move. You
hung out with the right people and avoided the creeps. Now it is time to reap the
benefits. You now have a hottie on your arm and are out the door, prize in tow. Just
don’t screw it up now by showing off your stamp collection.
6 Compared to earlier test takers, you are actually very fortunate. Before the June
2012 LSAT, all games were printed on a single page. After thousands of mental
breakdowns and complaints, the LSAC has decided to print each game on two pages.
This is how you will see them printed in this fine publication.
7 Warning: For demonstration purposes only. We are aware that the variables in this
game do not match those in the setup. But good catch.
7/theQUESTIONS
WHAT DO THEY WANT?
When you worked through that horrible game a few chapters back, you saw
that games will present you with a variety of different questions. In order to
answer the questions correctly, you must understand the qualification for the
correct answer choice and the criteria that allows you to eliminate the
incorrect answers. If you don’t understand the question, it can be rather
complicated to find the answer. Consider the following exchange:
Maria: Hey honey, how was your day?
Bob: Gorilla brains.
You don’t want to be Bob. If you answer the wrong question, then you will
choose the wrong answer. For example, if you are searching for an answer
that must be true when the question actually requires one that could be true,
you’re going to have a tough time.
The LSAT intentionally
includes answer choices designed to
catch people who misread the
questions. As you practice, paraphrase
the criteria for the correct and incorrect
answer choices.
BP Minotaur:
Correct answer choices will meet one of four different
criteria:
(1) must be true; (2) could be true; (3) could be false;
or (4) must be false.
The same game will often include many or all of the different types. You
must be able to quickly adjust your mindset. This table outlines the criteria
for the correct and incorrect answer choices.
Correct Answer (Winner)
Incorrect Answers (Losers)
Must Be True
Could Be False
Could Be True
Must Be False
Could Be False
Must Be True
Must Be False
Could Be True
For example, say you are presented with the following question:
If Tony is assigned to the technology committee, then
which one of the following could be false?
Here, you should attempt to find an answer that could be false and eliminate
the four answers that must be true.
EXCEPT Questions
Some questions are complicated by the use of the word EXCEPT. Many
students miss the word altogether or get confused about the implications for
the answer choices. These questions just add one step to our approach.
Consider the following example:
If Tony is assigned to the technology committee, then each
of the following could be true EXCEPT:
This question flips things around a bit. Normally, a question will ask you for
an answer choice that could be true and you should dismiss answer choices
that must be false. Here, four of the answer choices could be true. You must
now dismiss any answer that could be true and pick the one that must be
false.
It sure seem like they
want sumpin’ that could be false.
Cleetus Comment:
Actually, no, Cleetus. And that can be a dangerous mistake. Since the right
answer choice to this question cannot be true, you are looking for the one that
must be false. Luckily, we have another table to help you out.
Question
Incorrect Answers
(Losers)
Correct Answer
(Winner)
Must Be True
EXCEPT
Must Be True
Could Be False
Could Be True
EXCEPT
Could Be True
Must Be False
Could Be False
EXCEPT
Could Be False
Must Be True
Must Be False
EXCEPT
Must Be False
Could Be True
Now that you have an understanding of the criteria for correct and incorrect
answers, it’s time to take a look at the specific types of questions you’ll see in
Logic Games.
There are only three basic types of questions in games:
Elimination, Absolute, and Conditional. Each type
requires a slightly different approach, so let’s take a
look.
ELIMINATION QUESTIONS
Like it or not, many decisions in life are made through the process of
elimination. From selecting schools to ordering dinner to choosing a spouse,
you’re often left picking the last option available (don’t worry, we won’t tell
him).
In general, we want to avoid using the process of elimination on Logic
Games. It’s more time- consuming than using your setup to anticipate and
locate the correct answer. However, there is one large exception to this
general rule. On some questions, the process of elimination is not only
always possible, but it is always the best approach. Say hello to elimination
questions.
An elimination question asks you to identify one
possible outcome in a game.
On elimination questions, you do not have to worry about what must be the
case. Rather, your task is to identify one combination that could work based
on the introduction and the rules.
Here are some examples of elimination questions:
Which one of the following could
be a complete and accurate list of
the types of cookies sold by the
bakery?
Which one of the following could
be a complete and accurate list,
from Monday through Friday, of
the contestants on the game show?
Nearly every game
starts with an
elimination question,
and some games
contain more than
Which one of the following could
be a complete and accurate
matching of the team captains with
the colors of their uniforms?
one. It’s a great
chance to visualize
how the rules
function in a game.
And pick up an easy
point.
Get ready for some good news. You’re going to fall in lust with elimination
questions.1 Since these questions are simply asking for one possible outcome,
the correct answer choice simply must not violate any of the rules in the
game.
The best way to find the correct answer is to use the rules to eliminate wrong
answer choices (hence the name of the question type). As soon as an answer
choice violates a rule, you can eliminate it. Run through the rules, and you
will quickly arrive at the correct answer.
The following is an example of the Blueprint strategy for an elimination
question. Turn the page to see how to quickly identify the correct answer.
Which one of the following could be a list of the factories
in the order of their scheduled inspections, from day 1
through day 6?
1 You were probably expecting us to say love, but let’s be real. This is the LSAT.
Fleeting attraction is the most you can hope for.
First, it’s important to recognize this as an elimination question. This type of
question asks you to identify one possible outcome, given the original
conditions of the game. Almost every game, including this one, starts with an
elimination question.
The next step is to work through the rules, one by one, and use each rule to
eliminate any answer choices that violate it until there is only one remaining.
The first rule states that F is inspected on either day 1
or day 6. In (D), F is inspected on day 5. That is not
day 1, nor is it day 6. Bye-bye to (D).
The second rule states that J is inspected on an earlier
day than Q. In (A), J is inspected on day 5 and Q is
inspected on day 2. That’s not earlier, so there goes
(A).
The third rule states that Q is inspected on the day
immediately before R is inspected. In (C), Q is on day
3 and R is on day 5. Close but no cigar, so (C) is outta
here.
The last rule: If G is inspected on day 3, Q must be
inspected on day 5. In (E), G is on day 3 but Q is on
day 4. Not so good. (E) goes away. (B) is our winner.
As you can see, we just eliminated the four incorrect answer choices without
referencing anything but the rules. The final step is to happily circle the
remaining answer.
In that example, each rule
eliminated one answer choice. That
happens frequently, but sometimes one
rule will knock out more than one
answer or no answers at all. Never fret;
collectively, the rules will always
eliminate four and leave you with just
Ninja Note:
one remaining answer.
Not too bad, huh? This approach will work for all elimination questions on
any type of game. No one knows why the LSAT allows this strategy to work;
it is one of life’s great mysteries.2 But don’t question it - just sit back and
enjoy.
Once you are comfortable with this strategy, you are never allowed to get an
elimination question wrong. Never.
ABSOLUTE QUESTIONS
Absolute questions constitute the second category of questions in Logic
Games. Although the name may sound intimidating (or like vodka), these
questions are fairly straightforward. An absolute question asks you to find
something that is true or false based on the original introduction and rules. In
a perfect world, you should be able to identify the answer to an absolute
question without doing any additional work. (Of course, in a perfect world,
you would have inherited billions of dollars from a long-lost relative and all
of this would be unnecessary.)
An absolute question tests your understanding of the
original conditions in a game. Here are some examples
of absolute questions:
Which one of the following
CANNOT be true?
Which one of the following is the
latest day on which Chuck could
appear on the game show?
Which one of the following must
be true?
Which one of the following is a
complete and accurate list of
teams, any of which CANNOT
wear the yellow uniform?
Want an easy way to
spot an absolute
question? As you can
see, they almost
always start with the
word “which.”
Absolute questions should
be less time-consuming than other
questions. If you are running short on
time, jump to the absolute questions.
BP Minotaur:
Absolute questions are a moment of reckoning for you. Since they are based
on the original conditions in the game, they are designed to test whether
you spotted important deductions. If the answers come easily, you are in
good shape. If not, is it time to panic? Not necessarily, but it might be time to
go back to the rules and look for more deductions.
Since they regard basic facts
about a game, answers to absolute
questions can be helpful for later
questions. For example, if you learn on
an early absolute question that
Theodore cannot be questioned later
than day 4, you can use this valuable
morsel of information on later questions.
Ninja Note:
CONDITIONAL QUESTIONS
Occasionally, the rules change. In life, you might have a different set of rules
that you follow on Halloween or a tropical vacation than you have at a formal
family dinner (Grandma might not appreciate the French maid outfit). In
Logic Games, you also have to be able to adapt and incorporate new rules.
Conditional questions are the third and final question type. They introduce
new information and require you to quickly determine how it interacts with
the previous rules. Unlike absolute questions, conditional questions normally
do require additional work. You will frequently need to construct a quick
hypothetical to help you visualize the new situation.
A conditional question tests your ability to incorporate
new information.
Here are some examples of conditional questions:
If peanut butter cookies are made
on Tuesday, then which one of the
following could be true?
If Lisa appears before Kendall,
then each one of the following
must be true EXCEPT:
If Tiffany’s team wears green, then
which one of the following
CANNOT be true?
Can you guess the
word that normally
begins a conditional
question? Look for
it... look for it. That’s
right - “if” is a good
indicator.
Conditional questions introduce new information, but this doesn’t negate the
importance of the original setup and rules. Rather, the new conditions build
off the foundation of the game.
On a conditional question, the
new information is only in play for that
one question. If they tell you that
Theodore is questioned on day 3 for
question #17, this does not need to be
true for question #18. When you move
on, the new information stays behind.
Ninja Note:
With that, we end our overview of the questions you are going to
face. It’s important to understand what you are looking for in the
correct answer and the most effective approach for each one.
There will be lots of variations as we work through different
types of games, but the general characteristics will be the same.
In the next chapter, we have a few more introductory comments,
and then it’s on to our first game type.
2 Other notable mysteries: How Ryan Seacrest works 29 hours each day, and the
notion of nonfat ice cream.
8/theBLUEPRINT
In the next chapter, we serve up the main course: We will start working
through actual games. But before we do, it’s important to understand how
this book is structured and the best way to work through it.
Blueprint Building BlocksTM
Inferior LSAT books will simply throw a game at you and then explain the
best way to solve it. At Blueprint, we think that is bupkis (excuse our
language). If you are learning to play the piano, you don’t want to give a
recital by yourself and then have some annoying teacher point out your
mistakes. After you have attempted the game, it’s already too late.
Blueprint has developed a revolutionary new approach called Blueprint
Building BlocksTM. When we introduce a new game type, we will use this
process to navigate the game together. We will give you small challenges and
offer advice and tips as you work through a game. This will help you
understand what you should be thinking and doing during a game, rather than
after the fact.
Here is a quick outline of the notations that you will see as we work through
games:
Correct answers are great, but it’s just as important to focus on how you get
there. Blueprint Building BlocksTM will allow us to teach you what you
should be thinking as well as when and why you should be thinking it.
We will primarily use our Blueprint Building BlocksTM approach when we
introduce new games and concepts. After a little practice, we’ll take off the
training wheels, and you’ll work through entire games on your own.
SCRATCH PAPER
Even though you will not be allowed to use scratch paper on the LSAT, we
highly recommend its use while you work through this hefty book. Here are a
few ways it will be helpful:
You will encounter a variety of drills and exercises. Scratch paper will
allow you to compare your work side-by-side with the answer keys
(which will generally fall on a separate page).
Believe it or not, some of the games in this book might not be a
smashing success on your first attempt. Scratch paper allows you to
retry the game without your ugly first attempt getting in the way.
The Blueprint Building BlocksTM approach allows us to work through
games together and give you challenges along the way. However, the
solution to said challenges will often appear on the next page or even
farther down on the same page. Don’t cheat yourself - use your scratch
paper to cover up the solution while you attempt to do the work on your
own.
MYBLUEPRINT VIDEOS
The Blueprint for LSAT Logic Games isn’t just a book; it’s an experience.
And an important part of that experience is found online. There are
streaming, high-definition video explanations for all games in this book. The
methods and techniques are the same, but it can be very helpful to watch a
game broken down on video. All of the videos feature Matt Riley, Blueprint
founder and Logic Games guru.
You can access the videos through the Blueprint website:
blueprintlsat.com/login1
Well, that’s enough for the preliminary remarks. Now
that we’ve laid the foundation, it’s time to get down to
business. In the next chapter, we will cover our first
type of game.
1 On this page, you can create a MyBlueprint online account. Or, if you already have
a MyBlueprint online account, you’ll simply login to access this extra content.
9/WORLDofORDERING
LINE THEM UP
Welcome to the exciting world of ordering games. While perhaps not as
exciting as watching the newest viral baby/kitten video, ordering games are
pretty thrilling in the LSAT world. We will spend the next several chapters
covering all the various types of ordering games. These games are all about
spatial relationships; they require you to determine the relative placement of
different players. You might have to assign tigers to consecutive cages, figure
out a batting order for a baseball team, rank performance on an exam, or
determine Stella’s dating schedule.
The following are examples of ordering games:
Each of seven television programs—H, J,
L, P, Q, S, V—is assigned a different rank:
from first through seventh (from most
popular to least popular). The ranking is
consistent with the following conditions:
The eight partners of a law firm are Gregg,
Hodges, Ivan, James, King, MacNeil,
Nader, and Owens. In each of the years
1961 through 1968, exactly one of the
partners joined the firm.
Exactly seven toy-truck models—F, G, H,
J, K, M, and S—are assembled on seven
assembly lines, exactly one model to a
line. The seven lines are arranged side by
side and numbered consecutively 1
through 7. Assignment of models to lines
must meet the following conditions:
During a single week, from Monday
through Friday, tours will be conducted of
a company’s three divisions—Operations,
Production, Sales. Exactly five tours will
be conducted that week, one each day. The
schedule of tours for the week must
conform to the following restrictions:
Exactly five cars—Frank’s, Marquitta’s,
Orlando’s, Taishah’s, and Vinquetta’s—
are washed, each exactly once. The cars
are washed one at a time, with each
receiving exactly one kind of wash:
regular, super, or premium. The following
conditions must apply:
Note the wording italicized in
these examples. That is the type of
wording that will indicate you are
doing an ordering game.
Ninja Note:
THE SETUP
An ordering game always contains at least two variable sets, such as students
and grades, monkeys and cages, products and aisles. Your task is to figure
out the arrangement of at least one of the variable sets. For example, you will
have to decipher whether Al is in seat 4 or seat 6 or whether the monkey is in
cage 3 or cage 7. The easiest way to visualize this task is to identify the
variable set with an inherent order and use that variable set as the base of
your setup.
In ordering games, the variable set with an inherent
order (rankings, days, lanes) should always be used as
the base of your setup.
A variable set with an inherent order is one that has a clear beginning and
end. Days of the week, order of finish from first to last, numbered chairs, or
appointment times all have an inherent order. A group of girlfriends,
countries in Europe, or car models do not. Here is one of the games from the
previous page:
Exactly seven toy-truck models—F, G, H,
J, K, M, and S—are assembled on seven
assembly lines, exactly one model to a
line. The seven lines are arranged side by
side and numbered consecutively 1
through 7. Assignment of models to lines
must meet the following conditions:
Toy trucks should bring back some fond memories. In this game, there are
two variable sets: toy-truck models and assembly lines. The assembly lines
have an inherent order (1 through 7), so we use them as the base. Then, our
job is to assign the models to the lines.
Building this setup early in the process will allow you to visualize the rules
and deductions. Try to keep your setups as consistent as possible. You will
learn through repetition.
A small number of ordering
games will be better suited to a vertical
setup (for instance, layers of a cake or
floors in a building). When this
happens, just organize the slots
vertically.
BP Minotaur:
Over the next four chapters, we will cover the four
types of ordering games: (1) Basic ordering, (2) 1 to 1
ordering, (3) Underbooked/overbooked ordering, and
(4) Tiered ordering.
10/basicORDERING
WHAT IS IT?
Ordering games will present you with a variety of rules that introduce spatial
relationships between players. The most basic type of ordering principle
simply tells you that one player must come somewhere before or after another
player. Basic ordering games only include this basic type of relationship.
A dash is used to show that one player must be placed
somewhere in front of another player.
Here are some basic ordering principles and the correct diagrams:
L is ranked higher than P.
K is assembled on a highernumbered line than M but a lowernumbered line than G.
Gregg joined the law firm before both
Hodges and Ivan.
Marquitta’s car is washed after both
Orlando’s and Frank’s.
Basic ordering games will give you a combination of these rules. Each of the
rules should be represented with the all-powerful dash. However, there is a
very important second step. You must learn how to combine such ordering
principles to make deductions.
Basic ordering principles
establish that one player comes before
another, but not necessarily
immediately before. If L is ranked
higher than P, L could be ranked one,
two, or five spots ahead of P. Don’t
assume that they have to be
consecutive.
BP Minotaur:
ORDERING CHAINS
In a basic ordering game, you can generally combine some (or even all) of
the rules to build an ordering chain. This will allow you to see how the rules
work together. This strategy will save valuable time and help avoid mistakes.
In a basic ordering game, try to build an ordering chain
with as many rules as possible.
To illustrate how to build an effective ordering chain, we will use a
hypothetical game involving a classic American television show: Saved by
the Bell. Here’s the intro:
Exactly six students at Bayside High
School—AC, Jessie, Kelly, Lisa, Screech,
and Zach—are each called into Mr.
Belding’s office, consecutively and one at
a time, in connection with the kidnapping
of Valley High School’s mascot. The
following conditions must apply:
When you see that the students are called in
“consecutively and one at a time” you know you have
an ordering game.
First things first - you have to build a good setup. There are two variable sets
here - the students and the order in which they enter Mr. Belding’s office.
The order they’re called in has an inherent order, so you should use that as
the base of your setup. At this point, you want to represent the variables and
build your setup.
Next up, check out the rules:
AC and Zach are both called into the office at some time
before Lisa.
Screech is called into the office at some time after Zach.
Jessie is called into the office at some time before AC.
Kelly is called into the office at some time after Lisa.
*Note that many players appear in more than one rule.
This is a clear sign that the rules can be combined.
Instead of dealing with each of these rules in isolation, it is much more
helpful to build an ordering chain. Here is the step-by-step process.
AC and Zach are both called into the office
at some time before Lisa.
Now, we want to attempt to add the later rules.
Screech is called into the office at some
time after Zach.
Since Zach is already part of the chain
from the first rule, we can just add this rule
onto the end. Both Screech and Lisa now
enter after Zach.
Two rules down. Let’s take a look at the next one and see if it can also be
added to the same chain.
Jessie is called into the office at some time
before AC.
Things that we know about AC: (1) he
has huge biceps, and (2) he is in front
of Lisa. AC is already part of the
chain, so we can add this rule by
placing Jessie in front of AC.
This is going well, but there is still one rule to go.
Kelly is called into the office at some time
after Lisa.
Lisa was part of the very first rule.
This rule tells us that Kelly is called
later than Lisa, so we can add Kelly
into our ordering chain behind Lisa.
That’s an example of how you can combine the rules in a basic ordering
game into one ordering chain. These ordering chains can be a little ugly in the
beginning, but they will grow on you (like Screech or so many people in your
own life).
In some games, there will be
rules that can’t be combined. No need
to stress - you can just deal with them
separately.
Ninja Note:
Once you build an ordering chain, it’s important to know how to use it. Here
are the rules to keep in mind:
1. Always Watch the Front and the Rear
On occasion, you will be able to severely limit the options for the first and
last slot. If so, you want to spot this restriction before you hit the questions.
Always check to see who could be first and who could be last.
Any player that does not have
someone who must be in front of
them could be first.
Either Jessie or Zach must be the first student to enter the office.
Same strategy for the rear: Just look
for players without anyone that must
be after them.
Either Kelly or Screech must be the last student into the office.
Deductions of this sort will be helpful
as you work through the game. Don’t
try to remember them - write them
directly into your setup. For example, if a question places Zach into the third
slot, you immediately know that Jessie must be first.
2. Stay Within the Lines
Creating an ordering chain will enable you to quickly draw conclusions about
the relative positions of different players. However, it is important to avoid
drawing unwarranted conclusions. The general rule is that you must have a
clear relationship between two players to decipher who must come first.
Here are some helpful conclusions that you can draw from the world of
Bayside:
These types of deductions will be very helpful as you attack the questions.
For instance, if an answer choice states that Lisa enters the office before
Jessie, you will immediately know that it must be false. Or if a question
states that Kelly enters the office fifth, you know that Zach must enter before
that point.
Life on these games, however, is not all candy and smiles. There are also
some tempting invalid inferences that you must avoid. We’ll explore those on
the next page.
If your ordering chain does not directly specify which of two players must
come first, don’t make erroneous conclusions.
There is no definite relationship
between Jessie and Screech, so it
could be true that Screech enters
before Jessie, or vice versa.
You are crazy. It totally looks
like Jessie is before Screech.
Ditz McGee:
Yes, Ditz, it does look that way. But that is precisely the type of mistake you
must avoid. Don’t fall for visual tricks - stick to the rules. In this ordering
chain, the only person who must enter before Screech is Zach, so it could be
true that Screech enters before Jessie.
3. Know the Limits for Each Player
In basic ordering games, you will be asked a number of questions about the
possible placements of various players. Can Zach be the second student?
Could Kelly enter fifth? To answer these questions, you should count the
number of players that must come before or after the player in question. Let’s
do two examples - one with Jessie and one with Lisa.
There are three students who
must head into the office after
Jessie: AC, Lisa, and Kelly.
Since there are three students that must follow Jessie, she can’t be fourth,
fifth, or sixth. The latest that Jessie could enter the office is third.
Now we are going in the
other direction. Jessie, AC,
and Zach are the only three
students who must enter
before Lisa.
There are three students that must enter the office before Lisa, so she can’t
be first, second, or third. Sorry, Lisa: The earliest spot possible for you is
fourth.
This process can help you
identify the most restricted players. If
someone is limited to two or three
spots, that can break open the game.
Ninja Note:
4. Watch Out for Randoms
You know that creepy guy who is always hanging out in front of the 7Eleven?1 No one really knows who he is or where he came from. Although
he seems harmless enough, he has that unpredictable look in his eye. There is
an analog to this vagrant in Logic Games. Randoms are players in a game
that have no restrictions placed on them. If Cecil is not mentioned in any of
the rules, then you should take note of this fact (normally we just circle the
variable). Cecil is very flexible and that makes him dangerous.
A random is a player that is not mentioned in any of
the rules in a game.
Noticing a random in a game can be helpful. The very fact that they have no
restrictions makes them good candidates to pop up in the answer choices to
could be true questions. Could our random 7-Eleven guy break into a power
ballad at any moment? Well, yeah. Could Cecil end up in lane 6? If he’s not
mentioned in the rules, there’s no reason why he can’t.
Randoms will be an issue in all types of games, but we
figured this was a good place to make the introduction.
Keep an eye out for them at all times.
Up next, we have a drill to help develop
your skills with ordering chains.
1 If you frequent a 7-Eleven parking lot, we apologize. However, chances are you are
not reading this book. And if you are, you might want to see if the 7-Eleven is hiring.
ORDERING CHAIN DRILL
For each of the following games, construct an appropriate ordering setup.
After you read through the rules, build an ordering chain by combining as
many rules as possible. When you think you have a handle on the game and
the rules, go ahead and attempt the questions.
Big Money
A magazine ranks the average salaries for eight affluent types of
professionals—astronauts, baristas, chefs, doctors, entertainers,
firefighters, gymnasts, and hairdressers—from highest (first)
through lowest (eighth). There are no ties. The following
conditions must apply:
The firefighters rank higher than the astronauts.
The gymnasts rank higher than the hairdressers.
The doctors rank lower than the baristas but higher than the
chefs.
Both the doctors and the astronauts rank higher than the
gymnasts.
The entertainers rank lower than the astronauts.
1. What is a complete and accurate list of all of the professions that could
have the highest average salary?
2. What is a complete and accurate list of all of the professions that could
have the lowest average salary?
3. What is the highest ranking that the gymnasts could have among the
eight professions?
4. What is the lowest ranking that the baristas could have among the eight
professions?
5. What is a complete and accurate list of all of the possible rankings for
the doctors?
6. If the chefs rank fourth, then what must be the ranking of the astronauts?
Cool Runnings
At the Winter Olympics, nine countries competed in the four-man
bobsled: Jamaica, Kazakhstan, Lithuania, Madagascar, Nicaragua,
Oman, Poland, Qatar, and Romania. The nine countries finish first
through ninth based on their overall time. There are no ties. The top
three teams win medals. The following conditions govern the
results:
Poland has a slower time than both Lithuania and Nicaragua.
Lithuania has a faster time than Oman but a slower time than
Romania.
Madagascar has a faster time than Qatar.
Poland has a faster time than Jamaica.
1. What is a complete and accurate list of the countries that could have the
fastest time?
2. If Poland is the fourth fastest team, then which teams must win medals?
3. What is a complete and accurate list of the countries that cannot win a
medal?
4. If Qatar has the second fastest time, then what is the best possible finish
for Oman?
5. What is the worst possible finish for Romania among the nine countries?
6. What is a complete and accurate list of the possible finishing placements
for Kazakhstan?
ORDERING CHAIN ANSWER KEY
Use the following key to check your deductions and your answers. Make sure
to go back and review anything that might have gone wrong along the way.
Big Money
1. The baristas and the firefighters are the only options for the top spot (B,
F).
2. The three professions that could be last are the chefs, hairdressers, and
entertainers (C, H, E).
3. The gymnasts must rank lower than the baristas, doctors, firefighters,
and astronauts, so the highest ranking the gymnasts could have is fifth
(5th).
4. The baristas must rank higher than the doctors, chefs, gymnasts, and
hairdressers, so the lowest ranking the baristas could have is fourth
(4th).
5. • The doctors only have to rank lower than the baristas, so the doctors
could rank as high as second.
• The doctors have to rank higher than the chefs, gymnasts, and
hairdressers, so their lowest possible ranking is fifth.
• The doctors could rank anywhere between second and fifth (2, 3, 4, 5).
6. • If the chefs rank fourth, the baristas and doctors both must rank in the
top three.
• The firefighters rank higher than any of the remaining professions, so
the firefighters must also place in the top three.
• The astronauts must rank higher than the entertainers, gymnasts, and
firefighters, so the astronauts come in at number five (5th).
Cool Runnings
In this game, one of the rules cannot be combined with the rest of the chain.
Also, Kazakhstan is not mentioned in any rules, so it’s a random (although
notable for being the home of Borat).
1. Romania, Nicaragua, and Madagascar could each place first according
to our chains. But don’t forget about Kazakhstan, which could place
anywhere, including first (R, N, M, K).
2. Romania, Lithuania, and Nicaragua must each be faster than Poland, so
they are the top three if Poland is fourth (R, L, N).
3. Poland and Jamaica are the only countries to have at least three other
countries that must be faster than them, so there are no medals for the
Poles or the Jamaicans (P, J).
4. • If Qatar finishes second, then the gold goes to Madagascar.
• Oman must also be slower than both Romania and Lithuania, so the
best Oman could finish is fifth (5th).
5. Romania always beats out Lithuania, Oman, Poland, and Jamaica, so the
worst finish for Romania is fifth (5th).
6. Kazakhstan is random and could finish anywhere from first to worst
(1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th).
Lessons we learn from this drill: (1) It pays to be a barista, (2) that movie
about the Jamaican bobsled team was a total fabrication, and (3) ordering
chains really help you visualize how the rules intersect in a basic ordering
game.
The time has finally arrived: Next up is our very first game together.
This is a basic ordering game from the September 2007 exam. Since this is
our first one, we will use the Blueprint Building BlocksTM method and walk
through it together. It will be helpful for you to grab some scratch paper so
we can work through the game side-by-side.
SEPTEMBER 2007: GAME 1 (1-7)
All right, team, no more messing around. It’s time to saddle up for our first
rodeo. The following was the first game on the September 2007 LSAT and
we are going to work through it together.
1. Setup
Workers at a water treatment plant open eight valves—G,
H, I, K, L, N, O, and P—to flush out a system of pipes that
needs emergency repairs. To maximize safety and
efficiency, each valve is opened exactly once, and no two
valves are opened at the same time. The valves are opened
in accordance with the following conditions:
Opening water valves? We told you that games are
exciting! (sarcasm)
This is a classic example of an
ordering game. You are presented
with eight valves. These eight valves
must be opened exactly once and
none of the valves can be opened at
the same time. The task here is to
ascertain the order in which the valves are opened.
At this point, there are two goals: (1) represent the eight valves and (2)
build the appropriate setup with eight slots. Check and check.
2. Rules
Now it’s time to jump into the rules. We will work through these one at a
time.
This game starts out with a basic ordering relationship. Since both K and P
are opened before H, we use dashes to represent this rule. Note that this rule
doesn’t specify if K comes before or after P. They both are opened before H,
but that’s all we know.
Lots more to come. Let’s take a look at the second rule in this game.
We are given another basic ordering relationship. First, you should
visualize what this rule looks like. It introduces an H-O-L sequence
because O is opened after H but before L. But there is something bigger to
note here: H was a player in the first rule, and H makes a quick
comeback in the second rule. That means this second rule can be plugged
onto the back of the ordering chain.
Keep on moving. Check out rule number three.
This one is relatively simple on its own: L is opened after G. Once again, we
are on the hunt to combine rules. L is involved in both the second rule and the
third rule. Thus, we can also add this rule onto our expanding chain. Since L
is opened after G, we can simply place G in front of L on our chain.
So that G has gots to be
opened pretty late?
Cleetus Comment:
Not so fast, Cleetus. Be careful to avoid those visual tricks. All we know is
that G must be opened before L. At this point, G could be opened first. Two
rules remain.
This is the third rule that involves H (which is going to be one important little
valve in this game). From the first rule, you know that both K and P are
opened before H. Now N joins that group. At this point, we know three
valves that are opened before H and two that are opened after it.
We are nearing the finish line - just one little rule to go.
This last rule can be added to complete our humongous ordering chain. The
very first rule in the game involved K, so now we can add a dash to show that
I must be opened after K.
If you do a quick head count,
you can see that all eight valves are in
the ordering chain. Most excellent.
Ninja Note:
Woohoo! That is the end of the rules. If you are able to combine all the rules
as we just did, you should be feeling very good. You will begin to get a faint
itch of accomplishment inside, but do not scratch. There is much more work
to be done, grasshopper. This ordering chain will help significantly with the
questions, but first...
3. The Mighty Deductions
In a basic ordering game, it can be difficult to deduce the exact positions of
any of the players. However, you should take a few moments to test the limits
of your ordering chain.
1. Which valves could be the first one opened?
There are actually a good number of options here. K,
N, P, and G could each be the first valve opened.
Many students forget about poor little G because it
does not appear to be near the front of our ordering
chain. Avoid that mistake. There are no valves that
must be opened before G, so it could be first.
2. Which valves could be the last
one opened?
There are not as many options available for the last
valve opened. According to the rules, it must be
either I or L. Students sometimes mistakenly think
that L must be the last valve opened, but I is another viable option. This is a
helpful deduction and should be plugged right into our setup.
Next up, you want to check out the possibilities that
exist for the valves. It’s always best to investigate
the most restricted valves first. You want to search
for valves that are involved in a number of rules.
To test how early a player
could be selected, count how many
other players must come before her. To
test how late a player could be
selected, count how many players must
come after her.
Ninja Note:
While working through the rules, we noticed
that H was mentioned in three separate rules.
That’s a good place to start. H must be opened
after K, N, and P, so the earliest H could be
opened is fourth. H must also be opened
before O and L, so the latest H could be
opened is sixth. H is likely to pop up in a
number of questions since it is so restricted.
L could be opened last, but L cannot be
opened very early. K, N, P, H, O, and G
must each be opened before L, so the
earliest L could be opened is seventh. I is
the only valve that could be opened after
L. That’s huge. L is the most restricted of
the valves with only two possible
placements.
O must be opened after K, N, P, and H, so
the earliest O could be opened is fifth.
Only one valve (L) must be opened after
O, so the latest O could be opened is
seventh.
The three valves outlined above (H, O, and L) are the valves that should first
catch your eye. Here’s a quick outline of the possibilities for the other valves:
G has to be opened before L, but that’s all.
So G could be opened anywhere from first to seventh. That’s called
freedom.
I is also pretty free. It must be opened after
K, but that means I could be opened from
second through last.
K must be opened before I, as well as H, O,
and L. Thus, K could be opened as early as
first but no later than fourth.
H, O, and L must be opened after N, so N
could be opened anywhere between first and fifth.
P is very similar to N. It must be opened before H, O, and L, so first
through fifth are all possibilities.
By the time test day rolls around, you won’t time to think through all of these
variables. However, you always want to identify the most restricted players in
the game. At this early stage in your preparation, timing is not a concern. Just
work to understand each element of the game. After you practice with a few
ordering chains, this process will speed up significantly.
4. Time for the Questions
At this point, we should be well equipped to jump into the questions.
Remember, it’s not just about right answers; it’s also about how you get
there. Make sure to pay attention to the moves we make along the way so you
can replicate them when you do games on your own.
One important skill is knowing when to do more work and when to stop and
go search for an answer. This takes time to develop - not too little and not too
much.1
Question #1
1. Which one of the following could be the order, from
first to last, in which the valves are opened?
First question! Very exciting!
This is the classic elimination question, and it is exactly what you should
expect from the first question in almost every game. If you remember, we
illustrated the strategy for these questions a few chapters back. You want to
use the rules to kick out the four incorrect answers.
For elimination questions,
don’t try to use deductions. Keep it
simple - just apply the rules one-byone to the answer choices.
Ninja Note:
We have done most of the work up to this point, but now it’s your turn. It’s
early, but see if you can work through this one on your own.
Challenge : Eliminate the four incorrect answer choices
using the rules, and find the last answer standing.
Both K and P are opened before H.
O is opened before L but after H.
L is opened after G.
N is opened before H.
I is opened after K.
(A) P, I, K, G, N, H, O, L
(B) P, G, K, N, L, H, O, I
(C) G, K, I, P, H, O, N, L
(D) N, K, P, H, O, I, L, G
(E) K, I, N, G, P, H, O, L
Don’t peek below! That’s where we explain the correct
answer. We will use a line to remind you not to look.
This is when you should use your scrap paper to cover
up the explanation and keep yourself honest. Come on;
don’t cheat yourself.
The first rule is not very fulfilling (much like eating a salad for dinner). If
you scan through the answer choices, K and P are opened before H in each
one. When you have five rules, this shouldn’t be very surprising. There
are, after all, only five answer choices. We still have four more rules to
help us out.
The second rule tells us O must be opened before L but after H. In answer
choice (B), O is opened after L. That’s definitely not going to work, so get
that loser out of there.
The third rule tells us that L is opened after G. In answer choice (D), L is
opened before G, so we get rid of that answer like the trash that it is.
Halfway home.
The fourth rule tells us that N is opened before H. Answer choice (C) is
discarded because N is opened after H. And the race is on: (A) or (E)?
The final rule tells us that I is opened after K. In answer choice (A), I is
opened before K. Get rid of that one.
We have eliminated everything except (E). Time to circle it, move on,
and be happy (or at least experience a small twinge of satisfaction).
Question #2
2. Each of the following could be the fifth valve opened
EXCEPT:
Next up is an absolute question.
We are given no additional
information, so we should be able to
identify the answer by using our
original deductions. There are two
possible ways we could prove a
certain valve can’t be opened fifth:
(1) Show that five valves must be
opened before it, or (2) show that
four valves must be opened after it.
So take a look at our ordering chain
and try to identify a valve that
cannot be opened fifth. When you
think you’ve spotted it, read on.
(A) H
(B) I
(C) K
(D) N
(E) O
Let’s take a look at the answers and see if you got it.
(A) H must be opened after K, N, and P, so the earliest it could be
opened is fourth. H also must be opened before O and L, so the
latest it could be opened is sixth. Since H can be opened anywhere
in between fourth and sixth, which includes fifth, (A) is no good.
(B) All we know about I is that it must be opened after K, so I could be
opened anywhere between second and eighth. That includes fifth,
so (B) is another loser.
(C) K must be opened before H, O, and L; it also must be opened
before I. There are a total of four valves that must be opened
after K. Now count backwards - K can’t go in spots eight,
seven, six, or five. The latest K could be opened is fourth. Try
as you might, we cannot squeeze K into the fifth slot, so (C) is
our answer.
(D) N must be opened before H, O, and L,
so the latest it could be opened is fifth.
But fifth is a possibility, so (D) is no
good.
(E) O must be opened after K, N, P, and H,
so the earliest it could be opened is fifth. But fifth is cool, so (E) is
also a loser.
Question #3
3. If I is the second valve opened, then each of the
following could be true EXCEPT:
Here’s the first conditional question. Time to do some
work.
Here is our first conditional question. New information is presented (“If I
is the second valve opened”), and you must see how it interfaces with the
original conditions. First, construct a hypothetical, plug I into slot 2, and see
if you notice anything.
We already made the deduction that either I or L must be the last valve
opened. Since I is now the second valve opened, L must bring up the rear.
Since I is now the second valve opened and K must be opened before I, it
is safe to conclude that K must be the first valve opened.
And then you get a little stuck.
If you check out your ordering
chain, it’s hard to spot any
more concrete deductions. For
example, either N, P, or G could be the third valve opened.
When you stop filling up slots
and you start thinking about things that
could be true, it’s time to check the
answers.
Ninja Note:
Challenge : Using the
ordering chain and
deductions, try to identify
the correct answer
(remember, you are
looking for the one that
must be false).
(A) G is the third valve opened.
(B) H is the fourth valve
opened.
(C) P is the fifth valve opened.
(D) O is the sixth valve opened.
(E) G is the seventh valve
opened.
(A) This is a popular answer, but be careful. It is not yet clear exactly
where G must end up in the order. All we know about G is that it is
opened before L. L is the last valve opened now, so G could be
assigned to any empty slot, including the third one. (A) could be
true and is no good.
(B) K is the first valve
opened, and I is the
second valve opened.
Both N and P must
still be opened before
H, so the earliest H
could be opened is fifth. Thus, (B) must be false and is going to
be the correct answer.
(C) P must be opened before H, O, and L. Therefore, the latest that P
could be opened is fifth. Working backwards, L could be last, O
could be seventh, H could be sixth, and P could be fifth, so (C)
could be true.
(D) O must be opened after K, N, P, and H. For this question, O must
also be opened after I (since I is opened second). That means that
the earliest O could be opened is sixth. O must be opened before L,
so the latest O could be opened is seventh. Therefore, the earliest O
could be opened sixth as long as G is opened seventh, so (D) could
be true.
(E) The only restriction on G is that it must be opened before L. Since
L is opened eighth, G could be opened seventh and (E) could be
true. (Also note that we already said G could be opened seventh in
order to prove (D) could be true.)
Question #4
4. If L is the seventh valve opened, then each of the
following could be the second valve opened EXCEPT:
This is another conditional question. We want to quickly construct a
hypothetical from our original deductions and plug in the new information.
As soon as you plug L into slot 7, your eyes should quickly be drawn to
our original deduction about the last valve. Since either I or L must be the
last valve opened (which has turned out to be the gift that keeps on
giving), I must now be last.
At this point, you want to check out your trusty
old ordering chain to see if we can fill out more
slots.
Hmmmm...not seeing much? That’s because there’s not much to find. Since
there are no more clear deductions, it’s time to move to the answers.
Well, look at that. The only
deduction we made in this
(A) G
question placed I in the eighth
(B) I
slot. That would, of course,
(C) K
disqualify it from being opened
(D) N
second. Once you glance at the
(E) P
answers, (B) should stick out
like a sore thumb (or a
wonderful prize, in this case). Since I cannot be opened second, (B) is
our winner.
Each of the other valves offered as answer choices (K, N, P, and G) could
easily be the second valve opened. Just check the ordering chain.
Ninja Note:
That little deduction about the last spot doesn’t
seem so little anymore, huh?
Question #5
5. Which one of the following must be true?
Here’s an absolute, must be true question. Our original deductions should get
us through the answer choices. This one is up to you.
Challenge : Spot the answer that must be true.
(A) At least one valve is opened before P
is opened.
(B) At least two valves are opened before
G is opened.
(C) No more than two valves are opened
after O is opened.
(D) No more than three valves are opened
after H is opened.
(E) No more than four valves are opened
before N is opened.
(A) There are no valves in front of P, so P could be the first valve
opened. Therefore, it does not have to be true that at least one valve
is opened before P; (A) is out.
(B) Again, there are no valves that must be opened before G, so G
could definitely be the first valve opened. Therefore, it does not
need to be true that at least one valve is opened before G; (B) is
gone.
(C) To evaluate this answer, we have to count the valves that could be
opened after O, not just those that must be opened after O. I, L, and
G could all be opened after O. Also, you could note that K, N, P,
and H all have to be opened before O, but any other valves could be
opened after O. Either way you slice it, three valves could be
opened after O so it is not true that no more than two valves are
opened after O; (C) is out.
(D) Just like (C), we have to count the valves that could be opened
after H to evaluate this answer. I, O, L, and G could all be opened
after H. Alternatively, K, N, and P all have to be opened before H,
but all the other valves could be opened after O. The conclusion,
however, is the same: Four valves could be opened after H. So it is
not true that no more than three valves are opened after H, and (D)
goes bye-bye. Now we are really hoping that (E) works out.
(E) There are two ways to evaluate this
answer. First, N must be opened
before only three other valves (H, O,
and L), so these are the only valves
that must be opened after N. Second,
all of the other valves (K, I, P, and
G) could be opened before N. Either
route gets you to the same conclusion: There is a maximum of
four valves that could be 3 opened before N. (E) must be true
and we have our answers.
BP Minotaur:
“No more than four” means four or less.
Question #6
6. If K is the fourth valve opened, then which one of the
following could be true?
Now, we head back to the land of conditional
questions. Time to incorporate this information into
a new hypothetical. At this point, you want to
visualize the restrictions placed on K.
A lot of people get stuck in the
proverbial mud right here.
Let’s not let that happen. On
previous conditional questions,
there were quick deductions to be made about other valves. Here, it’s not
quite as apparent. When you are stuck, always try to visualize the rules in
your setup.
K must be opened before I, and it also must be opened before H, O, and L.
Four valves must therefore be opened after K. Now that K is opened
fourth, there are only four openings after K. So we can deduce that I, H, O,
and L are going to fill the four slots that follow K.
Since the open slots after K have all been filled, the remaining valves (N,
P, and G) must be opened before K.
Note that you can’t place any of the
valves exactly - you just know
which valves are opened before K
and which ones are opened after K.
Challenge : See if you can identify the answer that
could be true.
(A) I is the second valve opened.
(B) N is the third valve opened.
(C) G is the fifth valve opened.
(D) O is the fifth valve opened.
(E) P is the sixth valve opened.
In basic ordering games, this
is a very common deduction. One
player is thrown into a position that
forces some players to the front and
others to the back.
Ninja Note:
(A) I must be opened after K, so there is no way I can be opened
before fourth. Thus, it must be false that I is the second valve
opened, and (A) is no good.
(B) According to our deductions, N must be opened before K.
However, we don’t know anything else about N, so it could be
opened first, second, or third. It could be true that N is the
third valve opened, and (B) is our winner.
(C) We deduced that G must be opened before K, and K must be
opened fourth. Therefore, G cannot be the fifth valve opened, so
(C) is garbage.
(D) We concluded that O must be opened after K. It seems like that
would include fifth, right? Not so much. Since H still must be
opened before O, O cannot be opened fifth. The earliest H can be
opened is fifth, so the earliest that O can be opened is sixth. (D) is
out because O cannot be opened fifth.
(E) We already deduced that N, P, and G must be opened before K
because all of the spots after K are filled. Therefore, P cannot be
the sixth valve opened, and (E) goes away.
Question #7
7. If G is the first valve opened and I is the third valve
opened, then each of the following must be true EXCEPT:
This is the final question. We are told two things here: G is the first valve
opened, and I is the third valve opened. Once again, it is time to build a
hypothetical and check out our ordering chain for other deductions.
The first big deduction should look familiar. Since I is now the third
valve opened, L must be the last valve opened.
There is a second deduction that should pop out when you look at the
ordering chain. K must be opened before I. Since G is first and I is third,
there is only one spot open in front of I. The second deduction is that K
must be the second valve opened.
We have done some solid work, and you might be tempted to just check out
the answers, but we’re not quite there yet. Four of the slots are filled, but
there are still four valves remaining. Check out those four valves, and see if
you can spot any other helpful deductions.
The four remaining valves are N, P, H, and O. Of these four valves, O
must be opened last. So O must take the latest spot that is still available.
For this question, O is opened seventh.
Now we are down to just N, P, and H. Of these three valves, H must be
opened third. Thus, H must take the latest remaining open spot, which is
the sixth one.
N and P are the two
remaining valves. One of
them must be opened fourth,
and the other is opened fifth.
As you can see, the deductions go fairly far on that question, allowing you to
fill out nearly all eight slots before checking out the answer choices. At this
point, it’s pretty easy. You just have to identify an answer that could be false
(the incorrect answers must be true).
(A) K is the second
valve opened.
(B) N is the fourth
valve opened.
(C) H is the sixth valve
opened.
(D) O is the seventh
valve opened.
(E) L is the eighth
valve opened.
(A) True - K must land in the second slot.
(B) N could be opened fourth... or N could be opened fifth. Since N
doesn’t have to be opened fourth, (B) doesn’t have to be true
and is therefore our answer.
(C) Yep, H does have to be the sixth valve opened.
(D) Yessir, O is right there in spot number seven.
(E) L must be the last valve that is opened.
In this classic game, building an accurate ordering chain
can lead to powerful deductions, simplifying the
questions greatly. That’s the goal in basic ordering games.
Well, that’s a wrap - quite a performance for our first game together. Now
take a breather, a quick power nap, or grab an iced tea to let your brain
recover. When you are ready, it’s your turn to try the game on the next page.
Don’t worry about timing; just try to build an ordering chain, make
deductions, and work through the questions in the most efficient way
possible.
Good luck...
1 This balancing act is similar to the scene at the end of a first date. You want to go
far enough to show the person that you are interested, but you don’t want to go too far
and look like a creep.
DECEMBER 2006: GAME 4 (16-22)
Questions 16-22
A courier delivers exactly eight parcels—G, H, J, K, L, M, N, and O. No two
parcels are delivered at the same time, nor is any parcel delivered more than
once. The following conditions must apply:
L is delivered later than H.
K is delivered earlier than O.
H is delivered earlier than M.
O is delivered later than G.
M is delivered earlier than G.
Both N and J are delivered earlier than M.
16. Which one of the following could be the order of deliveries from
first to last?
(A)
(B)
(C)
(D)
(E)
N, H, K, M, J, G, O, L
H, N, J, K, G, O, L, M
J, H, N, M, K, O, G, L
N, J, H, L, M, K, G, O
K, N, J, M, G, H, O, L
17. Which one of the following must be true?
(A) At least one parcel is delivered earlier than K is delivered.
(B)
(C)
(D)
(E)
At least two parcels are delivered later than G is delivered.
At least four parcels are delivered later than H is delivered.
At least four parcels are delivered later than J is delivered.
At least four parcels are delivered earlier than M is delivered.
18. If M is the fourth parcel delivered, then which one of the
following must be true?
(A)
(B)
(C)
(D)
(E)
G is the fifth parcel delivered.
O is the seventh parcel delivered.
J is delivered later than H.
K is delivered later than N.
G is delivered later than L.
19. If H is the fourth parcel delivered, then each of the following
could be true EXCEPT:
(A)
(B)
(C)
(D)
(E)
K is the fifth parcel delivered.
L is the sixth parcel delivered.
M is the sixth parcel delivered.
G is the seventh parcel delivered.
O is the seventh parcel delivered.
20. Each of the following could be true EXCEPT:
(A)
(B)
(C)
(D)
(E)
H is delivered later than K.
J is delivered later than G.
L is delivered later than O.
M is delivered later than L.
N is delivered later than H.
21. If K is the seventh parcel delivered, then each of the following
could be true EXCEPT:
(A)
(B)
(C)
(D)
(E)
G is the fifth parcel delivered.
M is the fifth parcel delivered.
H is the fourth parcel delivered.
L is the fourth parcel delivered.
J is the third parcel delivered.
22. If L is delivered earlier than K, then which one of the following
must be false?
(A) N is the second parcel delivered.
(B)
(C)
(D)
(E)
L is the third parcel delivered.
H is the fourth parcel delivered.
K is the fifth parcel delivered.
M is the sixth parcel delivered.
WHAT CAN BROWN DO FOR YOU?
Welcome back. Hopefully, your parcel arrived on time. This is a basic
ordering game with some similar challenges to the last one. There are lots of
questions, so this is a great chance to get some practice working with an
ordering chain. Let’s take a look.
1. Setup
The original setup establishes
that there are eight parcels
delivered in order. It also states
that there are no ties, which
guarantees there are eight slots
for the different parcels. At this
point, we need to represent the variables and visualize a good setup.
2. Rules
Up next, jump straight into the rules. The goal is to create a helpful ordering
chain that incorporates as many variables as possible. Let’s take a look at the
first four rules. They lead to two separate, smaller ordering chains.
The first and third rules can be combined to form a chain with H, L, and
M.
The second and fourth rules can also be combined to include G, K, and
O.
At this stage, you can’t combine the rules into one ordering chain. But just
when you were thinking it would be swell if there was a way to put it all
together, look at what comes next...
This rule is great. Since M is involved in one chain, and G is part of the other
chain, we can combine them.
This last rule can be added as well. It’s possible to combine all six rules into
one powerful ordering chain. Make sure your chain looks similar to this one.
3. Deductions
Even if you feel like you have a good grasp on a game, make sure to spend
some time searching for helpful deductions.
The first package delivered could be H, N, J, or K. That’s still a lot of
options.
Cleetus Comment:
I got one! O is in 8.
Close, but no cigar. It’s good that you are looking in that direction,
though. O might appear to be the only package that can be delivered
last, but L is another option. The only restriction on L is that it must be
delivered after H, which leaves open the possibility that it is delivered
last.
Ninja Note:
The fact that either L or O must be last is the
most helpful deduction in the game.
It can also be helpful to identify some of the more
restricted players in a game. These players will play a
big role as you run through the questions.
M doesn’t have a lot of wiggle room. M must be
delivered after three packages (H, N, and J) and
before two other packages (G and O). Therefore,
M must be delivered no earlier than fourth and
no later than sixth.
G is also very restricted. G must be delivered after H, N, J, and M, so
the earliest that G can be delivered is fifth. O must be delivered later
than G, so the latest G can be delivered is seventh.
O is definitely bringing up the rear here. O must be delivered after H, N,
J, G, and K. From the other direction, L is the only package that can
possibly be delivered after O, so O must be either seventh or eighth.
If you noticed all of that, you should stop reading this book and just go take
the LSAT right now (kidding). If you noticed most of that, you are in good
shape and ready to jump into the questions. Here is our final setup, rules, and
deductions.
4. Questions
There are some good challenges hidden in the questions. Make sure you
utilized the correct approach for each one.
Question #16 (elimination, could be true)
This is our elimination question, so we quickly use the rules to knock out the
four pretenders.
The first rule, that L is delivered later than H, eliminates nothing,
unfortunately.
The second rule, that K is delivered earlier than O, also eliminates a
whole lot of nothing. We are starting to feel a little uncomfortable.
The third rule, that H is delivered earlier than M, finally knocks out
something. (E) is gone because it has H later than M.
The fourth rule, that O is delivered later than G, knocks out answer
choice (C) because O tries to sneak in front of G.
The fifth rule, that M is delivered earlier than G, knocks out (B) because
it has M later than G. Almost home.
The sixth and final rule, that both N and J are delivered earlier than M,
lays the final knockout punch to (A) because J is later than M.
And that leaves us with... (D).
Question #17 (absolute, must be true)
Our amazing ordering chain is all we need on this absolute question.
(A) We noted that K could be the first parcel delivered, so it doesn’t
have to be true that at least one parcel is delivered earlier than K.
(B) We also noted that G could be delivered as late as seventh. Only O
must be delivered after G, so it does not need to be true that at least
two parcels are delivered after G.
(C) Check out that ordering chain and you
can see that L, M, G, and O must be
delivered after H. Do the math: There
are four packages that must be
delivered after H. Thus, it must be true
that at least four parcels are delivered
after H and (C) is our answer.
(D) J must be delivered earlier than M, G, and O, which totals three
parcels. However, that still falls short of four, so it does not need to
be true that at least four parcels are delivered later than J.
(E) There are three parcels that must be delivered earlier than M (H, N,
and J). However, all of the other parcels could be delivered later, so
it does not need to be true that at least four parcels are delivered
before M.
Question #18 (conditional, must be true)
Now, we must test to see what other inferences can be
drawn from the new condition that M is the fourth
parcel delivered. To attack this question, place M into
a hypothetical and visualize the other rules regarding
M.
There are three parcels that must be delivered earlier than M (H, N, and
J).
H, N, and J must be the first three parcels delivered. However, we still
do not know exactly which parcel is delivered first, second, and third.
Since the first three slots are now filled, we know that all of the other
parcels must be delivered after M. So L, G, K, and O must be delivered
later than M.
While you are learning, it is
incredibly important to note the
consistencies that exist between games.
In this question, M is used as a place
blocker in the same way that players
were used in the last game.
BP Minotaur:
We still know very little about the exact placement of the parcels. However,
to answer this question, the important thing is to note those parcels that are
delivered before M, and those delivered after. Remember, this is a must be
true question.
(A) G could be the fifth parcel delivered, but K or L could also be
delivered fifth.
(B) O could still be the seventh or eighth parcel delivered, so it does
not need to be true that O is the seventh parcel delivered.
(C) Among the first three parcels, the order is not determined. Thus, J
could be delivered earlier than H or vice versa, so it does not need
to be true that J is delivered later than H.
(D) N must be among the first three parcels delivered. K must be
delivered no earlier than fifth. Thus, it must be true that K is
delivered later than N. (D) is the winner.
(E) Among the last four parcels, G could be delivered later than L. But
L could be delivered last, so L could also be delivered later than G.
Thus, it does not need to be true that G is delivered later than L.
Question #19 (conditional, must be false)
In this question, we are given a different tidbit of
information, but the deductions are similar to the last
question. First, we want to visualize H as the fourth
parcel delivered.
There are four parcels that must be delivered later
than H (L, M, G, and O).
There are only four available slots after H. Therefore, L, M, G, and O
must be the only four parcels delivered after H. However, we still do not
know exactly which parcel is delivered in which slot.
Since all of the slots after H
have been filled, all of the
remaining parcels must be
delivered earlier than H. So
N, J, and K are the first three
parcels delivered, though not necessarily in that order.
The deductions on this question are strikingly similar to the last one. Now,
we hit the answer choices looking for one that must be false.
(A) Well, they don’t make us wait long for this one. We deduced
that K must be delivered earlier than H, so K must be one of
the first three parcels delivered. It must be false that K is the
fifth parcel delivered, and (A) is the early winner.
(B) If M is the fifth parcel delivered, then L could be the sixth. G and
O would then fill the last two spots.
(C) It could be true that M is the sixth parcel delivered if L slides into
the fifth spot, and G and O are delivered seventh and eighth,
respectively.
(D) As long as O is the final delivery, it could be true that G is the
seventh parcel delivered.
(E) If M is the fifth parcel delivered, G is the sixth parcel delivered,
and O is delivered before L, then it could be true that O is the
seventh parcel delivered.
Question #20 (absolute, must be false)
Our original ordering chain saves the day once again.
(A) Even though it might appear that H must be in front of K, K could
actually be the first parcel delivered. So it could be true that H is
delivered later than K.
(B) Using our ordering chain, it is clear that J is delivered earlier
than M, and M is delivered earlier than G. Thus, J must be
delivered earlier than G. It must be false that J is delivered
later than G, so (B) comes out on top.
(C) L could be the last parcel delivered, so it could be true that L is
delivered later than O.
(D) Both M and L have to be delivered later than H, but there is no
relationship between M and L. So it could be true that M is
delivered later than L.
(E) N and H both must be delivered earlier than M, but there are no
rules that establish whether N is delivered before H, or vice versa.
So it could be true that N is delivered later than H.
Question #21 (conditional, must be false)
There are lots of questions in this game, but that just means more
opportunities to rack up the points. On this question, the new condition is that
K is the seventh parcel delivered.
K must be delivered earlier than O, so O is pushed into the last spot.
You were probably expecting to fill in more slots on
this question, but those are the only parcels you can
figure out. Whenever that happens, don’t panic. At
this point, trust your deductions and rely on them to find the answer.
Remember, we need to locate the one that must be false.
(A) If we place L in the sixth slot, then G could be delivered fifth.
(B) G is the only remaining parcel that must be delivered after M, so
M could be delivered fifth as long as G is delivered sixth.
(C) Even with K and O determined, L, M, and G each must be
delivered after H. Since there are three parcels that still must
be delivered after it, the latest that H could be delivered is
third. It must be false that H is the fourth parcel delivered, and
(C) is the answer.
(D) The only restriction on L is that it must be delivered later than H,
so it could be true that L is the fourth parcel delivered.
(E) M and G must be delivered later than J, but J could still be
delivered as late as fourth. Thus, it could be true that J is the third
parcel delivered.
Question #22 (conditional, must be false)
Once in a while, the wonderful folks who make the LSAT throw you a curve
ball in the last question on a game. They will change a rule, add a rule, take a
rule away, or make your test explode (not really). When they do so, it can be
quite vexing because it throws your hard work on the game out the window.
This is such a question, and the new information changes our ordering chain.
Darn. Now L is delivered earlier than K.
Unless directly stated, these questions don’t
negate any of the original rules in the game.
Ninja Note:
On this question, it is
necessary to incorporate
the new condition by
drawing a new ordering
chain. Here is what your
new chain should look
like:
The new ordering chain is a bit ugly, but now you want to check for new
deductions.
Before, either L or O could be the last parcel
delivered. Now, L must be delivered before a
medley of other parcels, so O must be the final
parcel delivered.
Either K or G must be the seventh parcel delivered.
After those deductions, everything else is a little up in the air. It’s time to hit
the answer choices.
We need one that must be false.
(A) N could easily be the second parcel delivered if H or J is the first
parcel delivered.
(B) L must be delivered later than H. If either N or J is also delivered
before L, then L could be delivered third.
(C) With the new condition, L, K, O, M and G each must be
delivered later than H. Thus, there are a total of five parcels
that must be delivered later than H, so the latest that H could
be delivered is third. It must be false that H is the fourth parcel
delivered. (C) is perfect.
(D) K must be delivered after H and L, but K could be delivered
anywhere from third to seventh. It therefore could be true that K is
the fifth parcel delivered.
(E) G and O are the only two parcels that must be delivered after M, so
M could be the sixth parcel delivered.
Say good night to the parcel game. Once again,
constructing an ordering chain was crucial to success in
this game. Also, note that some of the smaller deductions (most notably that
either L or O had to fill the last slot) paid big dividends.
That will bring our first type of ordering game to a close. Hopefully, you are
feeling more comfortable with ordering chains and, more generally, the
process of working through a game. We have a long way to go, but all types
of games will break down in much the same way.
Up next, we have a surprise (and it’s not a
great one). We aren’t moving on to the
next type of game. Rather, there is another
skill set that we have to cover before
venturing any further. (Warning: It might
just be logic.)
The next chapter covers the important logical
relationships you will find in Logic Games (and
Logical Reasoning, if you happen to be studying that
stuff as well).
11/gamesLOGIC
WHAT IF...?
It’s a bummer to stop the momentum at this point. We just got through our
first chapter of games, but we have to interrupt for a commercial break. In
every other type of game you will face, you might be confronted with an ugly
type of rule called a conditional statement.
In the previous game, a rule told you that L is delivered before H. That is
an absolute statement - it was true for the entirety of the game. But what if
a rule said, “if L is delivered before H, then K is delivered before M”? This
doesn’t imply that L must be delivered before H, or that K must be
delivered before M. But if L is delivered before H, then you know
something else must occur. There’s an important distinction between
absolute rules, which state a fact, and conditional rules, which give a
relationship between conditions.
A conditional statement asserts a logical relationship
such that satisfying one condition guarantees that
another condition must follow.
Conditional statements are
prevalent in many types of games and
Logical Reasoning. A good LSAT
score requires a deep understanding of
such claims.
BP Minotaur:
Mastering conditional statements is one of the most difficult skills for
students to develop. When you see an absolute rule, it’s pretty simple. But
conditional statements are more complex. Take the last example - if L is
delivered before H, then K is delivered before M. But what if L isn’t
delivered before H? Or what if M is delivered before K? As you might
expect, there are rules associated with such statements, and they are
important to memorize.
This chapter is devoted to conditional logic in Logic Games. It’s lengthy, but
that is indicative of its importance. Here’s a look at what’s on tap:
1. Sufficient and
Necessary Conditions
2. Valid and Invalid
Conditional Inferences
3. Logically Denotive
Words
4. Conjunctions and
Disjunctions
5. Rules of Negation
These are just big, scary
words for concepts that we
are going to make easily
digestible.
SUFFICIENT OR NECESSARY?
Conditional statements assert the existence of a hypothetical relationship such
that if the first condition is met, then the second must follow.
If you drive a Rolls Royce, then you are rich.
This statement does not assert that anyone actually drives a Rolls Royce, or
even that anyone is rich. Rather, it asserts a relationship between driving a
Rolls Royce and being rich.
Conditional statements are always composed of two elements: a sufficient
condition and a necessary condition.
Sufficient Condition [ Necessary Condition
In the previous example, the sufficient condition is driving a Rolls Royce, and
the necessary condition is being rich. The distinction between sufficient and
necessary conditions will be tested repeatedly on the LSAT, so it’s very
important to understand the difference.
Sufficient Conditions
Satisfying a sufficient condition is enough to guarantee that a necessary
condition will follow. Nothing else is required. In our previous example,
the fact that a person drives a Rolls Royce is enough to conclude that he or
she is rich. No other evidence is needed and no other evidence can destroy
that conclusion. Even if this person also lives in a cardboard box and eats
bargain-brand cat food, you can still conclude that he or she is rich.
Necessary Conditions
A necessary condition is required in order to satisfy a sufficient
condition. Without a necessary condition, the sufficient condition cannot
occur. In our previous example, it is required that someone is rich for them
to drive a Rolls Royce. If someone is in the poor house, then they do not
drive a Rolls Royce.
Do not confuse sufficient
and necessary conditions. Being rich is
necessary for someone to drive a Rolls
Royce. However, it is not sufficient.
People can be extremely wealthy and
choose to drive a BMW, a Prius, or a
camel. They’re rich - a little
eccentricity is expected.
BP Minotaur:
I think I get it. While being a
D-list celebrity is sufficient for me to
date a guy, it is clearly not necessary.
Ditz McGee:
Actually, Ditz, that is precisely correct. In Logic Games, you will be
presented with a variety of conditional statements, and it’s important to know
what conclusions you can draw from them. Next, we are going to cover two
valid inferences that will be helpful, and two invalid inferences you must
avoid.
THE GOOD GUYS
John Wayne. Peter Parker. Bill and Ted.1 Ron Burgundy. Good guys, through
and through. In life, we love the good guys. The same is true with the LSAT.
Except on the LSAT, we call them valid inferences.
There are two valid inferences that can be drawn from any conditional
statement you confront in a game.
Valid Affirmation (Satisfying the Sufficient Condition)
The Valid Affirmation is the most basic inference that can be drawn from a
conditional statement. If a sufficient condition is satisfied, that is always
enough to conclude the necessary condition follows.
If Homer is at home, then Marge is
at home as well.
This rule would be part of a game in which you have to assess which family
members are at home (probably not Bart). If you know Homer is at home
(satisfying the sufficient condition), it is guaranteed that Marge is at home as
well.
Satisfying a sufficient condition guarantees that the
necessary condition follows.
As you can see, we use an arrow to represent
conditional relationships. Try to avoid any other
representations, as it is vital to quickly and accurately
diagram these rules in difficult games.
BP Minotaur:
Contrapositive (Denying the Necessary Condition)
There is a second valid inference that follows from any conditional statement.
It’s called the contrapositive, and it will be your best friend in Logic Games.
Think of the contrapositive as the flip side of the valid affirmation. In a
conditional statement, the necessary condition is required for the sufficient.
Thus, if a necessary condition is denied, then the sufficient condition
must be denied as well.
Denying a necessary condition guarantees that a
sufficient condition cannot follow.
You don’t realize it, but you
use the contrapositive every day. To
make your girlfriend happy, it’s
necessary that you bring her flowers.
No flowers? You are going to have an
unhappy lady.
Ninja Note:
To understand the contrapositive and how it works, let’s take a look at a rule
from a game:
If Pink Floyd performs on day 4, then
Led Zeppelin performs on day 6.
You might find this rule in a game that involves scheduling a music festival
for 70’s psychedelic rock bands. If Pink Floyd performs on day 4, then Led
Zeppelin must perform on day 6. But there is a second inference. If Led
Zeppelin does not perform on day 6 (the necessary condition is not met), then
Pink Floyd cannot perform on day 4 (the sufficient condition can’t follow).
To form the contrapositive, there are two steps:
(1) Switch the sufficient and
necessary conditions, and
(2) Negate both conditions.
If Led Zeppelin does not perform on day 6, you can immediately conclude
that Pink Floyd doesn’t perform on day 4.
To form the contrapositive, swap the sufficient and necessary conditions,
and negate both conditions. We will cover more complicated rules of
negation later, but for now, negating a condition means ruling it out as a
possibility. For instance, the negation of Al is in seat 3 is Al is not in seat 3.
The negation of Beatrice is not in seat 5 is Beatrice is in seat 5.
For every conditional statement, watch for two things: (1) If the sufficient
condition is affirmed and (2) if the necessary condition is denied. Valid
conclusions follow from both.
Here are more details about how this rule will function
in a game. In our example, a clear conclusion can be
drawn if Pink Floyd performs on day 4 (that Led Zeppelin must perform on
day 6). It can be harder to spot the contrapositive. If Led Zeppelin performs
on day 1, or day 5, or any other day besides 6, then you know that Pink Floyd
can’t perform on day 4. This can be easy to miss, so it’s imperative to write
out the contrapositive for a visual reminder.
Because the contrapositive is
so important, write it out every time
you are given a conditional rule. Every
time.
Ninja Note:
THE BAD GUYS
Darth Vader. The Joker. That weird fashion designer dude (Mugatu) who
wanted to rule all male models in Zoolander. In movies, the good guy always
comes out on top. But this is real life. On the LSAT, bad guys can easily win
unless you know how to defend against them.
The bad guys on the LSAT take the form of two very tempting invalid
conditional inferences. You must avoid these like the plague, or like a
heavyweight champ after you steal his tiger. They are very dangerous, and
you must stay away.
Converse (Satisfying the Necessary Condition)
The most common logical fallacy on the LSAT is what we call the converse.
The converse occurs when the sufficient and necessary conditions are
reversed but not negated.
Do not confuse the conditional
arrow with an equal sign. These
relationships do not go both ways.
(Think more Clint Eastwood, less
Andy Dick.)
Ninja Note:
This fallacy stems from the distinction between sufficient and necessary
conditions. Satisfying a sufficient condition guarantees that the necessary
condition follows, but satisfying the necessary condition does not
guarantee anything.
Satisfying a necessary condition is not enough to
conclude that the sufficient condition must follow.
Consider the following rule:
If Bigfoot is captured second,
the Loch Ness Monster is
captured third.
We are now entering the wonderful world of fake monsters. This game might
have you identify the order in which the creatures above, as well as King
Kong, Medusa, and Cher, are captured. The above rule informs you that if
Bigfoot is captured second, then the Loch Ness Monster must be captured
third. However, if the Loch Ness Monster is captured third, it is a mistake
to think Bigfoot must be captured second.
As you can see here, the two conditions are reversed but not negated. To
form the contrapositive (a valid inference), you need to both reverse and
negate.
This fallacy can be very tempting, and it will be offered to you in many
incorrect answer choices. Stay away.
Satisfying a necessary condition doesn’t
guarantee the truth of the sufficient condition, but it does
not preclude it, either. If the Loch Ness Monster is
captured third, Bigfoot could be captured second. It is just
a fallacy to conclude that Bigfoot must be captured
second.
BP Minotaur:
Inverse (Denying the Sufficient Condition)
Another common fallacy arises when both of the conditions of a conditional
statement are negated, but the conditions are not reversed. Even if a
sufficient condition is not met, that does not imply the necessary condition
cannot be satisfied.
Negating a sufficient condition is not enough to
conclude that the necessary condition will not occur.
Check this out:
If Channing is drinking merlot, then
Niles is drinking shiraz.
This example stems from an infamous Blueprint trip to Napa Valley (names
have been changed to protect the innocent). Here, a group of people are
assigned to different wine-tasting teams. If Channing is sipping on merlot,
then Niles must be throwing back the shiraz. However, don’t make the
mistake of thinking that if Channing is not drinking merlot, then Niles cannot
be drinking shiraz.
Here, the two conditions are negated, but they are not reversed as they are
with the contrapositive. This fallacy, like the converse, will be a tempting
answer choice, but don’t be fooled.
If a sufficient condition is
negated, the necessary condition may
or may not follow (we don’t know
either way).
Ninja Note:
THE REAL DEAL
Now let’s apply these abstract concepts to real logic games. Here is a rule
that could be part of a game dealing with parenting workshops.
If Britney attends the workshop on Tuesday, then Kevin
must attend on Friday.
The first step is always to diagram the rule and its contrapositive. If Britney
attends on Tuesday, then Kevin must attend on Friday. To form the
contrapositive, reverse both terms and negate them. So, if Kevin does not
go to the workshop on Friday, then Britney cannot attend on Tuesday.
Be sure to watch out for the two common fallacies we just learned. If Kevin
goes to the workshop on Friday, that does not mean that Britney must attend
on Tuesday (converse). Also, if Britney does not learn some parenting skills
on Tuesday, we cannot conclude that Kevin cannot attend the workshop on
Friday (inverse).
Once you diagram the
conditional claim and its
contrapositive, train your eyes to only
focus on the sufficient conditions.
These will lead to valid conclusions,
while looking for necessary conditions
will only lead to confusion.
BP Minotaur:
The following chart illustrates the conclusions that can be validly drawn from
this rule:
What You Know
Conclusion
Britney attends on Tuesday.
Kevin attends on Friday.
Britney does not attend on
Tuesday.
No conclusion
Kevin attends on Friday.
No conclusion
Kevin does not attend on
Friday.
Britney does not attend on
Tuesday.
Let’s do one more example.
If the bearded lady is not on stage 2, then the sword
swallower is on stage 1.
Step one: Diagram the rule and its contrapositive. If the bearded lady is not
on stage 2, the sword swallower must be on stage 1. We also know that if the
sword swallower isn’t on stage 1, the bearded lady will be found on stage 2.
Even though the structure of this rule looks
different, you form the contrapositive with the exact same
steps: (1) Switch the sufficient and necessary conditions,
and (2) negate both.
Ninja Note:
Here is the table of valid conclusions from our circus example:
What You Know
Conclusion
The bearded lady is on stage 2.
No conclusion
The bearded lady is not on stage 2.
The sword swallower is on stage
1.
The sword swallower is on stage 1.
No conclusion
The sword swallower is not on
stage 1.
The bearded lady is on stage 2.
At this point, you should have a rudimentary grasp of the rules regarding
sufficient and necessary conditions. It’s important because these rules will be
the difference between right and wrong answers. Time for a drill. Turn the
page when you are ready.
SUFFICIENT/NECESSARY DRILL
This drill is designed to test your understanding of conditional statements. In
the table, you’ll find four different columns. Your job, if you choose to
accept it, is to diagram these statements and identify when (and what) valid
conclusions can be drawn.
1. Read the conditional rule and create an arrow diagram in the
second column. (Remember to always diagram the contrapositive.)
2. The third column gives you an additional piece of information. This
is similar to what you will find in the middle of a game. Use the
fourth column to fill in any valid conclusions that can be drawn. As
always, avoid those ugly fallacies.
Note : In a number of examples, you will not be able to
draw any valid conclusions. If this occurs, write “no
conclusion” in the fourth column.
Conditional Rule
Arrow
Diagram
What You
Know
1. If Suzy is selected, then
Esther is selected.
Suzy is
selected.
2. If the zoo purchases a panda,
then it also purchases a zebra.
The zoo
purchases a
zebra.
3. If Antoine is on TV, then
Bernadette is not on TV.
Antoine is
not on TV.
Conclusion
4. If G is onstage, then H is not.
H is on
stage.
5. If Peggy does not run the
third leg, then Rita runs the
second leg.
Rita runs the
second leg.
6. If X does not survive, then Z
does survive.
X does not
survive.
7. If Sly is not on the pink team,
then Ty is not on the pink team.
Ty is on the
pink team.
8. If there is no diamond, there
is no engagement.
There is a
diamond.
Answer Key
Here are the answers to the previous drill. If you committed any mistakes,
make sure to feel very ashamed (or just understand the mistake you made and
avoid it next time).
1)
s [ e (Esther is selected.)
2)
p [ z (No conclusion)
3)
a [ B (No conclusion)
4)
g [H (G is not on stage.)
5)
P 3 [ r 2 (No conclusion)
6)
X [ z (Z survives.)
7)
S p [ T p (Sly goes pink.)
8)
D [ E (No conclusion2)
Now, young caterpillar, you should feel comfortable with sufficient and
necessary conditions, the twin pillars of LSAT logic. You should also have a
basic understanding of the contrapositive and the two common logical
fallacies (converse and inverse). These are difficult and important concepts,
so you might want to bookmark these pages for later. But there’s more to
learn...
INDICATOR WORDS
As you can see from our discussion of valid and invalid inferences, if you
can’t correctly identify sufficient and necessary conditions, then you will
have a difficult time mastering games. To make it even more challenging,
conditional rules are not always phrased as if/then statements. Because of
this, the following rules will teach you how to diagram the most common
expressions of conditional rules in Logic Games.
Conditional rules are often
written in a convoluted manner.
However, you should translate them
into “if/then” statements. This is the
simplest form.
Ninja Note:
There are a number of keywords used to introduce sufficient or necessary
conditions.
Hint : You should memorize the following indicator
words (just replace the part of your brain still holding
onto the lyrics of “Ice, Ice, Baby”).
If (All, Any, Every, When)
If is the most common word the LSAT uses to introduce a sufficient
condition. The condition that follows “if” forms the sufficient condition,
and the remaining clause forms the necessary condition. This type of
conditional rule is the most straightforward to diagram. Luckily for you, it is
also the most common.
If there are slurpees at the convenience
store, then there is also beef jerky.
The Louvre cannot be visited on day 5 if
Versailles is visited on day 7.
Be aware that it does not
matter whether “if” introduces the first
or second claim in the rule. It is always
the sufficient condition, regardless of
where it appears.
Ninja Note:
The words all, any, every, and when also can be used to introduce a sufficient
condition. As with the word if, these words are enough to guarantee an
outcome. These rules should be diagrammed in the same way as “if”
statements.
All of the students in the Mathletes are also in
the Glee club.
Pigs in a blanket are baked on every day that
custard is not.
“If” (and similar words) always introduces a sufficient
condition.
Before we move on to the next indicator word, it’s time for some practice.
You will find a couple conditional rules below. Diagram each of them
correctly by spotting the keywords. The answer key can be found at the end
of this section (page 89).
1. If Tina is on the wetlands committee,
then Ralph is on the forest committee.
2. Jamie must drive on the fourth day if
Ken does not drive on the second day.
3. If disco is played, then reggae is not.
4. The quarterback is in every class that the
cheerleader is in.
5. All of the employees that work on
Thursday also work on Friday.
Only (if)
One small word can drastically change the meaning of a conditional
statement. The word “only” and the phrase “only if” always introduce a
necessary condition. The referent of only (which nearly always immediately
follows the word or phrase) forms the necessary condition, while the other
clause forms the sufficient condition.
Kipp performs during the show only if Biff
also performs.
Only members of team X can serve as the
grand master.
“Only” always introduces a necessary condition.
Now it’s time for you to practice. Just like last time, diagram these
conditional rules by identifying the keywords. Yet again, the answer key is at
the end of the section.
1. Tiagra is a winner only if Salim is a
winner.
2. K is placed in slot 5 only if M is placed
in slot 7.
3. Only if the Bruins make the tournament
can the Trojans make the tournament.
4. Only those who get a rose move on to
the next round.
5. Gregg places fourth only if Haley does
not place first.
If and Only If
The phrase if and only if is not used commonly in games, but it is important
to understand in case you ever encounter it. Since the phrase “if” introduces a
sufficient condition and the phrase “only if” introduces a necessary condition,
the phrase “if and only if” introduces a condition that is both sufficient
and necessary. This type of rule should be represented with a reciprocal
(double-sided) arrow.
Dominique upgrades to a super wash if and
only if Charlotte upgrades to a super wash.
The contrapositive of this statement is a bit different than other conditional
rules. Because this is a reciprocal relationship, if one condition isn’t met,
then neither condition is met. The best way to interpret this type of rule is
that there are really only two possibilities: either both conditions are met, or
neither one is met.
“If and only if” introduces a reciprocal relationship.
Up next? That’s right, more practice. This phrasing is not very common, but
here are two examples for you to diagram. You will notice that the arrow is
missing, so you also need to fill in the appropriate arrow.
1. Lisa purchases a dishwasher if and only
if Pam purchases a toaster.
2. Marshawn is interviewed on day 4 if,
but only if, Lee is interviewed on day 6.
Unless
The word unless can be frightening to diagram, but we will get through it
together. Logically, the word “unless” introduces a necessary condition.
However, to diagram one of these statements, it is much easier to replace
“unless” with the phrase if not. The term following “unless” is negated to
form the sufficient condition, and the other term forms the necessary
condition. This may sound complicated, so let’s consider the following
examples:
The vacuum is not in the closet unless the
umbrella is in the closet.
To diagram this statement, replace unless with if not. Thus, the claim can be
restated as follows: If the umbrella is not in the closet, then the vacuum is not
in the closet.
You should practice this maneuver until your eyes begin
to bleed (not really, but please memorize the rule). Let’s try another.
Unless Barbie performs on stage 3,
Chastity must perform on stage 5.
Follow the same steps on this questionable example.
Replace unless with if not and you are left with the
following claim: If Barbie does not perform on stage 3,
then Chastity must perform on stage 5.
To diagram an unless statement, replace “unless” with
the phrase “if not.”
Your turn. Use the rule outlined on the last page to diagram the following
conditional rules.
1. The drums will not be included unless
the saxophone is included.
2. Unless Andre is on the green team,
Benji cannot be on the green team.
3. D must be invited unless F is not
invited.
That brings us to the end of our discussion of conditional keywords.
However, make sure to check this answer key and look over any errors.
“If” statements
1)
t w [r f
2)
K 2 [j 4
3)
d[R
4)
c[q
5)
th [ f
“Only” statements
1)
t[s
2)
k 5 [m 7
3)
t[b
4)
mo [ r
5)
g4[H1
"If and only if” statements
1)
ld \ pt
2)
m4\l6
1. “Unless” statements
1)
S[D
2)
A g [B g
3)
f[d
In the third “unless” example,
you have to negate the condition that F
is not invited. Thus, “F is invited”
becomes the sufficient condition.
Ninja Note:
If you got the majority of those correct, you are well on your way to
becoming a conditional rock star. Let’s move on to a bigger drill.
1 For those of you born after 1988, Bill and Ted were the protagonists of a riveting
film starring Keanu Reeves before he went insane. They also later embarked on a
bogus journey that was captured on the big screen. We highly recommend both films.
GET YOUR DIAGRAM ON
This drill is designed to test your ability to diagram conditional rules. For
each of the following statements, diagram the sufficient and necessary
conditions. Then, diagram the contrapositive. Hint: You will have to alter
some of the arrows.
1. If Batman is seen at the crime scene, then Robin will
also be present.
2. Mr. T will pity anyone who is a fool.
3. Conifers are not in the forest if spruces are not in the
forest.
4. If St. Nick is not on the red team, then the Easter Bunny
must be on the red team.
5. Only champions from the first round make it to the
Showcase Showdown.
6. The male supermodel will wear a speedo only if he has
thoroughly waxed.
7. Brandi will be at the party only if there is jungle juice.
8. Only a member of the first panel can be the chairman on
the second panel.
9. Jameer bids on the vacation if, but only if, Dwight bids
on the living room set.
10. Elizabeth will get married if and only if Lydia gets
married.
11. Gertrude cannot win an award unless Henrietta wins
an award.
12. X is selected unless Y is not selected.
13. Any symptom of pneumonia is also a symptom of
narcolepsy.
14. Only people from France shower irregularly.
15. Model 1 wears the purple dress unless model 3 wears
the yellow dress.
16. If the USA wins the gold medal, then Russia wins the
bronze.
17. The turtle is exhibited at the zoo only if the unicorn is
not exhibited.
18. If the brownie is the sixth course during dinner, then
the tilefish must be the third course.
19. Peter is accepted if, but only if, Rachel is also
accepted.
20. The Horn Toad Race is the third destination if the
Monster Truck Rally is not third.
21. If brain matter is found at the crime scene, then
fingerprints cannot be.
22. The fourth toy is green only if the sixth toy is purple.
23. Joey cannot attend the party unless Ross also attends
the party.
24. If curling is scheduled for Wednesday, then skeleton is
scheduled for Friday.
25. Jessica is featured in every skit that Katie is not.
How’d you do?
1)
b[r
R[B
If introduces a sufficient condition. So when the caped crusader shows up, his
loyal sidekick will always be there with him. If Robin is not present, then you
know Batman is nowhere to be found. However, does Batman also have to
follow Robin? No way, as any self-respecting Christian Bale fan knows.
2)
f[p
P[F
“I pity the fool.” Little did you know that Mr. T is a big fan of the conditional
statement. Anyone introduces a sufficient condition, so being a fool is
sufficient to receive some pity. The contrapositive demonstrates that if Mr. T
shows you no pity, you are not a fool. But do only fools receive pity from Mr.
T? Of course not: He could also show pity to those with small biceps and
meager jewelry collections.
3)
S[C
c[s
Remember: No matter where if appears in the sentence, the sufficient
condition will always follow. If spruces are not in the forest, then conifers are
not in the forest. Also, if conifers are in the forest, then spruces must be in the
forest. But the forest could contain spruces without conifers.
4)
Nr[br
Br[nr
What does if introduce again? Oh yes, a sufficient condition. If St. Nick is not
on the vaunted red team, then the Easter Bunny must be on the red team. And
if the Easter
Bunny is not on the red team, St. Nick must be. This rule can also be
interpreted to mean that either St. Nick or the Easter Bunny must always be
on the red team. It is also acceptable for both St. Nick and the Easter Bunny
to be on the red team. Now there’s a power couple!
5)
s[c
C[S
Only always refers to a necessary condition. You might recognize this
example from a classic TV show that rewarded contestants with living room
sets and wave runners. It is necessary for a contestant to be a first- round
champion to advance to the showcase showdown. However, it is far from
sufficient. Many first-round champions lose on the wheel and never make the
big showdown.
6)
s[w
W[S
Only if always introduces a necessary condition. There are many
requirements for a male supermodel to wear a speedo (spray tanning comes
to mind), and waxing is definitely one of them. Does waxing guarantee
anything? Not even close. There may be other issues that intervene, such as
heat rash or self respect. But if a supermodel does not wax, there will be no
speedo.
7)
b[j
J[B
Only if still introduces a necessary condition. Brandi likes her jungle juice
and will only attend the party if it is in attendance. If the party takes place in a
state where Everclear is illegal (and thus you cannot make jungle juice), then
Brandi will not be there.
8)
c2[m1
M1[C2
One more time: Only introduces a necessary condition. In order to become
the chairman on the second panel, a person must have been on the first panel.
So if a person (be it Bob, Steve, or Laquisha) was not on the first panel, they
cannot be the chairman on the second panel.
9)
jv\dl
If, but only if signifies a reciprocal relationship. Here, we return to bidding on
questionable prizes. This rule means there are two options: either Jameer bids
on the vacation and Dwight bids on the living room set, or Jameer does not
bid on the vacation and Dwight does not bid on the living room set. Both or
neither.
10)
em \ lm
If and only if also signifies a reciprocal relationship. Here, the ladies are
sticking together. Either both Elizabeth and Lydia are getting married (thanks
to Mr. Darcy), or they will both grow to be cougars together. Don’t forget: If
and only if means we either have both conditions, or we have neither one.
11)
H[G
g[h
Remember: Always replace unless with the phrase if not. This rule can be
restated: If Henrietta does not win an award, then Gertrude does not win an
award. Also, if Gertrude wins an award (perhaps for most unfortunate name),
then Henrietta must also win an award.
12)
y[x
X[Y
Unless becomes if not, so we know that if Y is selected, then X must be
selected. Also, if X is not selected, then Y cannot be selected. Watch those
double negatives.
13)
p[n
N[P
Not a fun game when you have to deal with symptoms and diseases, but there
is a lesson here: The word any introduces a sufficient condition. So if
coughing or drowsiness or an ugly rash is a symptom of pneumonia, then it is
also a symptom of narcolepsy (and if a symptom is not found with
narcolepsy, then it’s not found with pneumonia). But if insomnia or priapism
is a symptom of narcolepsy, it need not be a symptom of pneumonia. Good to
know.
14)
i[f
F[I
Back to the world of only. Here, we visit the land of cheese-eating soap
haters. If the only people who shower irregularly are from France, then being
from France is necessary. Does everyone in France shower irregularly? Well,
no, even if it sometimes seems that way. If you aren’t from France, however,
then you shower regularly (hopefully just about every day).
15)
Y3[p1
P1[y3
After you replace unless with if not, this rule reads: If model 3 does not wear
the yellow dress, then model 1 wears the purple dress. And, of course, if
model 1 does not wear the purple dress, then model 3 is rocking a yellow
frock. (Yep, we said frock.)
16)
ug [ rb
Rb[Ug
Damn straight, commies! If America brings home the gold, then Russia earns
the bronze. The contrapositive tells us if Russia does not win the bronze, then
the US will not win the gold. However, if Russia wins the bronze, that does
not guarantee the US wins the gold.
17)
t[U
u[T
Is this a hallucinogenic zoo? Unicorns? What’s next, manticores? Only if
indicates a necessary condition, so a turtle exhibit implies that there is no
unicorn exhibit. And a unicorn exhibit would be clear evidence that there are
no turtles to be found. It is possible to have neither turtles nor unicorns
(boring), but the zoo never exhibits both. That would be sensory overload for
the children.
18)
b6[t3
T3[B6
Yummy. Nothing says haute cuisine like tilefish and brownies. If the
brownies are the sixth course, then the tilefish must be third. And if tilefish is
not third (perhaps it is second or, gasp, not served at all), then the brownies
cannot be the sixth course.
19)
p\r
Peter and Rachel are sticking together. If either one is accepted, then the
other one must be accepted as well. And if Peter or Rachel is not accepted,
then neither are accepted. In other words, you have to have both or neither.
This relationship is diagrammed with the reciprocal arrow.
20)
M3 [ h3
H3[m3
Horn Toad Race or Monster Truck Rally? Tough decision. If either one of
these destinations is not third, then the other one must be third. This rule is
best interpreted to mean that either the Horn Toad Race or the Monster Truck
Rally must be the third destination.
21)
b[F
f[B
Gross. Now you get to play the role of a crime scene investigator. If you
uncover some brain, then there are no fingerprints. And if you uncover some
fingerprints, don’t expect any brain matter to be found. This rule means that
you can never find both brain and fingerprints together. But they’re sure to
find some DNA. That stuff is everywhere.
22)
g4[p6
P6[G4
Only if indicates a necessary condition. So if the fourth toy is green, then the
sixth toy must be purple. Alas, if the sixth toy is not purple, then the fourth
toy cannot be green.
23)
R[J
j[r
Unless is replaced with if not, and the rule tells us if Ross does not attend the
party, then Joey will not attend the party. This also means that Ross must
attend if Joey attends. But what about Chandler and Monica? They’re
probably making out in the back room.
24)
cw[sf
Sf[Cw
In this rule, you are in charge of scheduling Olympic events. If curling (the
definition of excitement) is on Wednesday, then skeleton (a really safe
activity to pick up) is on Friday. Also, if skeleton is scheduled for any day
besides Friday, then curling is not on Wednesday.
25)
K[j
J[k
The word every indicates a sufficient condition. If Katie is not featured in a
skit, then Jessica must be. And if Jessica is not featured in a skit, then Katie
must be. Thus, if either one of them is not in a skit, then the other one must
be. This means either Katie or Jessica (or both) must be in every skit.
If diagramming seems slow
and arduous at this point, that is
natural. You will speed up with
practice. However, if you still have that
uncomfortable feeling (like walking
into a dentist’s office), be sure to
review these rules before moving on.
It’s going to get more complicated.
BP Minotaur:
At this point, you should feel comfortable with the following:
1. There are only two types of rules in Logic Games: absolute and
conditional.
2. A conditional rule doesn’t actually establish anything concrete rather, it introduces a relationship between two conditions such that
satisfying one guarantees another.
3. A sufficient condition is enough to guarantee the necessary
condition follows.
4. A necessary condition is required for a sufficient condition to follow.
5. There are two valid inferences from any conditional statement: the
valid affirmation (satisfying the sufficient condition) and the
contrapositive (denying the necessary condition).
6. Avoid two common invalid inferences when working with
conditional statements: the converse (satisfying the necessary
condition) and the inverse (denying the sufficient condition).
7. Always diagram conditional rules and their contrapositives.
That is a wrap on the basics. However, you will occasionally be confronted
with more complicated conditional rules. Guess what’s up next?
AND VERSUS OR
Alien versus Predator. The Yankees versus the Red Sox. Spy versus Spy. Mel
Gibson versus sobriety. These are classic battles. “And” versus “or” on the
LSAT may not have garnered as much hype over the years, but the difference
between the two is important for Logic Games. These claims are referred to
as conjunctions and disjunctions. Many rules will give you conjunctions or
disjunctions, and you must quickly know how to diagram them.
A conjunction is an “and” statement.
A disjunction is an “or” statement.
Conjunctions and disjunctions can be introduced in a variety of ways. The
following chart outlines the correct way to diagram the most common forms:
Rule
Correct Diagram
Both A and B
a+b
Either A or B
a or b
Not both A and B
A or B
Neither A nor B
A+B
“Not both” and “neither” cause lots of confusion for
students.
It looks like you switched
‘not both’ and ‘neither.’
Ditz McGee:
Many students think it looks that way. After all, not both A and B has and in
it and nor sounds just like or. But be careful. To satisfy the condition that you
don’t have both, you only have to be lacking one or the other. And to have
neither, you can’t have either one. We will introduce some helpful examples
in just a moment.
Enough chit chat. Let’s see this in action. Here are some hypothetical rules:
The fruit salad includes either bananas or
papaya.
According to this rule, either banana or papaya must be included in the fruit
salad. You must have one or the other (or you could have both). It’s
important to note that an “or” statements always allows for the
possibility of both.
Outside our LSAT bubble, when a friend says,
“I’d like filet mignon or a veal cutlet,” we usually take it
to mean that they desire only one of the two options. This
is not true on the LSAT. Unless stated otherwise, an “or”
statement allows for both (filet and cutlet), even though
BP Minotaur:
that would be quite a hefty load of
meat.
Let’s take a look at a couple more rules.
The squadron cannot include both Maverick
and Iceman.
Obviously, any squadron featuring both Maverick and Iceman would be
overflowing with testosterone and homoeroticism. This rule is commonly
misinterpreted to mean that both Maverick and Iceman are not on the
squadron. However, this rule allows either one of them to make it; it simply
rules out the possibility of having both. Thus, the rule should be
diagrammed to show that either Maverick or Iceman does not make the
squadron. (It is still possible that neither one makes the squadron, even
though no one would watch such a film.)
Neither Bunion Away nor Cellulite-B-Gone
is part of the Fortune 500.
When a rule says neither, it means that each condition does not happen.
This rule informs you that Bunion Away and Cellulite-B-Gone are both not
part of the Fortune 500. Shocking.
Exclusive Disjunctions (But Not Both)
Remember, on the LSAT, an “or” statement allows for both. So if a rule
states that Toby or Mary goes to the party, one or the other must go, but it
also could be true that they both go.
Occasionally, however, the test specifies that it must be one or the other and
not both. This is done through the use of the words “but not both” (not very
subtle). Here is an example:
Either Snoop Canine or
Dentist Dre, but not both,
is in the top 10 list on the
hip-hop charts.
Logically, there are two parts to this statement: (1) Either Snoop or Dre is
topping the charts, and (2) either Snoop or Dre is not included in the top 10.
It’s important to diagram both conditions to distinguish it from an inclusive
“or” statement. This is a bit ugly, but it represents the entire claim.
COMPLEX CONDITIONALS
When conditional statements include conjunctions and/or disjunctions, we
refer to them as complex conditionals. We already know the steps used to
form the contrapositive of any conditional statement: (1) Switch the
sufficient and necessary conditions, and (2) negate both conditions. The
same rules apply to complex conditionals, although you must be sure to
correctly negate any conjunctions and disjunctions.
To negate a conjunction or disjunction, “and” becomes
“or,” and “or” becomes “and.”
As always, some examples will help elucidate this process.
Clearly, something is going down
in the red states. The first step is
to diagram the rule as stated. If
the Senator from Oklahoma gives
a speech, then both the Senator
from Texas and the Senator from
Mississippi do not.
Here’s where it gets a bit
complex. For the big guy from
Oklahoma to give a speech, it is
necessary that both of the other
senators do not give speeches.
Thus, if either the Senator from
Texas or the Senator from
Mississippi gives a speech, then
the Senator from Oklahoma does
not give a speech.
That was so much fun, let’s do
If the Senator from
Oklahoma gives a speech,
then neither the Senator
from Texas nor the Senator
from Mississippi give a
speech.
1. Flipped the conditions
2. Negated everything
3. Changed “or” to “and”
another.
First, you hopefully spotted the
word “if,” which introduces the
sufficient condition. This rule tells
you that if either Marquez or Nando
is scheduled for Tuesday, then
Gomez must be scheduled for
Friday.
To form the contrapositive, follow
the same steps as before: Flip the
terms, negate them, and change
“or” to “and.” Since scheduling
either Marquez or Nando for
Tuesday is sufficient to guarantee
that Gomez is scheduled for Friday,
if Gomez is not scheduled for
Friday, then neither Marquez nor
Nando can be slotted in for Tuesday.
Gomez must be scheduled
for Friday if either
Marquez or Nando is
scheduled for Tuesday.
1. Flipped the conditions
2. Negated everything
3. Changed “and” to “or”
One more example! This is a complicated one...
Mmmm, now this is a tasty game.
Nothing spells gluttony like
Thanksgiving. This rule is more
complicated because it involves an
exclusive disjunction. If there is no
cranberry sauce on the table, then
there must be either pumpkin pie or
apple cobbler, but not both. In other
words, there must be one and not
the other. This should be
represented using the convention
If there is no cranberry
sauce, then there must be
either pumpkin pie or apple
cobbler, but not both.
discussed earlier.
To form the contrapositive, you
have to execute the same steps:
(1) Flip the sufficient and
necessary conditions, (2) negate everything, and (3) switch the operators
(and/or).
Contrapositive : If there is neither pumpkin pie nor apple
cobbler or if there is both pumpkin pie and apple cobbler,
then there must be cranberry sauce.
Conditional statements with conjunctions and/or disjunctions are among the
most challenging rules you will confront in Logic Games. If you can diagram
this tasty, Thanksgiving-inspired example, then you can diagram anything.
Make sure to review these rules often during your studies.
Next up : a drill to make sure that
everything is clicking.
DIAGRAMMING, SUPER-SIZED
In front of you lies a mix of complicated conditional rules. Diagram each rule
and its contrapositive.
1. If the dolphin is in tank 3,
then both the shark and the
whale are in tank
2. The Titans make the
playoffs only if the Steelers or
the Jaguars make the playoffs.
3. J cannot be selected if both
K and L are selected.
4. If the first and second
pregnancy tests are positive,
then the third one is positive as
well.
5. If Van takes chemistry, then
he takes physics but not
astronomy.
6. If the tarantula or the
rattlesnake is in the closet, then
neither little Susie nor her dolly
is in the closet.
7. If it is not the case that both
Ben and Vince win an Oscar,
then Will must win an Oscar.
8. If X is selected, then either
Y or Z, but not both, must be
selected.
9. Bill is having a good time in
his office if either his wife or
his intern, but not both, is with
him.
How’d it go this time?
1)
Diagram:
Contrapositive:
d3[s2+w2
S 2 or W 2 [ D 3
If the dolphin is in tank 3, that is sufficient to tell you that both the
shark and the whale are in tank 2. Because both conditions are
necessary, if either one does not happen (if the shark is not in tank 2 or
if the whale is not in tank 2), then the dolphin cannot be in tank 3. And
if you cannot find the dolphin in any tank, it’s time to call Ace Ventura:
Pet Detective.
2)
Diagram:
Contrapositive:
t [ s or j
S+J[T
It’s time for the NFL playoffs. Only if introduces a necessary condition,
so either the Steelers or the Jaguars must make the playoffs in order for
the Titans to make it. Thus, if both the Steelers and Jaguars do not
make the playoffs, then the Titans also do not make the playoffs.
3)
Diagram:
Contrapositive:
k+l [ J
j [ K or L
In this example, selecting both K and L is sufficient to conclude that J
cannot be selected. Thus, if J is selected, then it cannot be true that both
K and L are selected (either K or L cannot be selected). This rule can
also be interpreted to mean that K, L, and J cannot all be selected
together. Any two of the variables could be selected, but not all three.
4)
Diagram:
Contrapositive:
1p+2p[3p
3 p [ 1 p or 2 p
This is the stuff of nightmares. If the first test is positive, you are
allowed to take a second. But if both the first and second tests come up
positive, you know what is coming on number three. Thus, if the third
test is not positive, then either the first or the second (or both) must not
be positive. Note: This is not an effective means of contraception.
5)
Diagram:
Contrapositive:
c [ p+A
P or a [ C
This one can be a little confusing. If Van takes chemistry, then Van
takes physics but he does not take astronomy. To form the
contrapositive, you still follow the same rules: Flip the terms, negate all
of the conditions, and switch the operator. If Van does not take physics,
or if he does take astronomy, then he cannot take chemistry.
6)
Diagram:
Contrapositive:
t or r [ S+D
s or d [ T + R
This closet is more like a haunted house. If either the tarantula or the
rattlesnake is found in the closet, then both little Susie and her dolly are
not in the closet. Thus, if either little Susie or her dolly (or both) are in
the closet, then neither the tarantula nor the rattlesnake is in the closet.
7)
Diagram:
Contrapositive:
B or V [ w
W[b+v
The original claim here can be difficult to decipher. Here, our sufficient
condition says, “If it is not the case that both Ben and Vince win an
Oscar.” We have a not both statement which, if you remember from
our chart a few pages back, we diagram as not Ben or not Vince. If this
happens, then Will must win (he is, after all, kind of a big deal). And if
Will does not win an Oscar (some people just do not understand the
comic genius behind Sex Panther), then both Ben and Vince must have
brought home the shiny trophies.
8)
Diagram:
Contrapositive:
x [ (y or z) + (Y or Z)
(Y + Z) or (y + z) [ X
This is an ugly one. Make sure to always take note of the phrase but
not both. If X is selected, then either Y or Z must be selected and either
Y or Z must not be selected (one or the other, but not both). The
contrapositive would then state that if neither Y nor Z are selected or if
both Y and Z are selected, then X cannot be selected.
9)
Diagram:
Contrapositive:
(w or i) + (W or I) [ g
G [ (W + I) or (w + i)
“I did not have sexual relations with that woman.” In this example, if
either Bill’s wife or intern is in his office (which may or may not be
shaped like an oval) and either his wife or his intern is not there, then
Bill is having a good time. And if he’s not, then one of two things must
be true: (1) Neither of them is there, or the truly dangerous situation,
(2) both his wife and intern are in the office at the same time. Good
luck, Bill.
For some students, this drill feels worse than wearing a meat-scented
swimsuit in a South American river. The good news is that it gets better with
practice. Also, such complex rules don’t show up all the time. But when they
do, it’s important to be able to diagram correctly.
TRANSITIVE PROPERTY
There is one last important skill you need to develop - combining
conditionals to draw important deductions. Until now, we have been
dealing with conditional rules in isolation, but it is common to be presented
with more than one conditional rule in a game. When this happens, you
should see if any additional deductions can be made.
Let’s check out an example. Pretend that the wonderful people who write the
LSAT have been reading a bit too much Us Weekly, and they give you the
following rules:
Welcome to the world of celebrity
starlets, rife with stints in rehab,
rehab. parenting blunders, and
“accidentally” released video
footage. As always, the first step is
to diagram each rule.
If Britney goes to rehab,
then Paris goes to rehab.
If Paris goes to rehab, then
Nicole does not go to
rehab.
See how the necessary condition of
the first rule is the same as the
sufficient condition of the second
rule? Learn to love that. It means the
rules can be combined.
These two rules can be combined to uncover a helpful
deduction. If Britney goes to rehab, then Paris must go
to rehab. And if Paris goes to rehab, then Nicole
cannot go to rehab. These two rules tell you that if Britney goes to rehab, then
Nicole does not. The contrapositive will also be useful: If Nicole goes to
rehab, then Britney does not.
In logic, this is called the transitive property. Whenever the necessary
condition of one statement is the same as the sufficient condition of
another statement, they can be combined to form a transitive conclusion.
Transitive conclusions are
commonly the key to answering
questions, so there’s a big payoff when
you can combine conditional rules.
Ninja Note:
Congratulations! You’ve now learned how to diagram and understand
conditional statements. Be sure to review this chapter early and often. Now
that we have this stuff in our bag of tricks, it’s time to jump back to ordering
games. The next chapter covers games that will look like the basic ordering
games covered earlier, but the rules will be very different.
2 Men have a tendency to err on this example and conclude that a diamond is
sufficient for an engagement. However, any lady friend will quickly explain that the
ring is necessary, yet far from sufficient.
12/1:1ORDERING
WHAT IS IT?
The next set of games are similar to the last, but a little bit different.1 We are
now going to focus on 1:1 ordering games. These games will also introduce
two variable sets (ice cream scoops and children, basketball players and shoe
sizes). Each player in the game will be assigned to one and only one slot. The
big difference in this chapter will be a variety of new rules.
An ordering game with 1:1 correspondence has one
and only one slot for each player.
Assigning seven brides to seven brothers is a game with 1:1
correspondence. Arranging eight sprinters in eight lanes is a game with 1:1
correspondence. If you have six friends at a bachelorette party and only
two professional entertainers, you don’t have 1:1 correspondence (and you
should probably leave).
1:1 ordering games are
among the most common types of
games on the good old LSAT.
Frequently, though not always, the first
game in a section will be of this type.
BP Minotaur:
Note: As with all ordering games, remember to identify the variable set with
the inherent order (days, height, ranking), and use that variable set as
the base. The other variable set will then be arranged into the ordered
slots.
In this chapter, we will introduce a variety of new ordering rules. The setup
for 1:1 ordering games will look very familiar, but your challenge, if you
choose to accept it, is to learn to juggle these new rules effectively. Let’s go.
BLOCKS
Legos. Lincoln Logs. These were the staples of our youth. Before the
mindless days of computers and the world wide web, children used to spend
hours constructing empires out of miniature wooden blocks. Finally, all of
that practice pays off.
Blocks are used to represent spatial relationships that are more concrete than
those represented with dashes. When you know exactly where one player is
situated in relation to another, use a block to symbolize this relationship.
Here are some examples of rules correctly symbolized as blocks:
The shotgun must be placed immediately
in front of the crib.
The family room set is auctioned off
immediately after the vacation rental.
Jack arrives at the party immediately
before Karl but immediately after Mike.
Symbolizing these rules as blocks gives you an
important visual reminder to keep consecutive slots open.
Ninja Note:
On occasion, blocks will leave empty slots between
players. This can easily be incorporated into the
diagram. Here are a few examples:
Exactly one lucky winner
spins the big wheel after Betty but before
Tina.
The bus tour visits Chattanooga exactly
three days after Nashville.
You screwed up on that
last one. That’s only two days after.
Cleetus Comment:
While wrong, Cleetus does raise a good
point. You have to be very careful with
the wording that introduces blocks. The
last block does represent that Chattanooga
is visited three days after Nashville.
However, this is synonymous with a rule stating there are two cities between
Chattanooga and Nashville.
Reversible Blocks
There is one more trick when it comes to blocks. Sometimes, they can go
both ways.2 The previous blocks made it clear which player was in front, but
some blocks are reversible. Certain rules will tell you that two players must
be next to each other without defining who comes first. To represent this, we
will place a double-sided arrow over the block.
Here are some examples:
Sigfried and Roy must stand next to each
other in line for the buffet.
Exactly one clown performs in between
Bozo and Krusty.
You will grow to love blocks.
Since they give a fixed relationship,
you will commonly find that there are
only a few possible placements for a
block. This can lead to huge
deductions.
Ninja Note:
RESTRICTIONS
In ordering games, you always want to rule out possibilities. For example, if
both Ralph and Eduardo can’t sit in seat 4, there are two less people to worry
about in that slot. Rules will commonly dictate that a player cannot be
placed in a certain slot. We can easily represent these rules by placing a
restriction under that slot. Here are some examples:
Delilah cannot be
seated in chair 3.
Neither Monday nor
Friday is a day on
which the cheetah can
be wrestled.
You can also form restrictions from other types of rules, and they can give
you great visual reminders in a game. Both dashes (which you should
remember from basic ordering games) and blocks can be used to form helpful
restrictions. “How?” you ask. Here are some examples:
Sleepy must be in a lower-numbered room
than Dopey.
Grumpy must be in the room numbered one
lower than Bashful.
Since Sleepy must be in a lower-numbered room than Dopey, Sleepy
cannot be in room 6, and Dopey cannot be in room 1.
Since Grumpy must be in the room numbered one lower than Bashful,
Grumpy cannot be in room 6, and Bashful cannot be in room 1.
But wait, there’s more. You should always keep track of your restrictions.
When you are able to make a lot of restrictions about players that cannot go
in a certain slot, it is time to start thinking about who can go into that slot.
This can lead to big-time deductions.
For example, pretend you are attempting a 1:1 ordering game with five
players: T, U, V, W, and X. Here’s an example of how restrictions can lead
to a big deduction:
T cannot be first or fifth.
X must be third.
U and V must be in front of W.
T cannot be first or fifth, so we can make restrictions under those slots.
X gets plugged into the third slot.
Both U and V must come before W, so W cannot be first or second, and
both U and V cannot be fifth.
Here’s the big moment. Since T, U, and V cannot be fifth, and X must be
third, the big deduction is that W must be fifth because it is the only option
left.
Before we go any further, it’s time for a drill.
1 After writing this sentence, we realized that telling you something is the same, but
different, is not very helpful. Other platitudes that we really enjoy: “It is what it is,”
and the classic rejoinder, “Yeah, but still.”
2 Various drafts of this book had inappropriate comments inserted into this footnote.
But this one’s just too easy.
1:1 ORDERING DRILL
Peanut butter is great. Jelly is surely a treat. Put them together, though, and
the magic happens. Logic Games aren’t quite as delicious, but the same
principle applies. You must learn to use all of the rules together to form
deductions.
Represent the following rules using dashes and blocks.
Then, make as many restrictions as possible.
Note: Each game is a 1:1 ordering game with six players
and six slots.
1. Six daredevils on a reality TV show—Bam, Cam, Frank,
Johnny, Stevie, and Weenie—are ranked according to the number
of injuries they suffer from first (most injuries) to sixth (least
injuries). There are no ties.
Stevie has more injuries than Bam and Johnny.
Johnny ranks immediately ahead of Weenie.
2. An entirely different TV show features six characters— Brian,
Lois, Meg, Peter, Quagmire, and Stewie. They are ranked by their
coolness factor from first (coolest) to sixth (least cool). Again,
there are no ties.
Peter is ranked third.
Stewie ranks immediately ahead of Quagmire.
Brian is cooler than Lois but less cool than Meg.
3. Six animals are arranged from highest (first) to lowest (sixth) on
the food chain. There are no ties. The six animals are an elephant, a
hare, a panther, a snail, a turtle, and a wolf.
The elephant is lower than the hare.
The snail and the hare are separated by exactly one other animal.
The turtle is neither the highest nor the lowest animal.
The panther is the fourth highest animal.
The wolf is higher than the panther.
4. Six types of cereal—Cornies, Duffy, Eggish, Fantastic, Gruel,
and Hamandeggs—are part of a consumer survey that ranks the
cereals from most popular (first) through least popular (sixth). As
always in this drill, there are no ties.
Cornies are more popular than Duffy and Fantastic.
Gruel is ranked second.
Fantastic is more popular than Hamandeggs.
Eggish and Fantastic are ranked consecutively.
ANSWER KEY
There are two important things to check in this answer key. First, make sure
you diagrammed the rules correctly and combined them whenever possible.
Second, check the restrictions to see if you were able to spot them all.
The Johnny and Weenie block should be combined into the ordering
chain.
Stevie has more injuries than Bam, Johnny, and Weenie, so Stevie
cannot be ranked fourth, fifth, or sixth. Stevie must be ranked pretty
high.
Bam has fewer injuries than Stevie, so Bam cannot be ranked first.
Johnny has fewer injuries than Stevie but more than Weenie, so Johnny
cannot be ranked first or sixth.
Weenie has fewer injuries than Stevie and Johnny, so Weenie cannot be
ranked first or second.
Cam and Frank are randoms because they aren’t in any rules. No
restrictions there.
The first slot is the most restricted. Either Stevie, Cam, or Frank must be
ranked first.
Stewie must be ranked immediately higher than Quagmire, so Stewie
cannot be ranked second (because Peter is third) or sixth.
Quagmire must be ranked immediately below Stewie, so Quagmire
cannot be ranked first or fourth (because Peter would get in the way
again).
Meg ranks higher than Brian and Lois, so Meg can’t be ranked fifth or
sixth.
Brian ranks higher than Lois but lower than Meg, so Brian can’t be
ranked first or sixth.
Lois ranks lower than Meg and Brian, so Lois can’t be ranked first or
second.
The coolest character must be either Meg or Stewie.
The least cool character must be either Quagmire or Lois.
The elephant is lower than the hare, so the elephant isn’t first and the
hare isn’t sixth.
The panther is fourth, and the hare and the snail must be separated by
exactly one spot, so the hare and the snail cannot be second or sixth.
The turtle can’t be first or sixth.
Since the wolf is higher than the panther, the wolf must be no lower
than third.
The big deduction is about the lowest spot on the food chain. There are
restrictions for four of the six animals, and the panther also can’t be
sixth. The only player left is the elephant, so the elephant is lowest on
the food chain (shocking).
The first, third, and fourth rules can all be combined into a powerful
ordering chain.
Cornies are more popular than Duffy, Eggish, Fantastic, and
Hamandeggs. Since Gruel is second, Cornies must be the most popular.
The block with Eggish and Fantastic is more popular than Hamandeggs,
so neither Eggish nor Fantastic can be the least popular cereal. The
block must be placed third and fourth, or fourth and fifth.
Duffy can’t be ranked fourth because there would be no possible
placement for the block.
Hamandeggs is less popular than Cornies, Gruel, Eggish, and Fantastic.
The only cereal Hamandeggs could beat out is Duffy, so Hamandeggs is
either fifth or sixth.
That drill became increasingly difficult with each new example. By now,
however, you should be grasping how these new ordering rules work
together.
At this point, you should rate your confidence with the previous rules as
“strong to very strong.” But that’s not all. Say hello to more new rules...
DIVISIONS
Can’t we all just get along? Unfortunately, no. There are times when you just
have to keep people away from each other. Republicans and Democrats. One
spy and that other spy. Rednecks and educated folk. Some rules in games
perform the same function. Most commonly, they state that two players
cannot be next to each other. We call this type of rule a division. Here are a
couple examples:
The ham cannot be served immediately
before the matzah.
Rush and Obama cannot be
seated next to each other.
Divisions are much less
common than blocks.
BP Minotaur:
OPTIONS
Games are all about figuring out your options. There are rules that will
specify that a certain slot must be assigned to one of two players. When this
happens, you should scream for joy (figuratively, of course; you are not
allowed to scream during the actual exam). Only two players can possibly fill
a slot, and we represent this by writing an option in that slot. Here’s an option
in action:
Either Balthazar or Cryptonite
must be assigned to space shuttle
N.
Options can also be used to represent helpful deductions. Pretend you are
doing a 1:1 ordering game that involves scheduling the performances for five
acrobats—Alibaba, Barbie, Coinstar, Deuteronomy, and Elephantitis—from
first through fifth. Check out the following rules and how we can make an
option in our setup to represent a deduction:
Barbie must perform fourth.
Both Coinstar and Deuteronomy must
perform after Elephantitis.
Barbie must perform fourth. Coinstar and Deuteronomy cannot perform
first since they must follow Elephantitis. These restrictions only leave
two options for the first performance: Alibaba and Elephantitis. This can
be plugged right into the setup.
Linked Options
Linked options are a frequently-used variant of normal options. If you know
that two players must fill two specific slots, but you don’t know exactly who
fills each one, you should represent this situation with linked options.
Here is a quick example to show you how this works. Let’s play with five
acrobats again, but now the performers are named Vendetta, Wilma, Xavier,
Yolanda, and Zane.
Xavier must perform second.
Vendetta must perform fourth.
Vendetta must perform after both Yolanda
and Zane.
Since both Yolanda and Zane must precede Vendetta, and there are only
two available spots that precede her, Yolanda and Zane must fill those
two slots. However, it is not clear whether Yolanda goes in spot 1 and
Zane goes in spot 3, or vice versa.
Once you fill in slots 1 and 3 with linked options, it is easy to spot an
important deduction: Wilma must fill the last slot.
Linked options allow you to
visually represent that two slots are
reserved for two players. Otherwise, it
would be easy to make a mistake and
think another player could be assigned
to one of the slots.
Ninja Note:
ARCHES
Some rules will restrict a certain player to only two possible slots. For
instance, the big money must be behind door number 1 or door number 2.
These rules will be important because they place such a strong restriction on
one of the players. Instead of writing this rule off to the side and possibly
forgetting about it, we want to represent such rules directly in our setup. We
can do this by making arches to show that a player only has two options.
Here are two examples for your viewing pleasure:
Juan must be assigned to week 3
or week 5.
Divine must stand at the
front or the back of the
line.
That’s all, folks. To master 1:1 ordering games, you simply have to learn how
to use each of these types of rules. Here’s a quick overview of the rules that
we have covered so far:
1. Dashes - A is before B.
2. Blocks - A is immediately before B.
3. Restrictions - A is not in slot 4.
4. Divisions - A is not next to B.
5. Options - A or B is in slot 3.
6. Arches - A is in slot 1 or slot 6.
Your next task is to learn how to use these wonderful tools together. There is
a drill on the next page to help you do exactly that. Get in there.
1:1 ORDERING DRILL
This drill is designed to test your budding knowledge of ordering rules. For
each of the following games, build an appropriate ordering setup, represent
the rules, and try to make as many deductions as possible (mainly, try to
combine rules and note restrictions). When you feel ready, try to answer the
questions that follow. Game on.
Game #1: Cast of Characters
Exactly seven TV personalities—Alf, Beetlejuice, Chuck Norris,
Donatello (of Teenage Mutant Ninja Turtle fame), Mr. Ed,
Fantasia, and Geronimo—all attend an awards show. The
characters arrive only once and one at a time. The order for the
arrival is governed by the following conditions:
Either Chuck Norris or Donatello must be the fourth character to
arrive. Fantasia arrives earlier than Geronimo.
Donatello must arrive either immediately before or immediately
after Alf. Mr. Ed must arrive immediately before Beetlejuice.
1. If Alf arrives second, then when must Chuck Norris arrive?
2. If Geronimo arrives second, then when could Chuck Norris arrive?
Game #2: Eligible Bachelors
A women’s magazine ranks the top six bachelors in the world. The
finalists are Brad, Clooney, Depp, Jude, Timberlake, and
Zoolander. There are no ties. The following restrictions govern the
final results:
Timberlake ranks lower than Clooney.
Depp either ranks first or third.
Clooney ranks exactly three spots lower than Jude.
1. What is the complete and accurate list of the possible rankings for
Clooney?
2. If Timberlake is not the lowest ranking bachelor, then what is Depp’s
ranking?
Game #3: College
A university student has a busy week ahead. Between Monday and
Friday, he must go grocery shopping, play video games, organize
his DVD collection, play in a beer pong tournament, and do
homework. The student performs exactly one task each day. The
following rules must apply:
The student does not go grocery shopping on Monday,
Wednesday, or Friday.
The student must play video games before he plays beer pong.
At least one activity must be performed in between the days he
does homework and organizes his DVDs.
1. What is the complete and accurate list of activities the student could
perform on Monday?
2. If the student goes grocery shopping in between playing video games and
playing beer pong, what activities could the student perform on Wednesday?
Game #4: The Big Game
A NFL sportswriter visits the training camps of seven different
NFL teams on seven consecutive days. The writer will visit one
team on each day. The teams visited are the Arizona Cougars,
Baltimore Raptors, Chicago Beavers, Denver Baboons, Green Bay
Packrats, Indianapolis Caterpillars, and Jacksonville Jarheads. The
visits must conform to the following conditions:
Chicago is visited before Indianapolis.
Jacksonville is visited on the first or last day.
If Baltimore is visited on the third day, then Arizona is visited on
the seventh day. Denver is visited before Baltimore or before
Green Bay, but not both.
1. Which teams cannot be visited on the first day?
2. If the writer visits Baltimore on the third day, what is the complete and
accurate list of the days she could visit Denver?
Game #5: Happy Birthday
On a recent episode of Super Spoiled Sweet Sixteen, a Beverly
Hills teen opened six birthday presents. The presents were opened
consecutively and one at a time. The six presents were a Porsche, a
Rolex, a servant, tennis lessons, an actual unicorn, and a yacht. The
following rules govern the order in which the presents were
opened:
The Rolex is opened before both the Porsche and the tennis
lessons.
The tennis lessons are opened before the unicorn.
If the servant is opened before the yacht, then the yacht is
opened before the tennis lessons.
1. Among the six presents, what is the latest the Rolex could be opened?
2. If the servant is opened first and the Porsche is opened third, then which
present must be opened fourth?
Game #6: Busy Week
Jake must schedule one date each night this week. Seven lucky
ladies—Gwen, Harriet, Ivanna, Jackie, Krista, Lucy, and Mindy—
will go on a date with Jake on a different night, from Sunday
through Saturday. The schedule is consistent with the following:
Ivanna must be scheduled for Thursday.
Krista must be scheduled for an earlier day than Ivanna.
If Mindy is scheduled for Monday, then Jackie must be
scheduled for Wednesday. There must be exactly one day
between Jake’s dates with Lucy and Ivanna. Gwen and Jackie
must be scheduled for consecutive days.
1. If Mindy is scheduled for Monday, then which lucky lady is scheduled for
Friday?
2. If Gwen is scheduled for a later day than Ivanna, then what is the complete
and accurate list of the days that Jake could schedule his date with Mindy?
CHECK IT OUT
Here are the answers for the previous drill. Make sure you represented all of
the rules correctly and are starting to spot deductions. Review anything you
missed.
Game #1: Cast of Characters
Since Fantasia arrives before Geronimo, Fantasia cannot be the last
character to arrive and Geronimo cannot be the first character to arrive.
Mr. Ed must arrive immediately before Beetlejuice, so Mr. Ed cannot
arrive last and Beetlejuice cannot arrive first. Also, since either Chuck
Norris or Donatello must arrive fourth, Mr. Ed cannot arrive third, and
Beetlejuice cannot arrive fifth.
1. Fourth
If Alf arrives second, then Donatello must arrive first or third.
Since Donatello cannot arrive fourth, Chuck Norris must arrive fourth.
(Hide the women and children.)
2. Third, Fifth, Seventh
If Geronimo arrives second, then Fantasia arrives first.
Chuck Norris can’t arrive fourth because
there would not be enough room for both
blocks, so Donatello must arrive fourth.
If Alf arrives third, Chuck Norris could
arrive fifth or seventh (to allow space for
the block).
If Alf arrives fifth, Chuck Norris must arrive third.
Game #2: Eligible Bachelors
This game is a good example
of the ambiguous wording that can be
dangerous in Logic Games. Here,
ranking lower does not correspond to
lower numbers. Rather, slot 1 is the
highest rank, and slot 6 is the lowest
rank.
BP Minotaur:
In this game, there is a huge block that includes both Jude and Clooney. In
addition, you know that Timberlake must be placed somewhere after the
block. This combination could lead to tons of restrictions. However,
because there are so many places that these players cannot be placed, it
becomes more helpful to think about where they can be placed. The block
with Jude and Clooney has only two possible placements: first and fourth,
or second and fifth.
If Jude is ranked first and Clooney is fourth, then Depp must be third. In
addition, because Timberlake must rank lower than Clooney (fifth or
sixth), you should add an option for either Brad or Zoolander in the second
slot.
If Jude is ranked second and Clooney is fifth, then Timberlake must be
ranked sixth (cry me a river, Justin). Since Depp must be ranked first or
third, the fourth spot must be assigned to either Brad or Zoolander, leading
to an option.
1. Fourth, Fifth
Since the Jude/Clooney block only has two possible placements (above),
we deduced that Clooney must either be ranked fourth or fifth.
2. Third
If Jude is ranked second, then Timberlake must be ranked last. Thus, if
Timberlake is not ranked last, then Jude must be ranked first.
If Jude is ranked first, then Clooney must be ranked fourth and Depp is
ranked third.
Game #3: College
Since the student cannot go grocery shopping on Monday, Wednesday, or
Friday, grocery shopping must occur on Tuesday or Thursday.
The video games must come before beer pong, so the student cannot play
video games on Friday, and he cannot play beer pong on Monday (not a
good way to start the week).
Since there must be at least one activity in between homework and
organizing DVDs, that rule should be represented as a division.
1. Video Games, Homework, Organizing DVDs
The student can’t go grocery shopping or play beer pong on Monday, but
the other three options are acceptable because no other rules prevent them
from going on Monday.
2. Homework, Organizing DVDs
If the student goes grocery shopping in between the days he plays video
games and plays beer pong, he still could go grocery shopping on Tuesday
or Thursday.
If the student goes grocery shopping on Tuesday, then he must play video
games on Monday. Homework and organizing DVDs can’t be done on
consecutive days, so there should be linked options for these two activities
on Wednesday and Friday. Thursday looks like
a great day for playing beer pong.
If the student goes grocery shopping on
Thursday, then he must play beer pong on
Friday. Just like in the first situation,
homework and organizing DVDs form linked
options, this time on Monday and Wednesday.
Playing video games is slotted in for Tuesday.
The only possible activities for Wednesday are
homework and organizing DVDs.
Game #4: The Big Game
Since Chicago is visited before Indianapolis, Chicago cannot be visited on
the last day, and Indianapolis cannot be visited on the first day.
If Baltimore is visited on the third day, then Arizona is visited on the
seventh day, and Jacksonville has to be visited on the first day. Also, if
Arizona is not visited on the seventh day, then Baltimore cannot be visited
on the third day (contrapositive).
The last rule says that Denver must be visited before either Baltimore or
Green Bay, but not both. Therefore, Denver must be visited before one of
the cities and after the other. We represent this rule as an ordering chain
with linked options. From this chain, we see that Denver cannot be visited
on the first day or the last day.
1. Denver, Indianapolis
Indianapolis can’t be visited on the first day because it comes after
Chicago.
Denver must be visited after either Baltimore or Green Bay, so it also can’t
be visited on the first day.
2. Fourth, Fifth
If Baltimore is on the third day, then Arizona is on the seventh day.
Since the slot for the last day has been filled, Jacksonville must be visited
on the first day.
Denver must be visited after
Baltimore, but before Green Bay,
so Denver can’t be visited on the
second day or the sixth day.
Indianapolis must be visited after
Chicago, so Chicago is the only
option remaining for the second day.
The Denver visit could take place on the fourth day or the fifth day.
Game #5: Happy Birthday
The Rolex must be opened before the Porsche, tennis lessons, and the
unicorn, so the Rolex cannot be opened fourth, fifth, or sixth.
The Porsche must be opened after the Rolex, so the Porsche cannot be
opened first.
The tennis lessons must be opened after the Rolex but before the unicorn,
so the tennis lessons cannot be opened first or last.
The unicorn must be opened after the Rolex and the tennis lessons, so the
unicorn cannot be opened first or second.
The last rule states that if the servant is opened before the yacht, then the
yacht is opened before the tennis lessons. It is also important to form the
contrapositive: If the tennis lessons are opened before the yacht (negation
of the necessary condition), then the yacht must be opened before the
servant (negation of the sufficient condition). This rule implies that the
yacht must be opened before the tennis lessons or the servant, or both, so
the yacht cannot be opened sixth.
In 1:1 ordering games, there are no ties. If A is
not in front of B, then it must be behind, and vice versa.
Keep this in mind when you are negating conditions to
form the contrapositive.
BP Minotaur:
1. Third (The Porsche, tennis lessons, and unicorn must be opened after the
Rolex.)
2. Yacht
If the servant is opened first, then it must be opened before the yacht.
Thus, the yacht is opened before the tennis lessons.
The Rolex must be opened second
(before the Porsche). According to the
altered ordering chain, the yacht is
fourth, the tennis lessons are fifth, and the unicorn is last.
Game #6: Busy Week
Since Ivanna is on Thursday, and Krista must be scheduled before Ivanna,
Krista cannot be the date for Friday or Saturday.
Since Ivanna must be on Thursday and there is exactly one date between
Ivanna and Lucy, Lucy must be on either Tuesday or Saturday (an arch).
1. Harriet
If Mindy is on Monday, then Jackie is on Wednesday.
Gwen and Jackie are on consecutive days, so Gwen is on Tuesday.
Lucy can’t be on Tuesday, so
she gets the hot date on
Saturday.
Krista must be scheduled before Ivanna, so Krista is on Monday.
Friday goes to Harriet.
2. Sunday, Wednesday
Gwen and Jackie form linked options for Friday and Saturday.
Lucy can’t be on Saturday, so she must be on Tuesday.
Since Jackie isn’t on Wednesday, Mindy can’t be on Monday.
Mindy’s only options left are Sunday and Wednesday.
That’s a great drill to develop your ordering skills. It’s almost time to jump
into actual 1:1 ordering games, but there’s one more important point to cover.
ALL RULES ARE NOT CREATED EQUAL
When you throw a party, there are some guests that you really need to keep
an eye on, whether for good reasons (cute, charming guy) or not-so-good
reasons (sloppy drunk who likes to break things). The same principle can be
applied to Logic Games. Not all rules are created equal. When you work
through a game, it will commonly be the case that you will keep reverting
back to one or two rules, while the others are more like wallflowers.
Staring at an ugly setup with six or seven equally unattractive rules can be
quite overwhelming. However, true games gurus can spot the important rules
early, and thus have a good idea of where to start the work on each question.
There are two types of rules in ordering games that tend to be the most
helpful: block and arches.
When you are struggling to
find deductions or to work through a
question, it is a good rule to always
look for blocks and arches.
BP Minotaur:
As we work through more ordering games, we will introduce lots of
advanced strategies. But you will find that we will return to these rules over
and over again.
Now, it’s time for us to jump in and try some of these 1:1 ordering games. As
always, we are going to stress that you build a great setup and search for
deductions so that the questions are painless. If you build it, they will come
(the answers, that is).
We are going to walk through the first game together using Blueprint
Building BlocksTM. Eventually, we will cut you loose to try some on your
own. Put on your game face, and let’s go.
OCTOBER 2003: GAME 1 (1-7)
You might believe that you are taking the LSAT as one step in your quest to
become a rich and powerful attorney, but this game actually tests your hand
at organizing a closet. Here are our general objectives for this game:
1. Get a feel for all of the new types of rules that we just introduced;
and
2. See how important it is to quickly represent these rules and continue
to search for deductions.
1. Setup
A closet contains exactly six hangers - 1, 2, 3,
4, 5, and 6 - hanging, in that order, from left to
right. It also contains six dresses—one gauze,
one linen, one polyester, one rayon, one silk,
and one wool— a different dress on each of
the hangers, in an order satisfying the
following conditions:
Apparently, the people over at LSAC
are not exactly fashionistas. That is
quite a collection of dresses.1 The
introduction sounds very similar to the
games that we have already done.
Remember, the only difference in
these games will be the wider variety of rules. The game presents you with
six players (the dresses) and six slots (the hangers), so this is definitely a 1:1
ordering game. Our setup should be relatively straightforward.
2. Rules
Now it is time to represent the rules. Let’s work through these one at a time.
There’s our first arch. This will be important, so plug it right into the setup.
Mental note: Watch that rayon dress.
This rule gives you an option for the third slot, and it should also be plugged
right into your setup. Looking strong as we check out the last rule.
The last rule gives you a block. This rule doesn’t just state that the linen dress
is somewhere to the right of the silk dress; it is immediately to the right.
Ninja Note:
That block is going to be huge
in this game.
That’s the end of the rules, but this is just the start of our challenge. Resist the
urge to jump into the questions; it’s time for deductions.
3. Deductions Are King
It’s time to investigate how the ordering rules work together. The first
challenge is for you.
Challenge : Use the rules to form restrictions for some
of the dresses.
There aren’t any randoms in this game - all six dresses are involved in at least
one rule. It’s advisable to work through each rule and build restrictions when
possible.
Since the gauze dress must be on a lower-numbered hanger than the
polyester dress, the gauze dress cannot be on hanger 6.
Also, since the polyester dress must be on a higher-numbered hanger
than the gauze dress, the polyester dress cannot be on hanger 1.
Since the linen dress must hang
immediately to the right of the silk
dress, the linen dress cannot be on
hanger 1.
The silk dress must hang immediately
to the left of the linen dress, so the
silk dress cannot be on hanger 6. Also, since the linen dress cannot be
on hanger 3, the silk dress cannot be on hanger 2.
The placement of the block is going to be key for
the questions. The silk dress could be on hanger 3, which
would place the linen dress on hanger 4. Alternatively, if
the wool dress is on hanger 3, then there are a few more
options for the block. The silk and linen dresses could be
Ninja Note:
on hangers 1 and 2, 4 and 5, or 5 and 6.
Now you can see how the rules work together in a real game. With a good
setup, solid rules, and a nice set of deductions, we’re ready to jump into the
questions. It’s time to get some points.
1 A gauze dress? Gauze should be used for bandaging wounds, not for crafting
dresses. A rayon dress? Only if we are going back in time to attend a 70s prom. This
game is a legendary fashion faux-pas.
4. Questions
Question #1
1. Which one of the following could be an
accurate matching of the hangers to the
fabrics of the dresses that hang on them?
As you can expect, this game starts off with an elimination question. You
know the strategy.
Challenge : Use the rules to eliminate the four incorrect
answers.
The gauze dress is on a lower-numbered
hanger than the polyester dress.
The rayon dress is on hanger 1 or hanger 6.
Either the wool dress or the silk dress is on
hanger 3.
The linen dress hangs immediately to the
right of the silk dress.
(A) 1: wool; 2: gauze; 3: silk; 4: linen; 5:
polyester; 6: rayon
(B) 1: rayon; 2: wool; 3: gauze; 4: silk; 5:
linen; 6: polyester
(C) 1: polyester; 2: gauze; 3: wool; 4: silk;
5: linen; 6: rayon
(D) 1: linen; 2: silk; 3: wool; 4: gauze; 5:
polyester; 6: rayon
(E) 1: gauze; 2: rayon; 3: silk; 4: linen; 5:
wool; 6: polyester
It seems like they display the answer choices in the most complicated way
possible, but the elimination strategy will always get you to the finish line.
The first rule states that the gauze dress is on a lower-numbered hanger
than the polyester dress. This eliminates (C) because the polyester dress is
in front of the gauze dress.
The second rule tells us that the rayon dress must be on hanger 1 or hanger
6. This kills (E) because the rayon dress is on hanger 2.
The third rule limits hanger 3 to the wool dress or the silk dress. In (B), the
beautiful gauze dress is on hanger 3, and that is a no-no.
The final rule makes it pretty clear that the linen dress must be
immediately to the right of the silk dress. (D) shows no respect for that
rule and is thus eliminated.
Only (A) is left, and it looks pretty darn good at this point.
Question #2
2. If both the silk dress
and the gauze dress are
on odd- numbered
hangers, then which
one of the following
could be true?
Here’s a conditional
question - a great chance to
start using all of the rules.
This is a conditional question, and it comes early in the game. The new
condition is that both the silk dress and the gauze dress must be on oddnumbered hangers. The first step is to identify situations where this could be
true.
Challenge : Try to identify options for placing both the
silk dress and the gauze dress on odd-numbered
hangers (1, 3, or 5).
Many students get lost and don’t know how to start on this question. It
appears that there are multiple options for both dresses, so there seem to be a
lot of possibilities. This tempts students to start searching through answers,
and that’s when trouble begins.
Hanger 3 is very restricted, so let’s start there. The gauze dress can’t be on
hanger 3, so the only possible odd-numbered hangers for the gauze dress
are hanger 1 and hanger 5.
Since this is a could be true question, it’s
important to know all of the possible
outcomes. At this point, we want to use
the previous deduction to make two quick hypotheticals. You should make
one with the gauze dress on hanger 1 and another with it on hanger 5.
Challenge : Try to fill out both hypotheticals using the
other rules in this game.
Note: Remember, the silk dress must also
be on an odd-numbered hanger
for this question.
Here’s an outline of the deductions that follow if the gauze dress is on hanger
1:
The rayon dress cannot be on hanger 1, so it must be on hanger 6.
The silk dress cannot be on hanger 5 due to the SL block, and it must be on
an odd-numbered hanger, so the silk dress must be on hanger 3.
The linen dress must hang immediately to the right of the silk dress, so the
linen dress must be on hanger 4.
It’s not clear where the polyester and
wool dresses end up, so it is helpful to
make linked options on hangers 2 and
5.
And now let’s take a look at what happens if the gauze dress is on hanger 5:
The polyester dress must be on a higher-numbered hanger, so it must be on
hanger 6.
The rayon dress must be on hanger 1 since it can’t be on hanger 6.
The only odd-numbered hanger left for the silk dress is hanger 3, so the
silk dress must be on hanger 3.
The linen dress must hang immediately
to the right of the silk dress, so the
linen dress must be on hanger 4.
The wool dress takes the only empty hanger, which is hanger 2.
Note that the SL block is
always on hangers 3 and 4.
Ninja Note:
Now it’s time to find the answer.
This question requires a good
amount of work, but that will
happen. We need an answer that
could be true.
(A) The polyester dress is on
hanger 1.
(B) The wool dress is on
hanger 2.
(C) The polyester dress is on
hanger 4.
(D) The linen dress is on
hanger 5.
(E) The wool dress is on
hanger 6.
(A) The polyester dress could be on hanger 2 or hanger 5 in the first
situation and must be on hanger 6 in the second situation, so it
cannot be on hanger 1.
(B) The wool dress could be on hanger 2 in the first situation, and
the wool dress must be on hanger 2 in the second situation, so it
definitely could be true that the wool dress is on hanger 2. Ding,
ding, ding. (B) is the correct answer.
(C) The polyester dress could be on hanger 2 or hanger 5 in the first
situation, and it must be on hanger 6 in the second situation, so it
cannot be on hanger 4.
(D) The linen dress must be on hanger 4. Big loser.
(E) The wool dress could be on hanger 2 or hanger 5 in the first
situation, and it must be on hanger 2 in the second situation. No
way it’s on hanger 6.
Question #3
3. If the silk dress is on
an even-numbered
hanger, which one of
the following could be
on the hanger
immediately to its left?
This game requires a deep
understanding of odd and
even numbers. Study up.
In this conditional question, the silk dress must be on hanger 2, 4, or 6. The
restrictions are incredibly helpful on this one.
Our original deductions show that the silk dress can’t be on hanger 2 or
hanger 6, so hanger 4 is the only even-numbered hanger open for the
lovely silk dress.
Challenge : With the silk dress on hanger 4, try to find
more deductions.
Once you place the silk dress on hanger 4, there are a few quick deductions to
spot.
The linen dress must hang immediately to the right of the silk dress, so
the linen dress must be on hanger 5.
The wool dress must be on hanger
3 since the silk dress is busy on
hanger 4.
This question asks for a
dress that could be hanging
(A) the gauze dress
immediately to the left of the
(B) the linen dress
silk dress. We can do one
(C) the polyester dress
better and tell them the
(D) the rayon dress (E) the wool
dress that must be there: the
dress
wool dress. (E) is the answer.
Question #4
4. If the polyester dress is on hanger 2,
then which one of the following must be
true?
Yet another conditional question. On this one, we place the polyester dress on
hanger 2. By this point in the game, you should be more comfortable with the
rules.
Challenge : Try to place some of the other dresses on
hangers.
The gauze dress must be on a lower-numbered hanger than the polyester
dress, so the gauze dress must be on hanger 1.
The rayon cannot be on hanger 1, so it must be on hanger 6.
Ditz McGee:
So... then the block has to go on hangers 3 and
4?
Not exactly. It’s tempting to put the
block on hangers 3 and 4, but the block
could also land on hangers 4 and 5.
Since this is a must be true question, we should have enough already.
The rayon dress is on hanger 6.
That leads us very nicely to
(E). (A) must be false. (B), (C),
and (D) could be true, but they
also could be false.
(A) The silk dress is on hanger
1.
(B) The wool dress is on hanger
3.
(C) The linen dress is on hanger
4.
(D) The linen dress is on hanger
5.
(E) The rayon dress is on
hanger 6.
Question #5
5. Which one of the following CANNOT be true?
(A) The linen dress hangs immediately next to the gauze dress.
(B) The polyester dress hangs immediately to the right of the
rayon dress.
(C) The rayon dress hangs immediately to the left of the wool
dress.
(D) The silk dress is on a lower-numbered hanger than the gauze
dress.
(E) The wool dress is on a higher-numbered hanger than the
rayon dress.
It took a while, but we finally get an absolute question. For this one, you
can rely on your deductions and your work on prior questions.
Challenge : Try to locate the answer that must be false.
Use your hypotheticals from
previous questions to save precious
time. If an answer was true on a
previous question, then it could be true,
and it’s eliminated from contention.
Ninja Note:
(A) On question #2, we worked out a hypothetical with the linen dress
on hanger 4 and the gauze dress next to it on hanger 5.
(B) The rayon dress must be on hanger 1 or hanger 6. If it is on
hanger 6, there is nothing to its right. If it is on hanger 1, the
polyester dress cannot be on hanger 2 since the gauze dress
must be on a lower-numbered hanger than the polyester dress.
Thus, the polyester dress can’t be immediately to the right of
the rayon dress. (B) is the one.
(C) If the wool dress is on hanger 2, the rayon dress could be
immediately to its left, on hanger 1. This was true in a hypothetical
to question #2.
(D) The silk dress could be on hanger 1, so the silk dress could be on a
lower-numbered hanger than any other dress, including the gauze
dress. (Also, see question #2.)
(E) Since the rayon dress could be on hanger 1, any other dress
(including the wool ensemble) could be on a higher-numbered
hanger. (Again, check out #2.)
Question #6
6. Which one of the following CANNOT hang immediately next to
the rayon dress?
Here we run into another absolute question. On a question like this, it’s
tempting to just run headfirst into the answer choices, but running into things
headfirst can sometimes hurt. If you take a moment, it’s a good bet we can
spot a dress that can never hang next to the rayon dress.
The rayon dress must be on hanger 1 or hanger 6.
That’s a great place to start. Now, you want to visualize the two possibilities.
Below, you will see we have done just that. The rest is up to you.
Challenge : Try to spot the dress that can’t hang next to
the rayon dress.
(A)
(B)
(C)
(D)
(E)
the gauze dress
the linen dress
the polyester dress
the silk dress
the wool dress
The original restrictions
really pay off on this one.
BP Minotaur:
If the rayon dress is on hanger 1, the silk dress can never be next to it since
the silk dress cannot be on hanger 2.
If the rayon dress is on hanger 6, the silk dress cannot be on hanger 5
because the linen dress has to hang immediately to the right of the silk
dress.
The silk dress can’t be next to the rayon dress, regardless of whether
the rayon dress is on hanger 1 or hanger 6 (poor little silk dress).
There’s the big payoff. We quickly pick (D) for the silk dress and
move on.
Question #7
7. Assume that the
original condition that the
linen dress hangs
immediately to the right
of the silk dress is
replaced by the condition
that the wool dress hangs
immediately to the right
of the silk dress. If all the
other initial conditions
remain in effect, which
one of the following must
be false?
Ummm... apparently,
answering this question
requires reading a novel
first. This is way too long.
Oh crap. This is never fun. Just when we were getting nice and comfortable
with this game, they get crazy and switch things up on us.2 This rule takes
away our original block and replaces it with a new block. At this point, you
should definitely write out the new block and see what effect this new rule
has on our original deductions.
There is one huge thing to notice
here. Since either the wool or the
silk dress still must be on hanger
3, there are only two possible
placements for the new block. It
must go on either hangers 2 and
3 or hangers 3 and 4. When you
notice that, take a moment and jot down those two possibilities.
Somewhat surprisingly, there are no more helpful deductions to be found.
The rayon dress could still be on hanger 1 or hanger 6, and we have to
remember that the gauze dress must be on a lower-numbered hanger than the
polyester dress. But none of that leads to big breakthroughs. Since we’re at an
impasse, it’s time for the answers. We need something that must be false.
(A)
(B)
(C)
(D)
(E)
The linen dress is on hanger 1.
The gauze dress is on hanger 2.
The wool dress is on hanger 4.
The silk dress is on hanger 5.
The polyester dress is on hanger 6.
(A) Since our original block was suspended, the linen dress is now
totally random. As long as the rayon dress is on hanger 6, the linen
dress could be on hanger 1.
(B) In the second situation, the gauze dress could be on hanger 2.
(C) The wool dress is on hanger 4 in the second situation, so this one
could be true.
(D) Because the new block can only go in two places, the silk dress
must be on either hanger 2 or hanger 3. That means it cannot
be on hanger 5. (D) is the big winner.
(E) The polyester dress could be on hanger 6 in either situation, so (E)
is no good.
That was a good start to 1:1 ordering games. We worked
slowly through the setup and rules. Then, we took a few
moments to search for deductions. Once that was completed,
we jumped into the questions. In addition, note how we used
efficient methods to work through each question.
Now... repeat. We are going to continue with the Blueprint Building
BlocksTM technique on the next one, but you are going to do much more
work on your own.
2 This situation is akin to relationships. You know, when you have entered the
comfort zone, but then your partner decides they need to inject some new life into
your relationship? Spice things up? All of a sudden, you are traveling to Zimbabwe or
attending swingers’ parties. Not good.
DECEMBER 2004: GAME 1 (1-6)
We would like to introduce you to Patterson. She is a very busy woman, and
you get to play her personal assistant in this game. You must schedule
correctly or risk her wrath.
1. Setup
You should be feeling more comfortable with 1:1 ordering games, so this one
is mostly up to you.
Challenge : Read the introduction, symbolize the
variables, and build an effective setup.
On one afternoon, Patterson meets
individually with each of exactly five
clients—Reilly, Sanchez, Tang, Upton, and
Yansky—and also goes to the gym by
herself for a workout. Patterson’s workout
and her five meetings each start at either
1:00, 2:00, 3:00, 4:00, 5:00, or 6:00. The
following conditions must apply:
As we stated, this Patterson is a busy lady. There is a slight twist here. You
are given five clients, but, instead of a sixth client, Patterson must mix in a
workout at some point during the day.1 Since the workout occurs on the hour,
just like the other meetings, you can treat it the same as the rest of your
players.
The times, 1:00 through 6:00,
have an inherent order, so that
should be the base of your
setup. Also important to note:
Each of the clients meets
individually, so we are
guaranteed to have a 1:1
ordering game. Hopefully your setup looks something like this.
2. Check Out the Rules
Next up are the rules. You should expect a mix of the usual suspects (blocks,
chains, etc.).
Challenge: Symbolize the rules and, if possible,
combine them.
Patterson meets with Sanchez at
some time before her workout.
Patterson meets with Tang at
some time after her workout.
Patterson meets with Yansky
either immediately before or
immediately after her workout.
Patterson meets with Upton at
some time before she meets with
Reilly.
In this game, combining the rules is crucial. The first three rules can be
combined into a great ordering chain. The last rule has to stay separate, but
that’s not a huge problem.
Patterson meets with Sanchez before her workout, so we symbolize that
with a dash.
The meeting with Tang occurs after her workout, so we have another
basic ordering principle. This should be combined with the first rule to
show that Patterson does her workout after her meeting with Sanchez
but before her meeting with Tang.
The third rule is going to be the big kahuna in this game. Patterson
meets with Yansky either immediately before or immediately after her
workout. This is a reversible block, which shows that her meeting with
Yansky and her workout must occur consecutively.
Our wonderful new block should be
incorporated into the ordering chain,
since the Sanchez meeting must come
before both the meeting with Yansky
and her workout. The meeting with
Tang must follow the block.
The final rule gives us one last
ordering principle: Patterson meets with Upton before Reilly. Since
there are no variables in common with the ordering chain we built, this
rule has to stay separate.
3. Maximum Deductions
Now it’s time for deductions. Since we were able to combine most of the
rules, we should be able to uncover some powerful deductions. First up,
search for restrictions.
Challenge : Use the rules to make restrictions on the
setup.
All six players in the game (five clients plus the workout) are involved in at
least one rule, so there are no randoms. There are tons of restrictions to be
drawn from the ordering rules.
Patterson must meet with Sanchez before Yansky, Tang and her
workout, so she cannot meet with Sanchez at 4:00, 5:00, or 6:00.
Patterson’s workout is before her meeting with Tang and after her
meeting with Sanchez, so her workout cannot be at 1:00 or 6:00.
Patterson’s meeting with Yansky must be before her meeting with Tang
and after her meeting with Sanchez, so her meeting with Yansky cannot
be at 1:00 or 6:00.
Her meeting with Tang must come
after her meetings with Sanchez and
Yansky and after her workout, so
Patterson cannot meet with Tang at
1:00, 2:00, or 3:00.
Patterson meets with Upton before
Reilly, so she cannot meet with Upton at 6:00.
Patterson meets with Reilly after Upton, so she cannot meet with Reilly
at 1:00.
Tons of restrictions? Time to
start thinking about who actually could
be placed in those slots.
Ninja Note:
As you can see from the restrictions, the time slots you really want to check
out are 1:00 and 6:00. Patterson only has two options for the first meeting and
the last meeting. This will be a huge help.
At 1:00, Patterson must meet with
either Sanchez or Upton.
At 6:00, Patterson must meet with
either Tang or Reilly.
Those are some great deductions. We’re in good shape to hit the questions.
4. Conquer the Questions
Question #1
1. Which one of the following could be an
acceptable schedule of Patterson’s workout
and meetings, in order from 1:00 to 6:00?
Challenge : Find the answer to this elimination
question. You got this.
Patterson meets with Sanchez at some time
before her workout.
Patterson meets with Tang at some time
after her workout.
Patterson meets with Yansky either
immediately before or immediately after
her workout.
Patterson meets with Upton at some time
before she meets with Reilly.
(A) Yansky, workout, Upton, Reilly,
Sanchez, Tang
(B) Upton, Tang, Sanchez, Yansky,
workout, Reilly
(C) Upton, Reilly, Sanchez, workout,
Tang, Yansky
(D) Sanchez, Yansky, workout, Reilly,
Tang, Upton
(E) Sanchez, Upton, workout, Yansky,
Tang, Reilly
The first rule tells us that Patterson meets with Sanchez before her
workout. This knocks out (A).
The second rule tells us that Patterson meets with Tang after her workout.
This kills (B).
The third rule says that Patterson meets with Yansky either immediately
before or immediately after her workout. This knocks out (C). We are
working our way down.
The fourth rule tells us that Patterson meets with Upton before Reilly.
Continuing with the downward trend, this knocks out (D).
The only one left standing is answer choice (E).
1 Some students are tempted to represent the workout in a different fashion, with a
dumbbell or a thigh master, perhaps. However, we recommend sticking with a W.
Keep it simple.
Question #2
2. How many of the
clients are there, any
one of whom could
meet with Patterson at
1:00?
This one is all
deductions.
Challenge : Spot the answer to this absolute question.
(A) one
(B) two
(C) three
(D) four
(E) five
This question takes about 10 seconds with our deductions.
The only two options for Patterson at 1:00 are Sanchez and Upton, so
the correct answer is (B).
Question #3
3. Patterson CANNOT meet with Upton at
which one of the following times?
(A) 1:00
(B) 2:00
(C) 3:00
(D) 4:00
(E) 5:00
At first glance, this looks like it will be a breeze. We already made lots of
restrictions, so you just check out where Upton can’t be scheduled and find
that time in the answer choices.
Shoot. They need an
answer choice (F) for 6:00. So now
you just go and try all of the answers?
Cleetus Comment:
Not necessarily. While it sucks that they don’t give you the easy answer, you
can’t always expect an easy way out. There must be another time that Upton
can’t meet with Patterson. We just have to find it.
There will be times when
you miss a restriction or other
deduction. No need for stress - a little
more work will uncover it.
BP Minotaur:
At this point, you have to find another spot that doesn’t work for Upton. This
is, very likely, going to relate back to the block. Placing Upton near the start
or the end of the day doesn’t seem to be a problem, but there must be some
time slot in the middle that makes it impossible to find two consecutive spots
for the block.
Challenge : See if you can find a time that Patterson
can’t meet with Upton.
Hopefully you noticed a problem with putting Upton into the 3:00 time slot.
If Patterson meets with Upton at 3:00, then Patterson must meet with
Reilly after that.
Patterson must meet with Sanchez
before she meets with Yansky or does
her workout, so the block for the
workout and Yansky meeting cannot be
scheduled at 1:00 and 2:00.
If the Yansky meeting and the workout
are scheduled for later than 3:00, there
would be no time slots left for Tang
(who must follow the workout).
Patterson cannot meet with Upton at 3:00, so the answer is (C).
Question #4
4. If Patterson meets
with Sanchez the hour
before she meets with
Yansky, then each of
the following could be
true EXCEPT:
Here’s the first conditional
question.
For this question, you get a new condition: Patterson meets with Sanchez the
hour before she meets with Yansky. The first thing to notice is that this
changes the block.
If Patterson meets with Sanchez right before
Yansky, she must do her workout right after the
meeting with Yansky. Tang is somewhere after the new block.
At this point, you want to see where the new block could land. There are a
couple possibilities for this bad boy. The block could go in three different
places (1:00 to 3:00, 2:00 to 4:00, or 3:00 to 5:00). That deduction should
help us find the correct answer.
Challenge : Considering the new block, find the answer
that must be false.
(A) Patterson meets with Reilly at 2:00.
(B) Patterson meets with Yansky at 3:00.
(C) Patterson meets with Tang at 4:00.
(D) Patterson meets with Yansky at 5:00.
(E) Patterson meets with Tang at 6:00.
For this question, the key is to determine where the new block can land. As
long as you correctly visualize the three possibilities, the right answer follows
nicely.
(A) Patterson could meet with Reilly at 2:00 if she meets with Upton at
1:00.
(B) Sure, she could meet with Yansky at 3:00 and do her workout at
4:00. No problem.
(C) Patterson could meet with Tang at 4:00 if she meets with Sanchez
at 1:00, Yansky at 2:00, and does her workout at 3:00, so (C) also
could be true.
(D) Patterson must go to her workout and have her meeting with
Tang after her meeting with Yansky, so the latest she could
meet with Yansky is 4:00. (D) must be false and is our winner.
(E) Nothing has to follow her meeting with Tang, so Patterson could
meet with Tang at 6:00.
Question #5
5. If Patterson meets with Tang at 4:00,
then which one of the following must be
true?
This is a conditional question with lots of deductions. Give it a shot.
Challenge : With Tang at 4:00, try to fill in as many
slots as possible.
(A) Patterson meets with Reilly at 5:00.
(B) Patterson meets with Upton at 5:00.
(C) Patterson meets with Yansky at 2:00.
(D) Patterson meets with Yansky at 3:00.
(E) Patterson’s workout is at 2:00.
Patterson must meet with Sanchez at 1:00, and there are linked options for
her meeting with Yansky and her workout at 2:00 and 3:00.
Patterson must meet with Upton before Reilly, so Upton is at 5:00 and
Reilly is at 6:00.
At this point, you can just search the answers for one of our
deductions. Patterson must meet with Upton at 5:00, so (B) is our guy.
Question #6
6. Which one of the following could be the
order of Patterson’s meetings, from earliest
to latest?
This question only asks about the meetings, so the workout is irrelevant. This
should remind you of an elimination question; we just have to ignore one
player.
Challenge : Use the elimination strategy to identify the
correct answer.
Patterson meets with Sanchez at some time
before her workout.
Patterson meets with Tang at some time
after her workout.
Patterson meets with Yansky either
immediately before or immediately after
her workout.
Patterson meets with Upton at some time
before she meets with Reilly.
(A) Upton, Yansky, Sanchez, Reilly, Tang
(B) Upton, Reilly, Sanchez, Tang, Yansky
(C) Sanchez, Yansky, Reilly, Tang, Upton
(D) Sanchez, Upton, Tang, Yansky, Reilly
(E) Sanchez, Upton, Reilly, Yansky, Tang
While normally appearing as
the first question in each game,
elimination questions can reappear
later in a game. Whenever possible,
apply the elimination strategy to
identify the correct response.
BP Minotaur:
Patterson must meet with Yansky after Sanchez but before Tang. Say
good-bye to answer choices (A), (B), and (D). Excellent.
Patterson must meet with Upton before Reilly, and that kills answer choice
(C).
(E) is the only answer left, which is a strong indicator that it is correct.
Well, that went rather smoothly. Combining the rules and spotting some early
deductions was crucial to success.
Next up: You get to complete a 1:1 ordering game on
your own.
Try the next game from start to finish. Take your time and don’t rush. The
goal, at this early stage, is not to complete the game quickly. The goal is to
work through it accurately. If it takes you three days to complete this game,
but you do it correctly, that is great.2 We will cover a variety of timing
techniques later in the book.
2 For the record, that’s not true. Three days is too long. We are exaggerating for
effect. If it takes three days, there is a big problem (aside from the fact that you put
your life on hold for three days to complete a logic game). Twenty minutes, though, is
totally cool.
DECEMBER 2008: GAME 1 (1-6)
Questions 1-16
Individual hour-long auditions will be scheduled for each of six saxophonists
—Fujimura, Gabrieli, Herman, Jackson, King, and Lauder. The auditions will
all take place on the same day. Each audition will begin on the hour, with the
first beginning at 1 P.M. and the last at 6 P.M. The schedule of auditions
must conform to the following conditions:
Jackson auditions earlier than Herman does.
Gabrieli auditions earlier than King does.
Gabrieli auditions either immediately before or immediately after Lauder
does.
Exactly one audition separates the auditions of Jackson and Lauder.
1. Which one of the following is an acceptable schedule for the
auditions, listed in order from 1 P.M. through 6 P.M.?
(A) Lauder is scheduled to audition earlier than Herman.
(B) Lauder is scheduled to audition earlier than King.
(C) Jackson’s audition is scheduled to begin at either 1 P.M. or 5
P.M.
(D) Fujimura and Jackson are not scheduled to audition in
consecutive hours.
(E) Gabrieli and King are not scheduled to audition in
consecutive hours.
2. Which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
Fujimura, Gabrieli, King, Jackson, Herman, Lauder
Fujimura, King, Lauder, Gabrieli, Jackson, Herman
Fujimura, Lauder, Gabrieli, King, Jackson, Herman
Herman, Jackson, Gabrieli, Lauder, King, Fujimura
Jackson, Gabrieli, Lauder, Herman, King, Fujimura
3. The earliest King’s audition could be scheduled to begin is
(A)
(B)
(C)
(D)
(E)
5 P.M.
4 P.M.
3 P.M.
2 P.M.
1 P.M.
4. The order in which the saxophonists are scheduled to audition is
completely determined if which one of the following is true?
(A)
(B)
(C)
(D)
Herman’s audition is scheduled to begin at 4 P.M.
Jackson’s audition is scheduled to begin at 1 P.M.
Jackson’s audition is scheduled to begin at 5 P.M.
Lauder’s audition is scheduled to begin at 1 P.M.
(E) Lauder’s audition is scheduled to begin at 2 P.M.
5. If Fujimura’s audition is not scheduled to begin at 1 P.M., which
one of the following could be true?
(A)
(B)
(C)
(D)
(E)
Herman’s audition is scheduled to begin at 6 P.M.
Gabrieli’s audition is scheduled to begin at 5 P.M.
Herman’s audition is scheduled to begin at 3 P.M.
Jackson’s audition is scheduled to begin at 2 P.M.
Jackson’s audition is scheduled to begin at 5 P.M.
6. Which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
Gabrieli’s audition is scheduled to begin before 5 P.M.
Herman’s audition is scheduled to begin after 2 P.M.
Herman’s audition is scheduled to begin before 6 P.M.
King’s audition is scheduled to begin before 6 P.M.
Lauder’s audition is scheduled to begin before 5 P.M.
ACE THE AUDITION GAME
Who knew that scheduling some saxophone auditions could be so
complicated? There are lots of interesting rules in this one, and it is vital to
visualize them correctly. In the world of 1:1 ordering games, this one is
definitely on the tough side. Don’t worry if you stumbled a bit.
1. Setup
The setup on this one is pretty
similar to the previous games.
There are six saxophonists (the
players).1 Your job, even if you
grudgingly accept it, is to
determine the audition schedule
for the six players. The first
audition begins at 1 P.M., and the last one begins at 6 P.M. Since the time
slots have an inherent order, use those as the base.
Each audition features only one individual and begins on the hour, so this is a
1:1 ordering game.
2. Rules
Make sure you diagrammed the rules correctly. There are a few pretty large
blocks in this game that can be challenging. As always, you have to watch
out for players that are mentioned in more than one rule. Here’s an overview
of the rules:
Jackson auditions earlier than
Herman does.
Gabrieli auditions earlier than
King does.
Gabrieli auditions either
immediately before or
immediately after Lauder does.
Exactly one audition separates
the auditions of Jackson and
Lauder.
The second and third rules should be combined.
The fourth rule is very frustrating. This rule gives us another reversible
block. We love blocks, but this one is not very simple. Both Jackson and
Lauder are involved in previous rules. Unfortunately, there is no
effective way to combine this block with the other rules. The problem is
that both of the blocks are reversible, so there are still a number of ways
the players could be arranged.
3. Mighty Deductions
At this point, it’s crucial to investigate how the rules play together. Check to
make sure you were able to spot some, if not all, of these deductions.
First, Fujimura is random, since he is not involved in any of the rules.
You know what this means: Watch out for Fujimura.
Jackson auditions earlier than Herman, so Jackson cannot audition at 6.
Herman auditions later than Jackson,
so Herman cannot audition at 1.
Both Gabrieli and Lauder must
audition earlier than King, so Gabrieli
and Lauder cannot audition at 6.
KG King auditions later than both Gabrieli and L Lauder, so King
cannot audition at 1 or 2.
The auditions at 1 and 6 are
looking like the most restricted slots at
this point. Keep your eyes on the front
and the back.
Ninja Note:
If you visualized the rules correctly and used them to build some
restrictions, you are in good shape. The biggest rules are the two blocks.
You know those are going to play a large role, so you want to make sure to
watch the blocks.
1 As anticipated, there is some diversity in the mix of names. However, after
Fujimura, it seems like the authors lost some steam and settled on rather bland names
like Jackson and King.
4. Questions
The questions on this one test your ability to quickly incorporate new
information and find good placements for the reversible blocks. Even if you
got the answers correct, it’s important to review the questions, making sure
that you used the correct approach.
Question #1 (elimination, could be true)
As usual, they get the ball rolling with a classic elimination question.
The first rule says that Jackson auditions earlier than Herman, which
kicks out (D).
The second rule states that Gabrieli auditions earlier than King, which
kicks out (B).
The third rule gets rid of (A) because Gabrieli and Lauder must fill
consecutive time slots.
The final rule is that Jackson and Lauder must be separated by exactly
one audition, which kills (C).
Way down at the bottom, we find the only remaining answer choice:
(E).
Question #2 (absolute, must be true)
When a game presents you with an absolute, must be true question this early,
it is a challenge to see if you spotted all of the deductions.
(A) Lauder could audition earlier than Herman, but Herman could also
audition immediately after Jackson and earlier than Lauder, so (A)
does not have to be true.
(B) Combining those rules pays off again. Since Gabrieli and
Lauder must audition consecutively, and Gabrieli auditions
earlier than King, Lauder must audition earlier than King.
This was an early deduction, so (B) is the winner.
(C) Jackson could audition at 1 or 5... or 2 or 3 or 4. The Jackson and
Lauder block still has many possible placements, so this answer
does not have to be true.
(D) Fujimura is random, and
there’s no reason to believe
that Fujimura and Jackson
cannot audition consecutively. If this one gave you pause, just
throw down a quick hypothetical to prove that they could be next to
each other. Check out one to the right.
(E) King must audition after Gabrieli, and could audition immediately
after. You can see this in the hypothetical above.
Question #3 (absolute, could be true)
For this one, the deductions give you a
nice head start. King cannot audition at 1
or 2, but it appears that King could
audition at 3. You always want to check to make sure your chosen answer is
actually possible. Here’s one hypothetical with King scheduled at 3. Works
fine.
Now comes the nice part. Find 3 in the answer choices. There it is:
(C).
Question #4 (absolute, must be true)
This type of question is becoming much more common in the games section.
It asks you to identify an answer choice that determines the time slots for all
six auditions. Basically, one piece of information needs to set off a chain
reaction that completely fills out the setup.
Students typically jump into the answer choices and try each one until,
hopefully, they find the correct answer. That can be time consuming.
Remember, you don’t always have to attack the answer choices starting from
the top.
For questions like this, you
want to focus on the player(s) that will
be hardest to determine. In ordering
games, reversible blocks pose a huge
challenge. Starting there is a good idea
- look for answers involving Gabrieli,
Jackson, or Lauder.
Ninja Note:
Another important point about strategy : To save time, you have to know
when to eliminate answer choices. As soon as any player is up in the air (X
could go in 4 or 5), there’s no way that all the players are going to be
determined. Get rid of it.
(A) If Herman auditions at 4, then the Jackson and Lauder block would
have to go in slots 1 and 3. However, it is not clear whether
Jackson auditions at 1 or 3. As soon as you notice that, you know
(A) is not going to work.
(B) If Jackson auditions at 1, then Lauder must audition at 3. However,
Gabrieli could then audition at either 2 or 4. (B) is going to come
up way short.
(C) If Jackson auditions at 5,
then Herman must audition
at 6. There must be one slot
between Jackson and Lauder, so Lauder must go at 3. King
must audition after Lauder, so King must audition at 4.
Gabrieli must audition next to Lauder, so that takes up our
time slot at 2. And that only leaves Fujimura to audition at 1.
Well, look at that. We have a complete schedule for the
auditions, so (C) is correct.
(D) If Lauder auditions at 1, then Jackson must audition at 3. Also,
Gabrieli must be next to Lauder, so that means 2 for Gabrieli.
However, we are left with the powerful trio of Fujimura, Herman
and King. None of the three are determined, so this one falls short.
(E) If Lauder auditions at 2, then Jackson must audition at 4. The next
one we take a look at is Gabrieli. And that will be the end of this
answer. Gabrieli could still audition at either 1 or 3. Since we can’t
determine when Gabrieli auditions, we can eliminate this answer.
Question #5 (conditional, could be true)
This one is a doozy. The new condition is that Fujimura does not audition at
1. As you might have noticed, that little morsel of information isn’t very
helpful.
We deduced at the onset that Herman and King cannot audition at 1.
When you add our man Fujimura to that list, that only leaves Jackson,
Lauder, or Gabrieli to audition at 1.
I got that first part. But then it
still seemed like a couple of the
answers could be true.
Ditz McGee:
That happens to a lot of students on this question. Remember, anticipation is
key. It is advisable to go one step further and quickly jot down hypotheticals
for the three possibilities at 1.
If Gabrieli auditions at 1, then Lauder
must audition at 2, and Jackson
auditions at 4.
If Jackson auditions at 1, then Lauder
must audition at 3, and Gabrieli could
audition at 2 or 4.
If Lauder auditions at 1, then Gabrieli
must audition at 2, and Jackson
auditions at 3.
The big thing to notice is
that Gabrieli, Jackson, and Lauder all
audition at 4 or earlier.
BP Minotaur:
Now you can go find an answer that could be true.
(A) Herman could audition at 6 in any of the three possibilities
that we outlined. All we know is that Herman must audition
after Jackson. This one could be true, and we found an early
winner. But let’s discuss the rest.
(B) The latest that Gabrieli could audition is 4, so 5 is not going to
work.
(C) Herman must audition after Jackson, so Herman cannot audition at
3 in the first and third situations above. In the second one, Lauder is
already filling up the time slot at 3. So Herman cannot audition at
3.
(D) This one is tempting. Jackson can audition at 1, 3, or 4. And all of
those are close to 2. However, they are not 2, so this answer also
must be false.
(E) Jackson cannot audition any later than 4, so this is yet another
loser.
Question #6 (absolute, must be true)
They don’t let you go quietly into the night on this game. This last question
actually requires a new deduction that few students spot (or need) before this
point. Occasionally, you will be forced to make additional deductions as you
work through the questions.
(A) Gabrieli cannot audition at 6. If
Gabrieli auditions at 5 (to the
right), all of the other players fall
into place. But the combination works fine, so this could be true.
(B) Herman cannot audition at 1, but 2 is
still an option. Jackson must audition
before Herman, so Jackson would have
to audition at 1. Lauder must audition at
3 and Gabrieli auditions at 4. Fujimura and King still have plenty of
room, so this could be true.
(C) Always use prior work. The correct answer to the last question
(#5) is that Herman could audition at 6. We’ve already proven that
(C) doesn’t have to be true.
(D) Check out the hypothetical we made for (A). King is pretty flexible
in this game and could easily audition at 6.
(E) Lauder can’t audition at 6. However, in our deduction phase,
we didn’t note any other restrictions on Lauder. At first glance,
Lauder auditioning at 5 doesn’t look like a problem. But check
it out:
If Lauder auditions at 5, then King
must audition at 6.
Gabrieli must audition at 4.
Jackson must be two slots in front
of Lauder, so Jackson is at 3.
Houston, we have a problem.
Herman is supposed to audition
after Jackson, but there’s no
room.
Lauder can’t audition at 5 or 6. Thus, Lauder must audition
before 5. (E) must be true.
And with that, the music stops. Our saxophone game
is finished. You have now completed a number of
games with setups that look eerily similar. But you
can see that each one presents different challenges in
the form of the rules and deductions.
In the saxophone game, the blocks were key. It was
hard to combine the rules or even visualize how they
work together. But with a little work up front, we
were able to work through the questions efficiently.
More importantly, that’s going to wrap up this chapter. We will show you the
stats later, but you should be aware that 1:1 ordering games are among the
most common types of games on the LSAT. They also lay the foundation for
the different types of ordering games to be covered in the following chapters.
Make sure to review the ordering rules before moving on. Here are some
important points for review:
1:1 ordering games have two variable sets. There is an equal number of
players and slots. Each player is assigned to one and only one slot.
Always use the variable set with an inherent order as the base of your
setup.
The rules will present you with a variety of dashes, blocks, divisions,
options, arches, and conditional rules. It’s crucial to diagram the rules in
an accurate fashion.
Deductions are king. In 1:1 ordering games, always try to combine rules
and form restrictions before attempting the questions.
There are now two types of ordering games behind us: Basic ordering and 1:1
ordering games. That’s a big accomplishment, but there are some twists and
turns in our future. When you are ready, check out the next chapter.2 Meet
you there...
2 At this point, and at the conclusion of later chapters, you may still feel a bit
uncomfortable with this type of game. It’s natural to feel like you could benefit from
more practice. That is a very common reaction. We don’t expect you to ace a game
type after one or two practice runs. Feel free to do some additional games before
moving to the next chapter. We’ve got you covered. In the back of this book, there is
an appendix with information about every game in the recent history of the LSAT.
This will allow you to find similar game types and get a few more under your belt
before forging ahead. We even went ahead and highlighted the best games to practice
in each category.
13/under/OVERbooked
WHAT IS IT?
Up to this point in our adventure, we have dealt exclusively with ordering
games that have a property called 1:1 correspondence. Eight clowns and eight
stages. Seven donkeys and seven lanes. Six contestants and six chairs.
Everything fits nicely. However, the LSAT is not always so nice. Many
ordering games do not have 1:1 correspondence between players and
slots.
These games commonly show up in two forms. Here is an example of the
first:
The Rodriguez parents are out of town
from Monday night through Friday night.
Each night, one of the three older siblings
—Carla, Debra, or Ephram—must stay
home to watch their newborn sister.
In this game, there are three players (the siblings) that must fill the five slots
(Monday through Friday). This game has fewer players than slots.
An underbooked ordering game has fewer players than
slots.
And here is an example of the second:
Nine students—A, B, C, D, E, F, G, H, and
I—each must give an oral report in history
class on a historical figure of their
choosing over a span of six weeks, week 1
through week 6.
This one goes the other way. Now, there are actually nine players (students)
and only six slots (weeks). This game has more players than slots.
An overbooked ordering game has more players than
slots.
Underbooked and overbooked ordering games add a level of complexity
since not everything fits very nicely. However, we will show you the light.
As we move into more complex games, there will be more factors to juggle.
The previous games have been solely concerned with the question of who
goes where, but these new games will introduce questions related to how
many times players can appear or how many players can be assigned to each
slot.
UNDERBOOKED ORDERING
As always, your first task is to identify what type of game you are staring at.
As of this moment, we are adding a second step to this process.
Step 1: Does the game involve Ordering? Grouping?
Both?
Step 2: What is the ratio of players to slots? 1:1?
Underbooked? Overbooked?
Here’s how it works: You will read a very familiar-sounding ordering game,
but then you will notice a twist. There are too many players. There aren’t
enough players. It is important to notice this hurdle, because then it is time to
react.1
Always read the introduction
slowly. Missing a feature of a game
will spell disaster down the road.
BP Minotaur:
When you recognize that you are grappling with an underbooked ordering
game, there are normally two ways the game can be structured. Here is an
example of the first:
Each of five people—Alfonso, Babar,
Calvin, Demar, and Ernie—graduated from
high school in a different one of seven
years: 1992 through 1998.
This game is underbooked because you have to order five students in seven
years. Each person will only fill one slot, so this game will end with some
empty slots. For instance, it might be the case that no one graduated in 1996.
More frequently, however, you will be given a game that is more similar to
the one on the previous page. Remember the Rodriguez family?
The Rodriguez parents are out of town
from Monday night through Friday night.
Each night, one of the three older siblings
—Carla, Debra, or Ephram—must stay
home to watch their newborn sister.
In this game, you can’t have any empty slots. The newborn would break into
the liquor cabinet, order some Cinemax, and never be the same. Thus, some
of the siblings (Ephram, perhaps2) must watch the newborn on more than one
night.
In underbooked ordering games, either (1) some slots
will remain unoccupied, or (2) some players will fill
multiple slots.
1 This surprise and change in direction is akin to real life situations: (1) You are
speeding along the freeway when you see blinking lights and the cone zone up ahead;
or (2) You are having a nice night with your lady when she tells you it is time to talk
about the status of your relationship. No time for indecision; you must react.
2 A person named Ephram Rodriguez would be, undoubtedly, the most interesting
man in the world.
NEW RULES
Most of the rules in underbooked ordering games will be identical to those
we have already covered. You will continue to be confronted with basic
ordering principles, blocks, divisions, arches, and Megatrons (that last one
isn’t real, but it would be awesome if it were).
There are, however, a few additions. There can now either be (1) empty spots
or (2) players that go more than once, and this allows them to offer some
additional types of rules.
Let’s take a look at the new rules.
If you are working through a game
in which some slots are going to be
unoccupied, the new rules will
relate to which slots can or cannot
remain empty. Let’s use our old
high school graduation game as an
example.
No one graduated in 1995.
One of the students
graduated in 1993.
These will be incredibly important
rules, and they should be
represented directly into your setup.
Here is the correct representation:
If you have a game in which players can fill more than one slot, the rules can
be more varied (and complicated). You will generally find restrictions on the
number of slots one or more players can fill.
Check out some rules:
Each of the older siblings must watch the
newborn at least once.
Ephram must watch the newborn on more
nights than
Sometimes, you have to do a
little bit of math to figure out how
many times each player can appear.
Ninja Note:
It’s go time, team. In the next game, our task is to do some travel planning.
We will work through the first underbooked ordering game using the
Blueprint Building BlocksTM technique. Then, you’re on your own.
SEPTEMBER 2006: GAME 1 (1-5)
Off we go. We will work through this one as a team. Here you will see the
first game from the September 2006 LSAT. The greater purpose here is to
ascertain the meaning of life. Not really. Actually, the purpose is to get some
experience working through an ordering game that does not have 1:1
correspondence.
1. Setup
At each of six consecutive
stops—1, 2, 3, 4, 5, and 6
—that a traveler must make
in that order as part of a
trip, she can choose one
from among exactly four
airlines—L, M, N, and O
—on which to continue.
Her choices must conform
to the following
constraints:
Yikes. Talk about a terrible
travel day. Can anyone say
airport bar?
If you ever have to make six stops and choose from four airlines on a trip,
you need to fire your travel agent. Where is she headed? Antarctica? Once
you get past the natural sense of astonishment and outrage, it’s time to get to
work.
Take note of the normal clues that tell us we have an ordering game. The
game states that there are six consecutive stops (numbered 1 through 6) at
which she has to choose an airline, which guarantees that we have to figure
an order for her stops. Since the stops have an inherent order, they will form
the base of our setup. However,
there are only four airlines from
which to choose.
This is an underbooked ordering game. The traveler is going to have to
choose some of the airlines more than once. For example, she might choose
M at both stop 2 and stop 5. This is important to notice, and now we are
ready to build our setup.
But she has to choose
each airline at least once, so it’s not
that bad.
Cleetus Comment:
Actually, that’s not the case. At least not yet. Don’t ever make assumptions
about a game. Many ordering games state that each player must fill exactly
one spot or at least one spot. But check out the introduction to this game
again. It only says she has four airlines to choose from, not that she has to
choose each airline at least once.
2. Rules
Now that we have our setup, it’s time to charge into the rules. Most of these
will sound familiar; however, there will be a few surprises since this game is
underbooked.
When you are doing a game
with repeating variables, it is very
common to see a rule similar to this
one.
Ninja Note:
As the resident ninja so eloquently stated, this is a very common type of rule.
The LSAT really doesn’t like putting the same letter in consecutive slots.
We represent this as a division (“A” stands for airline). For example, if we
choose airline O at stop 3, we know that the traveler cannot choose O at stop
2 or stop 4. Next rule.
These two rules are similar to each other, but new for us. First, she must
choose the same airline at stops 1 and 6. Second, she must choose the same
airline at stops 2 and 4.
Stop! Read back over the first
three rules. Looks like they might work
together, so it’s time to look for some
deductions.
Ninja Note:
On the next page, we are going to push you to search for some gold from
these rules.
Challenge :
Based on the
first three rules,
identify
combinations of
stops at which
our traveler can’t choose the same airline.
Deductions often pop up
during the rules, not just at the end. It’s
a fluid process.
BP Minotaur:
The traveler must choose different airlines at stops 1 and 2. This means
that she must also choose different airlines at stops 4 and 6.
The stop to watch is big number 5. Since the traveler cannot choose the
same airline at consecutive stops, the airline at stop 5 must be different
than those at stops 4 and 6. Therefore, stops 4, 5, and 6 have to be three
different airlines.
Back to the rules...
The wording of this one can be a little tricky. It is best to visualize the
ordering combinations forbidden by this rule. The traveler cannot choose L
immediately before she chooses N; she also cannot choose M immediately
before N.
Can’t I make one of those
arrow things? If L or M, then no N.
Ditz McGee:
That’s not wrong, and that is how a lot of students are tempted to diagram
this rule. However, a conditional diagram doesn’t represent the spatial
dimension to this rule, and that’s crucial. At this point, there’s another key
deduction. Let’s see if you can spot it.
Challenge : There are now a
few rules about airline N. See
if you can turn them into a
deduction.
Did you spot it? Let’s discuss the options for the airline that could be chosen
before N.
N cannot be chosen at consecutive stops. Also, according to the last
rule, L and M cannot be chosen immediately before N. The only option
left is O. If N is chosen at any stop with a stop before it, O must be the
airline chosen at that stop.
Make sure to use a conditional arrow for a deduction like this. Otherwise,
you might think that anytime the traveler chooses O, she must choose N at
the next stop. That isn’t true. This deduction is purely based on the traveler
choosing N at one of the stops. Three cheers for us. One more rule:
This last rule gives us a nice option for slot 5 - it must be either N or O. This
is a very important rule because it finally starts to fill in a slot. A game will
commonly provide you with the most important rule last. They are hoping
that you are blacked out from anxiety at this point and won’t notice. Don’t let
that happen.
3. Do or Die Deductions
This game is pretty difficult if you don’t make some helpful deductions.
Luckily, we have already started the process. We have a great deduction
about airline N, and a ton of stops that can’t use the same airline. But there
are more...
After the first three rules, we noticed that stop 5 was going to be important
since there are so many other stops that cannot use the same airline. Then, the
last rule gives you an option for stop 5 - it must be either N or O. This is a
huge opportunity.
BP Minotaur:
The next step in this game is a watered down
version of an advanced technique
called making scenarios. We will
describe the strategy in much greater
detail in later chapters.
In this game, it’s helpful to jump the gun and investigate what follows from
choosing each of the two options at stop 5. We are going to push you to make
deductions in both situations.
Challenge : Find deductions that follow if N is chosen
at stop 5.
Check to see if you were able to uncover the following deductions.
Our earlier deduction pays off here. If N is chosen at stop 5, then O must
be chosen at 4.
Stop 2 must be the same airline as stop
4, so O must also be chosen at stop 2.
Stop 1 cannot be O. Stop 6 cannot be N.
Since stops 1 and 6 must be the same airline, the airline chosen at those
stops must be either L or M.
That was great, but you’re not done. Now you need to test the other
possibility. What happens if the traveler goes in the other direction and
chooses O at stop 5?
Challenge : If O is chosen at stop 5, apply the other
rules. (Hint: Look for restrictions.)
Same drill here - check your deductions against the following:
Since O is chosen at stop 5, O cannot be chosen at stops 1, 2, 4, or 6.
If the traveler chooses N at a certain stop, she must choose O at the
previous stop. Since O can’t be chosen at stops 1 and 2, N can’t be the
airline chosen for stops 2 and 3.
Stops 2 and 4 must be the same, so stop
4 also can’t be N. Stops 2 and 4 must be
either L or M.
You missed a big one.
Since O is at 5, N must be at 6.
Cleetus Comment:
No, no, and one more big NO. That is the fallacy of the converse. If the
traveler chooses N at a stop, then she must choose O at the previous stop.
But if she chooses O at a stop, this does not mean she must choose N at the
next stop. She could choose N at stop 6, but she doesn’t have to.
As you can see, this process greatly simplifies the game. We will practice
making scenarios repeatedly later in our journey together. Here’s the
awesome part about what we’ve done: The two situations above are not just
two of the ways the game can work; they are the only two ways the game
can work. On each question, we will simply refer back to these deductions
and work from there. Let’s jump in.
4. Attack the Questions
Question #1
1. Which one of the following could be an
accurate list of the airlines the traveler
chooses at each stop, in order from 1
through 6?
This one is a good old elimination question. Same old strategy here: Just use
the rules to destroy the four terribly wrong answer choices.
Challenge : Identify the correct answer.
Whichever airline she chooses at a stop,
she chooses one of the other airlines at the
next stop.
She chooses the same airline at stop 1 as
she does at stop 6.
She chooses the same airline at stop 2 as
she does at stop 4.
Whenever she chooses either L or M at a
stop, she does not choose N at the next
stop.
At stop 5, she chooses N or O.
(A) L, M, M, L, O, L
(B) M, L, O, M, O, M
(C) M, N, O, N, O, M
(D) M, O, N, O, N, M
(E) O, M, L, M, O, N
The first rule kicks out (A) since M is chosen at both stop 2 and stop 3.
Those would be consecutive, and that is not allowed.
In (E), the traveler chooses O at stop 1 and N at stop 6. Those are different
airlines, so this violates the second rule.
Stops 2 and 4 must be the same. This rule is broken in (B), so there are
only two left.
The fourth rule tells us that the traveler cannot choose N after either L or
M. In (C), the traveler chooses M at stop 1 and then chooses N at stop 2.
Say no to (C).
The last answer standing is (D).
Question #2
2. If the traveler chooses N at stop 5, which
one of the following could be an accurate list
of the airlines she chooses at stops 1, 2, and
3, respectively?
N at stop 5? We
saw that
coming.
On this one, we are given the condition that the
traveler chooses N at stop 5. Luckily, we already did the work. It’s as though
we were able to predict the future. Here’s a recap of what we know if N goes
in 5 (this is just for review - you shouldn’t actually do the work again) :
O must be chosen at any stop before N is chosen. If N is chosen at stop
5, then O must be chosen at stop 4.
Stop 2 must be the same airline as stop 4, so O must be chosen at stop 2.
Stop 1 cannot be O. Stop 6 cannot be N.
Since stops 1 and 6 must be the same
airline, the airline for those stops must
be either L or M.
(A) L, M, N
(B) L, O, N
(C) M, L, N
(D) M, L, O
(E) N, O, N
With deductions in hand, you are
ready to search the answer choices.
This question could easily burn a
couple minutes, but note that we did
no additional work. Can you spot the
right answer?
Airline O must be chosen at stop 2, which eliminates (A), (C), and
(D). Either L or M must be chosen at stop 1. This kicks out (E).
Hopefully, you made your way to (B).
Question #3
3. If the only airlines the traveler chooses for
the trip are M, N, and O, and she chooses O
at stop 5, then the airlines she chooses at
stops 1, 2, and 3, must be, respectively,
Now it’s O at
stop 5?
We are the Miss
Cleo of the
LSAT.
The first new condition is that M, N, and O are the
only airlines that the traveler chooses. The second
condition is that she chooses O at stop 5. It’s much
more helpful to start with the second condition since we can rely on our
deductions. It’s up to you to make more.
Challenge : Try to find more deductions using the new
information provided by question #3. (Remember that
L cannot be chosen.)
The traveler never chooses airline L, so both stop 2 and stop 4 must be
airline M.
She cannot choose the same airline at consecutive stops, which means
that M cannot be selected at stop 1 or stop 3.
Since O and M cannot be selected at
stop 1, the traveler must choose N.
Stops 1 and 6 must be the same, so the
traveler must choose N at stop 6.
Since N and M cannot be chosen at
stop 3, the traveler must choose O.
The deductions gave you a
good head start, and now the
correct answer pops off the
page. The first three stops
are N, M, and O,
respectively. (C) is looking
like a great choice at this
point. Cha-ching.
(A) M, O, and N
(B) M, N, and O
(C) N, M, and O
(D) N, O, and M
(E) O, M, and N
Question #4
4. Which one of the following CANNOT
be an accurate list of the airlines the
traveler chooses at stops 1 and 2,
respectively?
It’s easy to burn time and energy on a question like this. This is an absolute
question, and you might think that the best approach would be to just jump
into the answers and try each one. Bad idea. Rather, use your deductions to
identify the possibilities that could work for stops 1 and 2. Then, look for a
combination that you know doesn’t work.
Challenge : Check out the two situations and find an
answer that doesn’t work for stops 1 and 2.
(A) L, M
(B) L, O
(C) M, L
(D) M, O
(E) O, N
Regardless of which airline the traveler chooses at stop 5, she can’t
choose O at stop 1 and she can’t choose N at stop 2. That means (E)
definitely doesn’t work.
Question #5
5. If the traveler chooses O at stop 2,
which one of the following could be an
accurate list of the airlines she chooses at
stops 5 and 6, respectively,
This game closes out with one more conditional question. Yet again, the
deductions save the day. The new condition here is that the traveler chooses
O at stop 2. The first thing to check is whether she chooses N or O at stop 5.
If you can figure that out, then the rest should be easy.
Challenge : First, isolate a situation where she can
choose O at stop 2. Then, look for a combo that could
work for stops 5 and 6.
(A) M, N
(B) N, L
(C) N, O
(D) O, L
(E) O, N
The traveler can only choose O at stop 2 if she chooses N at stop 5.
If she chooses N at stop 5, she must choose either L or M at stop 6.
The only answer choice that offers N at stop 5 and either L or M at
stop 6 is (B). Fill in the bubble and put your pencil down.
That was fun - our first underbooked ordering game. There are two big
lessons to be learned from this one. First, underbooked ordering games are
complicated by players that can fill more than one slot. It’s crucial to track
the variables that can repeat. Second, always watch for deductions as you
work through the rules, not just once you are done with the rules.
After a quick sob or snack (whichever you feel is most appropriate), attempt
the next underbooked ordering game on your own.
OCTOBER 2004: GAME 1 (1-6)
Questions 1-6
In the course of one month Garibaldi has exactly seven different meetings.
Each of her meetings is with exactly one of five foreign dignitaries: Fuentes,
Matsuba, Rhee, Soleimani, or Tbahi. The following constraints govern
Garibaldi’s meetings:
She has exactly three meetings with Fuentes, and exactly one with each of
the other dignitaries.
She does not have any meetings in a row with Fuentes. Her meeting with
Soleimani is the very next one after her meeting with Tbahi.
Neither the first nor last of her meetings is with
Matsuba.
1. Which one of the following could be the sequence of the meetings
Garibaldi has with the dignitaries?
(A)
(B)
(C)
(D)
(E)
Fuentes, Rhee, Tbahi, Soleimani, Fuentes, Matsuba, Rhee
Fuentes, Tbahi, Soleimani, Matsuba, Fuentes, Fuentes, Rhee
Fuentes, Rhee, Fuentes, Matsuba, Fuentes, Tbahi, Soleimani
Fuentes, Tbahi, Matsuba, Fuentes, Soleimani, Rhee, Fuentes
Fuentes, Tbahi, Soleimani, Fuentes, Rhee, Fuentes, Matsuba
2. If Garibaldi’s last meeting is with Rhee, then which one of the
following could be true?
(A)
(B)
(C)
(D)
(E)
Garibaldi’s second meeting is with Soleimani.
Garibaldi’s third meeting is with Matsuba.
Garibaldi’s fourth meeting is with Soleimani.
Garibaldi’s fifth meeting is with Matsuba.
Garibaldi’s sixth meeting is with Soleimani.
3. If Garibaldi’s second meeting is with Fuentes, then which one of
the following is a complete and accurate list of the dignitaries with
any one of whom Garibaldi’s fourth meeting could be?
(A)
(B)
(C)
(D)
(E)
Fuentes, Soleimani, Rhee
Matsuba, Rhee, Tbahi
Matsuba, Soleimani
Rhee, Tbahi
Fuentes, Soleimani
4. If Garibaldi’s meeting with Rhee is the very next one after
Garibaldi’s meeting with Soleimani, then which one of the
following must be true?
(A)
(B)
(C)
(D)
(E)
Garibaldi’s third meeting is with Fuentes.
Garibaldi’s fourth meeting is with Rhee.
Garibaldi’s fifth meeting is with Fuentes.
Garibaldi’s sixth meeting is with Rhee.
Garibaldi’s seventh meeting is with Fuentes.
5. If Garibaldi’s first meeting is with Tbahi, then Garibaldi’s meeting
with Rhee could be the
(A)
(B)
(C)
(D)
(E)
second meeting
third meeting
fifth meeting
sixth meeting
seventh meeting
6. If Garibaldi’s meeting with Matsuba is the very next meeting after
Garibaldi’s meeting with Rhee, then with which one of the
following dignitaries must Garibaldi’s fourth meeting be?
(A)
(B)
(C)
(D)
(E)
Fuentes
Matsuba
Rhee
Soleimani
Tbahi
GARIBALDI GETS BEAT DOWN
Hopefully your journey was a success. This is another underbooked ordering
game with some new twists. So there once was an important lady named
Garibaldi, and she had to set up a bunch of meetings...
1. Setup
The actual challenge is hidden initially, but it surfaces eventually. You have
Garibaldi, a regal name for a regal character. In appropriate fashion, she
needs to set up a slew of meetings with dignitaries. You are told that there are
seven meetings, yet you are only provided with five dignitaries.1 Fewer
characters than slots? We’re underbooked. The order of the meetings should
definitely be used as the base.
There is a bunch of stuff up
in the air at this point, but
your setup should look like
this. You should note that
the game hasn’t specified if
Garibaldi must meet with
each of the dignitaries. (She
might deem Matsuba unworthy of her time.) However, that changes when
you hit the rules.
2. Rules
The first two rules in this game really help to clarify the situation and the
distribution of players to the seven slots.
These two rules are huge. The first rule defines the distribution for you - there
are three meetings with Fuentes and one with each of the other dignitaries.
This is helpful because now you know exactly the set of seven meetings that
must be arranged for the powerful Garibaldi.
From the first rule, you know that Fuentes is going to be an important player.
Then, the second rule makes this point even clearer. It’s similar to a rule
you’ve seen before. In the last game, the traveler could not choose the same
airline at consecutive stops. In this one, Garibaldi cannot have consecutive
meetings with Fuentes.
You can’t deduce the exact
place for any of the Fuentes meetings,
but they must be spread out among the
seven slots.
Ninja Note:
Have we mentioned that blocks can
be very useful in ordering games?
Oh, we did. Great. Well, the block
presented in the third rule is going
to be very useful.
The final rule gives you two restrictions for Matsuba - no chance on the first
or last meeting. You know what time it is...
3. Deductions
There are some important deductions to spot, which will help you cruise
through this one.
Rhee is not involved in any of the rules, so Rhee is a random.
The next thing you want to do is make some restrictions.
Soleimani must have a meeting immediately after Tbahi, so Tbahi
cannot be the last meeting, and Soleimani cannot be the first meeting.
This helps since we also know Matsuba cannot be first or last.
Once you make those quick deductions, most students just hit the questions.
However, there is more to discuss. While this game does not allow any
deductions about when Garibaldi must meet with certain dignitaries, it is
really important to visualize how the three meetings with Fuentes are going
to work.
Garibaldi has three meetings with Fuentes,
but they cannot be consecutive. In other
words, some other meeting or meetings
must occur in between the meetings with
Fuentes. If you check out your options, there are only three buffers to
separate the meetings with Fuentes: (1) the meeting with Matsuba, (2)
the meeting with Rhee, and (3) the block with Tbahi and Soleimani.
Most of the questions revolve around these deductions. The key is to quickly
place the Fuentes meetings and then figure out which buffers are used to keep
them apart.
1 If you could easily pronounce three or more of the names of the dignitaries, you
deserve a medal. If you are still stumped at how to choke out the name Tbahi, join the
club.
4. Questions
It’s always great to arrive at the correct answers. But even if you did, make
sure you utilized the most effective path to get there.
Question #1 (elimination, could be true)
Nothing shocking on the first question - just your standard elimination
question.
The first rules sets the distribution of meetings. Fuentes gets three
meetings, and everyone else gets one each. (A) only has two meetings
with Fuentes, so it’s gone.
The next rule tells us that the Fuentes meetings cannot be consecutive.
In (B), Garibaldi has her fifth and sixth meetings with Fuentes, so (B) is
gone.
The third rule tells us that Tbahi must be immediately followed by
Soleimani. In (D), Tbahi is followed by Matsuba, so (D) is gone.
The fourth and final rule tells us that Matsuba cannot be the first or last
meeting. In (E), Matsuba is scheduled as the last meeting, so (E) is the
final goner.
The ultimate survivor here is (C), our correct answer.
Question #2 (conditional, could be true)
Next up is a conditional question that interfaces nicely with our deductions.
The new tidbit of information is that Garibaldi has her last meeting with
Rhee.
If Rhee is the last meeting, then Rhee cannot be used to separate the
Fuentes meetings.
That only leaves Matsuba and the Tbahi and Soleimani block to separate
the Fuentes meetings.
Either Matsuba squeezes between the first and second Fuentes meetings
and the block goes between the second and third, or vice versa.
(A) Soleimani must be either the
fifth meeting or the third
meeting.
(B) Matsuba must be either the
second meeting or the fifth
meeting.
(C) Soleimani must be either the fifth meeting or the third meeting.
(D) This is the one we’ve been looking for. In the second situation,
Matsuba is the fifth meeting. (D) is the winner in a landslide.
(E) Garibaldi’s sixth meeting must be with Fuentes, so it cannot be
with Soleimani.
Question #3 (conditional, could be true)
This is a new conditional question, but it relies on our same old deductions.
Keep track of which meetings squeeze in between the meetings with Fuentes
and you will find the light. In this one, Garibaldi’s second meeting must be
with Fuentes.
If the second meeting is with Fuentes, then Tbahi cannot be the first
meeting since that would break up our block. Matsuba also cannot have
the first meeting, so Rhee is the only option remaining for the first
meeting.
The only two things that can now
separate the Fuentes meetings: the
meeting with Matsuba, and the Tbahi
and Soleimani block. So just like on
the last question, either Matsuba
squeezes between the first and
second Fuentes meetings and the block fits nicely between the second
and third, or vice versa.
On this question, we need to give them a complete list of the people
that could have the fourth meeting. Luckily, it’s a pretty short list.
It could be Fuentes, or it could be Soleimani. And that’s all. So we
go with (E).
Question #4 (conditional, must be true)
Yet another conditional question. This one tells you that the meeting with
Rhee must immediately follow the meeting with Soleimani.
This new condition changes our
block by adding Rhee onto the back
of the block.
Once again, our big deduction
revolves around spacing out the
Fuentes meetings. Now there are
only two things to use as buffers: (1)
our new, hefty block, and (2) the
meeting with Matsuba. Therefore,
the new block must start in the second slot (with Matsuba in the sixth)
or in the fourth slot (with Matsuba in the second).
From these two situations, there are only two things that must be
true (i.e. happens in both hypotheticals). Garibaldi’s first meeting is
with Fuentes, and her last meeting is with Fuentes. Search for one of
these in the answers and it leads you down to (E), which is going to
be the credited response.
Question #5 (conditional, could be true)
By this point in the game, you should have spotted the repetitive nature of the
deductions. Once again, we just have to play with the buffers separating the
Fuentes meetings. This time, the first meeting is with Tbahi.
Since Soleimani must immediately follow Tbahi, if Tbahi is first, then
Soleimani must be the second meeting.
The three meetings with Fuentes
cannot be consecutive, so they must
be third, fifth, and seventh.
There should be linked options for Matsuba and Rhee for the fourth and
sixth meetings.
On this question, it’s all about Rhee. We need an answer that gives a possible
meeting for Rhee.
Rhee must have either the fourth or sixth meeting. Answer choice
(D) is going to be the winner since Rhee could be the sixth meeting.
The fourth meeting is suspiciously missing from the answers
because it would also be correct.
Question #6 (conditional, must be true)
One last conditional question - same old trick about Fuentes and the buffers.
On this one, the meeting with Matsuba is the very next one after the meeting
with Rhee.
If the meeting with Matsuba is immediately after the meeting with Rhee,
then we have a new block. Very exciting.
There are now only two entities that can separate the three meetings
with Fuentes.
These two blocks are going to have to
separate the three meetings with Fuentes.
Therefore, Garibaldi’s meetings with
Fuentes must be first, fourth, and seventh.
This question asks for the dignitary
that must be scheduled for the fourth
meeting with Garibaldi. Well, we are
looking pretty strong.
Fuentes must have the fourth
meeting, so (A) is going to be
our final answer on this game.
Say goodbye to Garibaldi - that’s the end of our fancy meeting with
dignitaries. As you can see by now, having fewer players than slots
complicates a game to some extent. In this game, they defined the number of
times that each player went. That was a pleasant surprise. Other games will
leave that as an open question, and you will have to determine how many
slots each player could fill.
By noting that Fuentes was the most restricted player in the game, we
could visualize the solution to each question very quickly.
Now that we’ve taken a look at underbooked games, you can
probably guess what’s next. But let’s not ruin the surprise. Turn
the page when you’re ready for the next challenge.
OVERBOOKED ORDERING
Now we head in the other direction. First, we covered games with the same
number of players and slots. We just worked through games with less players
than slots. Up next are overbooked ordering games, in which there are more
players than slots. In life, these are never fun situations. When you have
more people than seats in your car, you have to cram in the backseat, sit on
some smelly guy’s lap, or just ride in the trunk. On the LSAT, it’s not quite
that bad, but you do need to know how to simplify these game.
In an overbooked ordering game, some slots will be
assigned more than one player.
In overbooked ordering
games, every player is assigned to one
of the slots. There will be games in
which some players aren’t assigned to
any of the slots and must be left out,
but those are combo games. We get to
enjoy such challenges in a later
chapter.
BP Minotaur:
Let’s see how this works. Consider the example that we used to introduce
overbooked ordering games earlier in this chapter:
Nine students—A, B, C, D,
E, F, G, H, and I—each
must give an oral report in
In this game, you must assign all nine
students to one of the six weeks. Thus,
there are going to be some weeks in
history class on a historical
figure of their choosing
over a span of six weeks,
week 1 through week 6.
which more than one student gives an
oral report. For instance, it could be
true that both C and E give oral
reports during week 5.
In overbooked ordering games, it will be very important to keep track of the
restrictions on how many players can occupy each slot.
Does someone have to give an oral report each week?
What is the maximum number of oral reports assigned
to any week? Which weeks cannot have more than one
report?
These are the questions that will be crucial during an overbooked ordering
game. You will still find a number of restrictions on the actual players (G
can’t go in week 4, D is before F). But there will also be deductions about
how many students could give reports in a given week (week 3 must have at
least two reports, week 5 only has one).
You want to keep your setup
consistent, but now we will stack
variables vertically to keep track of
everyone. In this game, a completed
hypothetical could look like this
(kinda ugly, we know).
Up next, let’s discuss the rules.
RULES
Most of the rules on overbooked ordering games will sound very familiar they will be identical to 1:1 ordering games. But, just like underbooked
games, the nature of the new challenge opens up a few new types of rules.
Let’s take a look.
There must be at least one report given
each week.
No more than two reports can be given in
any one week.
Only one report is given during the second
and fourth weeks.
Since each slot has the possibility of housing zero, one, two, or more
players, there will likely be restrictions to limit these possibilities. Here are
a few principles that you might see in our good old game about oral
reports:
For the first two principles, you just want to represent them with your other
rules. They set a minimum and a maximum for each week. The third rule can
be inserted directly into your setup by using a box (or any other appropriate
visual representation) on the second and fourth slots.
Ninja Note:
In some overbooked ordering games, ties are
possible. If X is not before Y, it is not
necessarily after. They could be
assigned to the same slot.
Now it’s time to jump in. On the next page, you will be confronted with an
overbooked ordering game. We are going to run through this one together
using the Blueprint Building BlocksTM technique. Let’s do it!
SEPTEMBER 2006: GAME 4 (18-22)
This one is going to be our first overbooked ordering game. It was the fourth
and final game on the September 2006 LSAT. It’s a tough one, but we will
work through it together.
Before we begin, flash back to your childhood. Remember those fun games
that you would play with your food? On this game, all of that experience pays
off. And to think, your mom used to tell you to stop.
1. Setup
A child eating alphabet soup notices that the only
letters left in her bowl are one each of these six
letters: T, U, W, X, Y, and Z. She plays a game
with the remaining letters, eating them in the
next three spoonfuls in accord with certain rules.
Each of the six letters must be in exactly one of
the next three spoonfuls, and each of the
spoonfuls must have at least one and at most
three of the letters. In addition, she obeys the
following restrictions:
Alphabet soup
with T, U, W,
X, Y, and Z?
Not very
realistic.
In this game, the spoonfuls have an inherent order to them, so they will be
used as the base for our setup. There are three spoonfuls and six letters, so
this game is overbooked.
Clearly, this little girl must shove more than one letter into some of her
spoonfuls. (Hopefully, she still chews with her mouth closed.) There is one
more huge thing to notice in the introduction to this game. They give you a
pretty strong restriction on how many letters can go into each spoonful. Each
spoonful must have between one and three letters.
In a truly inconsiderate
maneuver, some games will present
you with rules in the middle of the
introduction. Make sure to treat these
principles just like other rules.
BP Minotaur:
Many students build incorrect setups for this game. Remember to always
identify the variable set with an inherent order (spoonfuls) and use that as the
base.
2. Rules
There are some tricky rules in this game. It’s very important to simplify the
rules without changing their meaning. At this point, you should expect two
different types of rules:
1. Ordering restrictions about the specific letters, and
2. Rules about which letters can be in the same
spoonful.
Let’s jump in.
Well, that’s not a bad start. This rule is one of our basic ordering principles.
The T must be in an earlier spoonful than the U. We represent this rule in the
normal fashion.
With only three slots,
deductions will come quickly. Already
we know that the T is in the first or
second spoonful, and the U is in the
second or third. Yummy.
Ninja Note:
Slow down. When a rule in a game
sounds a little bit weird, that’s
normally because it is. Always be
careful when you see an irregular rule
like this one.
Since the U is not in a later spoonful than the X,
that means it must be in an earlier spoonful. Duh.
Ditz McGee:
Not exactly, Ditz. That is the pitfall we need to avoid.
In a 1:1 ordering game, if one variable doesn’t come
after another, then it must come before. However, you
always have to remember what type of game you are
doing. In this overbooked game, remember our
hungry little girl can eat up to three letters in one
spoonful.
The U could come before
the X, or they both could
be in the same spoonful. So
just represent that the U can’t come after the X.
You can also turn this rule
around a bit. It’s preferable to deal
with what must happen in a game, not
what is forbidden.
BP Minotaur:
Excellent point. If you cringed slightly when looking at the last rule, that is
understandable. Here’s a better way to visualize this rule:
Since the U cannot come in a later
spoonful than the X, it must come in an
earlier spoonful, or they must be in the
same spoonful. So we can represent that
the U is either before the X or with the X.
This will be much easier to deal with as we move through the game.
Let’s keep moving.
Back to the basic ordering principles. The Y must be in a later spoonful than
the W.
This rule sets off
the “oh crap”
radar.
The words “but not both” should always catch your eye.
We covered these statements in our discussion of
conditional statements (exclusive ‘or’). However, we
haven’t seen it in action until now. According to this
rule, we can make a block with the U and either the Y or
the Z. However, the U cannot be with both the Y and
the Z, so we should also make a note of that.
Did you notice that the U is a
major player in three of the four rules?
We should probably keep our eyes on
the U.
Ninja Note:
The rules on this one are tough, but we survived. Just imagine the terrible
slog that can result if you mischaracterize one of these complicated rules.
Never rush through the rules. Working slowly will always save you time
(not to mention heartache and nightmares) in the long run.
3. Deductions
This one isn’t pretty. There are a bunch of rules, and they are hard to
visualize. There’s no easy way to combine the ordering principles, and the
clock is ticking.
There’s just too many
rules. I never know where to look.
Cleetus Comment:
That is actually a common problem. Rather than staring at all of the rules
together, which can be intimidating, just work through the rules one-by-one.
If you think through the effect of each rule on the game, deductions will
unfold in front of you. And you are going to get a chance to practice just that.
A few of the rules lead to great restrictions. Let’s see if you can spot them.
Challenge : Use the rules to build restrictions for the
letters.
The first rule states that the U is in a later spoonful than the T. Thus, the
U cannot be in the first spoonful and the T cannot be in the third
spoonful.
The third rule states that the Y is in a later spoonful than the W. Thus,
the Y cannot be in the first spoonful and the W cannot be in the third
spoonful.
Those restrictions are going to be helpful, but there’s one more big one that
is tough to spot. Did you spot it?
We already know the U can’t be in the first
spoonful. Since the U must either be before the X
or they must be in the same spoonful, there is no
way the X can be in the first spoonful.
At this point, you have a good number of restrictions.
However, there are some other important deductions
about the U. It’s all about the U.
The U cannot be in the first spoonful, so it must be in either the second
or third spoonful.
The T must be in an earlier spoonful than the U. Thus, if the U is in the
second spoonful, then the T must be in the first spoonful.
The X must
either be in
the same
spoonful or
in a later spoonful than the U. So if the U is in the third spoonful, then
the X must also be in the third spoonful. In addition, the U must always
be with either the Y or the Z, but not both, so there would then be three
letters in the third spoonful.
4. Questions
Now let’s hit the questions and get some points.
Question #18
18. Which one of the following could be
an accurate list of the spoonfuls and the
letters in each of them?
This game starts off in a very predictable manner. We get our feet wet with a
nice elimination question. By this point, you should feel completely
comfortable with these questions.
Challenge : On the next page, hunt down the correct
answer.
The U is in a later spoonful than the T.
The U is not in a later spoonful than the X.
The Y is in a later spoonful than the W.
The U is in the same spoonful as either the
Y or the Z, but not both.
(A) First: Y second: T, W third: U, X, Z
(B) first: T, W second: U, X, Y third: Z
(C) first: T second: U, Z third: W, X, Y
(D) first: T, U, Z second: W third: X, Y
(E) first: W second: T, X, Z third: U, Y
The first rule says that the U is in a later spoonful than the T. In (D), both
the T and the U are in the first spoonful. So (D) is eliminated.
The second rule states that the U is not in a later spoonful than the X. In
(E), the X is in the second spoonful and the U is in a third spoonful. That
would be a later spoonful, so (E) is eliminated.
The third rule states that the Y is in a later spoonful than the W. In (A), the
Y is in the first spoonful and the W is in the second spoonful, so (A) is
eliminated. But don’t stop there. In (C), both the W and the Y are in the
third spoonful, so (C) is also eliminated.
Here, we don’t even need to use the fourth rule. The only answer left
standing is (B).
Question #19
19. If the Y is the only letter in one of the
spoonfuls, then which one of the following
could be true?
Here is the first conditional question for this game. The new piece of intel is
that the Y has to be alone in one of the spoonfuls.
The Y can’t be in the first spoonful, but it can be in the second or third
spoonful.
At first glance, it looks like the Y could be the only letter in either spoonful.
That might turn out to be the case, but you won’t know until you try. It is
therefore necessary to work through two hypotheticals on this question.
Challenge : On the next page, work through both
hypotheticals. In the first one, the Y is the only letter in
the second spoonful. In the second one, the Y is alone
in the third spoonful.
There are many deductions in both hypotheticals. First, let’s look at what
happens if the Y is the only letter in the second spoonful.
The U cannot be in the first spoonful, and it
can’t be with the Y. Both the U and the Z must
be in the third spoonful.
The W must be in an earlier spoonful than the Y,
so the W is in the first spoonful.
The X cannot be in an earlier spoonful than the
U, so the X must also be in the third spoonful.
The T cannot be in the second spoonful (the Y must be alone). The third
spoonful is full, so the T is in the first.
And now, let’s check what happens if the Y is alone in the third spoonful.
The U cannot be in the first spoonful, so both the
U and the Z must be in the second spoonful.
The T must be in an earlier spoonful than the U,
so the T is in the first spoonful.
The X can’t be in an earlier spoonful than the U,
and it can’t be in the third spoonful, so it must
be in the second.
The second spoonful is full, so the W is in the first spoonful.
That was a fair bit of work, but it will pay off. Now we just need to find an
answer that could be true. Take a look and see if you find anything
appealing.
(A) The Y is in the first spoonful.
(B) The Z is in the first spoonful.
(C) The T is in the second spoonful.
(D) The X is in the second spoonful.
(E) The W is in the third spoonful.
Both the T and the W must be in the first spoonful. Additionally, neither
the Y nor the Z can be in the first spoonful. There goes (A), (B), (C),
and (E). Sweet.
In the second hypothetical, the X is in the second spoonful. Thus,
(D) could be true and it’s our answer.
Question #20
20. If the Z is in the first spoonful, then
which one of the following must be true?
Up next is another conditional question. On this one, the Z must be in the first
spoonful. However, even though this question talks about the Z, it’s still all
about the U. Here’s the first round of quick deductions:
The U cannot be in the first spoonful. Since the Z now is in the first
spoonful, the U and the Z can’t be together. The U must be with the Y.
The U and the Y could be in either the second spoonful or the third
spoonful.
Now it’s time for you to do some work. Just like the last question, this one
will require two quick hypotheticals. Draw one with the U and the Y in the
second spoonful and one with the U and the Y in the third spoonful.
Challenge : Work through both hypotheticals.
As you have experienced, there are a couple questions in this game that
require two hypotheticals. It takes a little bit of work to get through them, but
it would be difficult to visualize the answer choices without everything at
your disposal.
If the U and the Y are in the second spoonful:
The T must be in an earlier spoonful than the U, so the T must be in the
first spoonful.
The W must be in an earlier spoonful than the Y, so the W must be in
the first spoonful.
Each spoonful must have at least one letter, so the X must be in the third
spoonful.
If the U and the Y are in the third spoonful:
The X cannot be in a later spoonful than the U, so the X must be in the
third spoonful.
The T and the W could each still be in either the first or second
spoonful, but at least one of them must be in the second spoonful.
At this point, you have two hypotheticals, and you are ready to go. The
answer choice here must be true, so you want to look for a letter that doesn’t
move between the two situations.
(A) The T is in the second
spoonful.
(B) The U is in the third spoonful.
(C) The W is in the first spoonful.
(D) The W is in the second
spoonful.
(E) The X is in the third spoonful.
X must be in the third spoonful in both hypotheticals. Our answer is
(E).
Question #21
21. Which one of the following is a complete list
of letters, any one of which could be the only
letter in the first spoonful?
After a few
conditional
questions, this is
the first absolute
one.
We already deduced (very astutely) that the U, the
X, and the Y can never be in the first spoonful. So
they certainly cannot be the only letter in the first
spoonful.
So that means that the answer
must be the other three letters (T, W,
and Z)?
Ditz McGee:
Not so fast. Even though we know that those three letters could be in the first
spoonful, we don’t yet know whether they could be the only letter in the first
spoonful. We still have to do a little work.
Ninja Note:
First step: Go back and check
your previous work.
Thank you, Ninja. That is exactly what we were going to advise. At this
point, you can just scan through your previous work to see if we have already
proven that the T, the W, or the Z could be the only letter in the first
spoonful. Let us refresh your memory.
On the last question (#20), we constructed a hypothetical where the Z
could be the only letter in the first spoonful as long as both the T and the
W are in the second spoonful.
That still leaves us with the T and the W to test. That part is up to you.
Challenge : Test whether the T or the W could be the
only letter in the first spoonful.
Did you find ways to make both of those work? You should have.
If the T is the only letter in the
first spoonful, both the W and
(A) T
the Z are in the second spoonful,
(B) T, W
and both the X and the Y are in
(C) T, X
the third spoonful. The U could
(D) T, W, Z
be in the second or third.
(E) T, X, W, Z
If the W is the only letter in the
first spoonful, the T is in the
second spoonful and both the U and the X are in the third spoonful. The Y
and the Z make linked options for the second and third.
The T and the W are added to the list. We easily spot our three
candidates (the T, the W, and the Z) in answer choice (D). Circle that
lovely answer and keep moving.
Question #22
22. If the T is in the second spoonful, then
which one of the following could be true?
The last question on this game is also a conditional question. At this point,
we have this alphabet soup down to a science. This one is all you. The new
condition is that the T is in the second spoonful. Take a moment to draw a
hypothetical, and try to spot the answer that could be true.
Challenge : Find that beautiful correct answer.
(A) Exactly two letters are in the first
spoonful.
(B) Exactly three letters are in the first
spoonful.
(C) Exactly three letters are in the second
spoonful.
(D) Exactly one letter is in the third
spoonful.
(E) Exactly two letters are in the third
spoonful.
After a few of the previous questions, you might be expecting huge
deductions. But you can’t figure everything out on this one.
On a could be true question, it
is normal for there to be some
variables that you can’t completely nail
down.
Ninja Note:
The U must be in a later spoonful than the T, so the U must be in the
third spoonful.
The U must be in the same spoonful as either the Y or the Z, but not
both.
The X cannot be in a later spoonful than the U,
so the X must be in the third spoonful.
There are three letters in the third spoonful.
Since there has to be at least one letter in each
spoonful, the W or the Z must go in the first
spoonful.
(A) Both the W and the Z could be in the first spoonful as long as
the Y is in the third spoonful. We didn’t have to wait long, and
(A) is the correct answer.
(B) There must be three letters in the third spoonful. That only leaves
three letters to split between the first and second spoonful. There is
no way for any spoonful besides the third to have three letters.
(C) See explanation for (B).
(D) The third spoonful must have three letters, so (D) must be false.
(E) See explanation for (D).
That was quite an adventure. Here are some important lessons to learn from
this game:
1. Always stay consistent with your setup.
Rather than concocting some unique
setup for the alphabet soup game, we
stuck with the normal ordering setup,
and that made the game much easier to
visualize.
2. Small deductions add up to big
deductions. We noticed that the U was involved in three different
rules. We inspected each of those rules, and that gave us a big head
start when we attacked the questions.
3. Drawing out a couple hypotheticals on a question is not a waste of
time. On a number of questions, we had to work through two
different situations to find an answer. But this is still much more
efficient than randomly testing answer choices.
4. Alphabet soup is still delicious.
It’s your turn. The next game is another overbooked ordering game for you to
try. Before you begin, we want to give you one quick hint. Most ordering
games deal with arranging variables from best to worst, or first to last, or
Monday through Friday, and we always set these games up horizontally.
However, there are some ordering games where you are arranging variables
from bottom to top, or lowest to highest. When this happens, you want to
use a vertical setup. We don’t want to give it away, but the following game
fits into that category.
We will see you at the finish line.
JUNE 2008: GAME 2 (6-12)
Questions 6-12
A critic has prepared a review of exactly six music CDs- Headstrong, In
Flight, Nice, Quasi, Reunion, and Sounds Good. Each CD received a rating
of either one, two, three, or four stars, with each CD receiving exactly one
rating. Although the ratings were meant to be kept secret until the review was
published, the following facts have been leaked to the public:
For each of the ratings, at least one but no more than two of the CDs
received that rating.
Headstrong received exactly one more star than Nice did.
Either Headstrong or Reunion received the same number of stars as In
Flight did.
At most one CD received more stars than Quasi did.
6. Which one of the following could be an accurate matching of
ratings to the CDs that received those ratings?
(A) one star: In Flight, Reunion; two stars: Nice; three stars:
Headstrong; four stars: Quasi, Sounds Good
(B) one star: In Flight, Reunion, two stars: Quasi, Sounds Good;
three stars: Nice; four stars: Headstrong
(C) one star: Nice; two stars: Headstrong; three stars: In Flight,
Sounds Good; four stars: Quasi, Reunion
(D) one star: Nice, Sounds Good; two stars: In Flight, Reunion;
three stars: Quasi; four stars: Headstrong
(E) one star: Sounds Good; two stars: Reunion; three stars: Nice,
Quasi; four stars: Headstrong, In Flight
7. If Headstrong is the only CD that received a rating of two stars,
then which one of the following must be true?
(A) In Flight received a rating of three stars.
(B) Nice received a rating of three stars.
(C) Quasi received a rating of three stars.
(D) Reunion received a rating of one star.
(E) Sounds Good received a rating of one star.
8. If Reunion received the same rating as Sounds Good, then which
one of the following must be true?
(A) Headstrong received a rating of two stars.
(B) In Flight received a rating of three stars.
(C) Nice received a rating of two stars.
(D) Quasi received a rating of four stars.
(E) Sounds Good received a rating of one star.
9. If Nice and Reunion each received a rating of one star, then which
one of the following could be true?
(A) Headstrong received a rating of three stars.
(B) Headstrong received a rating of four stars.
(C) In Flight received a rating of three stars.
(D) Sounds Good received a rating of two stars.
(E) Sounds Good received a rating of three stars.
10. Which one of the following CANNOT be true?
(A) Quasi is the only CD that received a rating of three stars.
(B) Quasi is the only CD that received a rating of four stars.
(C) Reunion is the only CD that received a rating of one star.
(D) Reunion is the only CD that received a rating of two stars.
(E) Reunion is the only CD that received a rating of three stars.
11. If Reunion is the only CD that received a rating of one star, then
which one of the following could be true?
(A) Headstrong received a rating of four stars.
(B) In Flight received a rating of two stars.
(C) Nice received a rating of three stars.
(D) Quasi received a rating of three stars.
(E) Sounds Good received a rating of two stars.
12. Which one of the following CANNOT have received a rating of
four stars?
(A) Headstrong
(B) In Flight
(C) Quasi
(D) Reunion
(E) Sounds Good
OVERBOOKED GETS OWNED
Congratulations! That was your first overbooked ordering game by yourself.1
This one popped up on the June 2008 LSAT. There are a couple entertaining
things to note: (1) the attempt by the LSAT to be “hip” by leaving behind
alphabet soup and clothes hangers and instead entering the music industry,
and (2) the equally “hip” names that they create for the albums, including the
infamous Sounds Good. Can anyone say triple platinum?
1. Setup
The setup to this game commonly causes some confusion. Always seek out
the basic process driving the game. Here, we have to arrange CDs by their
star rating. There is clearly an order to the star ratings, so we are doing an
ordering game. Moreover, the ratings should be used as the base of our
setup since they have an inherent order.
You can successfully complete this game using our classic horizontal setup,
but we think this is one of the rare ordering games that benefits from a
vertical setup.
Since there are six players (CDs) to fit
into the four slots (star ratings), this is
an overbooked ordering game. There
will be some CDs that receive the same
rating.2 Here is our setup:
Note: At this point, it hasn’t been
specified that at least one CD
receives each rating. From the intro, it could be true that no CDs
receive three stars, or it could be true that three of the CDs receive
three stars.
If you utilize a vertical setup
on an ordering game, make sure to
adjust your rules accordingly. For
instance, many ordering principles and
blocks will now be represented
vertically as well.
BP Minotaur:
2. Rules
There are still a lot of issues up in the air when we get to the rules in this
game, so it’s important to work through them slowly. You should be
expecting normal ordering rules as well as restrictions on how many CDs
could receive each of the star ratings.
So here we go!
Sweet. This is just what we wanted. Either one or two of the six CDs must be
assigned to each star rating. At this point, it is helpful to do a little bit of math
(we know, scary, but it’s really not that bad). Since there are six CDs and
four star ratings, there must be two pairs of CDs that receive the same
rating and two other CDs that don’t receive the same rating as any other
CD.
Later in our journey, we will
practice this process and explore how it
can lead to powerful deductions. For
now, you are still a white belt, so just
know that a little math can be very
helpful.
Ninja Note:
The next two rules give ordering principles. Always make sure to represent
the rules just as they will appear in the setup.
Both rules should be symbolized as blocks, but make sure they are consistent
with your setup. One should be vertical, and the other one should be
horizontal. Also, note that Headstrong is involved in both rules. This might
just turn into a deduction.
One more rule, and this one can be a doozy...
The difference between “exactly one” and “at most
one” can make or break a whole game.
There are various ways
to represent this rule,
but make sure you
understood its
meaning. Either zero or
one CD receives more
stars that Quasi. Quasi must be pretty good.
That brings us to the conclusion of the rules.
However, you should have the feeling that
deductions are in the air.
3. Deductions
The deductions on this one are huge. Here are the first ones to notice:
Sounds Good is not involved in any of the rules, so Sounds Good is a
random.
Headstrong must receive exactly one more star than Nice, so this block
tells us that Nice cannot receive four stars and Headstrong cannot
receive one star.
The last rule is a huge sign that Quasi is going to be an important CD in this
game. Since you know that at most one CD can receive a higher rating than
Quasi, you know that Quasi is going to be rated pretty highly. Investigating
further leads to helpful deductions.
Since only one CD can receive a better rating
than Quasi, the lowest rating that Quasi can
receive is three stars.
If Quasi does receive a rating of three stars,
then only one CD can receive four stars.
If two CDs receive a rating of four stars,
Quasi must be one of them.
The third rule tells us that In Flight must receive the same number of
stars as either Headstrong or Reunion. Since only one CD can receive a
better rating than Quasi, In Flight cannot receive four stars.
The final item to notice before jumping into the questions is the relationship
between the two blocks. When you are mentally prepared, we review on the
next page.
Headstrong must receive one
more star than Nice. And In Flight
must receive the same rating as
either Headstrong or Reunion.
This creates two possibilities: (1) In Flight receives the same rating as
Headstrong, creating an uber-block, or (2) In Flight receives the same
rating as Reunion, and then you must find places for both independent
blocks.
You can save valuable time by
anticipating how important rules are
going to play together.
Ninja Note:
Make sure to review these deductions. They are tough to spot, they pay off in
the questions.
1 List of firsts in your life that likely outweigh this one: the first time you tied your
shoes, the first time you realized that you could clear the history in your web browser,
and the first time that you tasted a snickerdoodle. But we are trying to celebrate your
accomplishments here.
2 Anyone born prior to 1985 should be having Star Search flashbacks at this point.
“Jamie receives three-and- three-quarters stars.”
4. Questions
Question #6 (elimination, could be true)
Pick a rule and use it to knock out an answer choice. Then repeat. You know
the drill.
The second rule stated that Headstrong must receive exactly one more
star than Nice. In (D), they do not even come close to following this
rule. (D), get outta here.
The third rule stated that In Flight must receive the same rating as either
Headstrong or Reunion. In rotten answer choice (C), In Flight receives
the same rating as Sounds Good, so (C) is gone.
The last rule told us that a maximum of one CD receives a better rating
than Quasi. In (B), Quasi only receives two stars and there are two CDs
that receive better ratings. And in (E), Quasi receives three stars but
there are two CDs that receive four stars. Both answers are eliminated.
That only leaves us with (A).
Question #7 (conditional, must be true)
Time to test our deductions on a conditional question. The new condition is
that Headstrong is the only CD that receives a rating of two stars.
Headstrong must receive exactly one more star than Nice, so Nice must
receive a lowly rating of one star.
Since In Flight must receive the same rating as either
Headstrong or Reunion, In Flight must now receive the
same rating as Reunion. Also, In Flight can never receive
a rating of four stars (because there would be too many
CDs with better ratings than Quasi), so In Flight and
Reunion receive three stars.
Quasi receives a perfect four-star rating.
The only wildcard at this point is Sounds Good, which
could receive one star with Nice or four stars with Quasi.
We have to identify an answer choice that must be true. Luckily,
that will be one of our deductions. In Flight must receive a rating of
three stars, so (A) is the correct answer.
Question #8 (conditional, must be true)
The return of the conditional question. Now, Reunion receives the same
rating as Sounds Good.
In Flight must receive the same rating as
Headstrong, so now we are dealing with
the bigger block that also involves Nice.
Reunion and Sounds Good cannot receive
four stars because then both of these CDs
would receive better ratings than Quasi.
There are two ways these pair of blocks
can fit together: Reunion and Sounds
Good could receive three stars and the big block could fit underneath, or
Reunion and Sounds Good could receive only one star and the big block
could squeeze on top.
In both situations, Quasi must be the big winner with four stars.
Thus, our lovely answer is (D).
Question #9 (conditional, could be true)
On this conditional question, both Nice and Reunion receive
one star.
Headstrong receives exactly one more star than Nice, so
Headstrong receives two stars.
In Flight cannot receive the same rating as Reunion, so
it must receive two stars along with Headstrong.
That only leaves Quasi and Sounds Good. One of them
gets three stars and the other one is going to receive four stars, so you
can represent that with linked options.
Sounds Good could receive a rating of either three stars or four
stars, so (E) is our answer.
When you are doing a could
be true question, look towards any
linked options. Odds are the answer
will follow nicely.
BP Minotaur:
Question #10 (absolute, must be false)
Here’s a change of pace. First, this is an absolute question. Second, this one
is looking for an answer that must be false. This question is going to be a pure
and simple test of our deductions - gotta find something that doesn’t work.
(A) Quasi could be the only CD that receives three stars, or four stars
for that matter. (Note: The hypothetical we did for #9 proves that
this answer could be true.)
(B) See explanation for (A).
(C) If Reunion is the only CD that receives a rating of one star, In
Flight and Headstrong could both receive three stars, and Nice
could get two stars. Quasi would have to receive four stars and
Sounds Good could receive either two or four stars, so (C) could be
true.
(D) If Reunion is the only CD that receives a
rating of two stars, then In Flight and
Headstrong must receive the same rating.
However, they would have to receive four
stars so that Nice could receive one star
less than Headstrong. This violates our
original deduction that In Flight cannot
receive four stars (because too many CDs
now receive better ratings than Quasi).
(E) If Reunion is the only CD that receives a rating of three stars, then
the block could fit under Reunion. In Flight and Headstrong could
receive two stars, Nice could receive just one star, and Quasi gets
four stars.
Question #11 (conditional, could be true)
Now we venture back into the world of conditional questions. By this point,
you should feel confident because we have a good amount of experience
playing with the rules. The new condition: Reunion is the only CD that
receives one star.
On the very last question, we
had to do a hypothetical with Reunion
as the only CD with one star to
eliminate (C). Done.
Ninja Note:
If Reunion is the only CD that receives a rating of one
star, then In Flight must receive the same rating as
Headstrong. Once again, we are dealing with the big
block.
In Flight cannot receive a rating of four stars, and
Headstrong must receive exactly one more star than
Nice, so In Flight and Headstrong receive three stars.
Nice must receive a rating of two stars.
Since two CDs already receive a rating of three stars,
Quasi must receive four stars.
Sounds Good could receive a rating of either two stars or four stars,
so (E) is our guy.
Question #12 (absolute, must be false)
We finish things off here with an absolute question. The deductions we made
give us a big payoff to finish off this one. The challenge is to identify a CD
that can’t receive a rating of four stars.
We deduced that both Nice and In Flight cannot receive four stars.
Nice is missing from the answer choices, but In Flight is right there
for us in (B).
That’s all for our CD game. There are three big
obstacles to success on this game: (1) building an
effective vertical setup and staying consistent with the
rules, (2) making restrictions related to Quasi, and (3)
visualizing how the two blocks work together. If you
complete all three, you are in good shape. But you
can’t miss any of the steps.
That wraps up our trek through underbooked and overbooked ordering
games. Make sure to review this chapter, but here’s a quick overview:
In underbooked ordering games, you have fewer players than slots.
Some games will leave slots unoccupied, but more often some of the
players will fill more than one slot. Look for deductions about which
players can show up more than once.
In overbooked ordering games, you have more players than slots.
Some of the slots will be assigned more than one player. Search for
deductions about how many players could be assigned to each of the
slots.
We have one more very important type of ordering game to cover in the next
chapter. It’s go time.
14/addingTIERS
WHAT IS IT?
In the first couple chapters of ordering games, we have developed a deep
relationship with games that have two variable sets: parcels and delivery
times; dresses and hangers; saxophonists and auditions. We are now
departing from that comfort zone.
Many ordering games will present you with more than
two variable sets. Some games feature up to four or
five!
When you have to order more than two variable sets, the best way to organize
the information is with different tiers in your setup. It’s crucial that you
continue to identify and use the variable set with an inherent order as the base
of your setup. Here are the correct steps to build a good tiered ordering setup:
1. Always identify the basic process in the game. Watch out for
the common phrases that indicate an ordering game (consecutive,
order, sequential).
2. Identify the variable set with an inherent order (days of the
week, rankings, time slots), and use that variable set as the base
of your setup.
3. New step: Build a tier for each additional variable set.
Here’s an example, in real time:
During a span of five weekdays— Monday
through Friday—five rooms in a house
must be cleaned, one room per day. The
rooms in question are the parlor, a
restroom, the saloon, a TV room, and the
unicorn stable. Five people— Vivica,
Wilma, Xena, Yavar, and Zela—must each
be assigned to clean one of the rooms. The
following conditions govern the cleaning
assignments:
1. We’re tasked with making a cleaning schedule,
assigning rooms and people to days.
2. The days of the week have an inherent order and
should be used as the base.
3. There should be two tiers for the other two variable
sets: rooms and people.
The goal for your setup is to create something in
which you can easily visualize any rule they throw in your
direction. Thus, it’s very important to build the
BP Minotaur:
appropriate number of tiers.
When you are doing games, you never want to force yourself to remember
things. Ever been to the grocery store without a list? You probably returned
with cereal but no milk, or meat but no bread. Sucks. Trust us, you have
enough to think about when you are taking the LSAT. Make sure to nicely
organize all of the information by creating tiers.
The setups for tiered ordering games will all look very similar. However,
these games are introduced in a couple of different ways. Let’s take a look.
Different Categories
There are ordering games in which the players belong to different categories
(boys and girls; 7th graders and 8th graders; right-handed or left-handed;
boxers or briefs). When this occurs, it is best to form a tiered ordering setup.
There should be one tier for the player and another tier for the category to
which that player belongs. Consider the following game:
At a pet store, exactly four puppiesBubba, Chucky, Ducky, and Eddie—and
three kittens- Mindy, Nina, and Oprah—
are displayed in seven consecutive cages
along the display wall. Each cage holds
only one animal.
This game sounds pretty similar to a 1:1 ordering game. However, the
players are split into two categories: puppies and kittens. Our setup needs to
be able to incorporate this information.
I don’t think we need a
second tier. We already know which
ones are dogs and which ones are cats.
Cleetus Comment:
Cleetus, that’s a common reaction by students, and many set this game up
with only one tier. However, imagine you were given the following rules in
this game:
A puppy is displayed in cage 5.
A kitten must be displayed in either cage 2
or cage 6, but not both.
Bubba is displayed in between two kittens.
These rules would be very hard to visualize and work into the setup if we did
not have tiers. It’s much easier to visualize these rules if you have a second
tier to track the categories (puppies and kittens).
When the players in an ordering game belong to
different categories, always form a second tier for the
categories. Always.
Here’s the appropriate setup for
this game. The animal’s name is
placed in the bottom tier and
whether the animal is a puppy or
kitten occupies the top tier. Very
cute.
On to the next version of tiered ordering.
Players with Characteristics
A slightly different type of game will ask you to order a set of players and
also track a certain characteristic of each one. In these games, it is best to
have a tier for the player (children, for example) and each characteristic that
is introduced (tall or short; green eyes or blue eyes; favorite brand of cereal).
Here is an example:
Over the span of one month, Juliette reads
seven novels: Abigail, Beauty and the
Beefcake, Chesapeake Children, Dances
with Dunces, Edward Scissorfeet, Forrest
Gimpy, and George and the Grasslands.
Some of these novels are romances and
some are action-adventures. Juliette reads
the seven novels consecutively.
The basic process in this game is to order the books according to when
Juliette reads them. However, we also have to track whether each book is a
romance (probably Edward Scissorfeet) or an action- adventure (most likely
Dances with Dunces). We need two tiers, and our setup should resemble the
beautiful setup you see to the left.
But wait, there’s more. This setup can actually grow larger. Check this one
out:
The top five contestants in a beauty show
are V, W, X, Y, and Z. Each contestant
performed one of the following during the
talent portion of the show: baton twirling,
gymnastics, singing, dancing, or magic.
During the evening gown competition,
each contestant wore one of the following
colors: red, blue, silver, gold, and violet.
The final ranking of the contestants must
meet the following conditions:
Everyone loves a beauty pageant. And the winner is... our tiered setup. On
this one, we have to determine the ranking of the contestants. However, we
also have to track the talent each one performs and the color of their evening
gown.
For this game, we need a setup with three tiers : one for the contestant, one
for his or her impressive talent, and
one for the color of his or her
evening gown.
They could add another
characteristic (such as swimwear),
but that can be easily incorporated as
well.
Independent Variable Sets
The third and final version of tiered ordering occurs when you are presented
with two (or more) independent variable sets that must be ordered. The setup
will look identical to the previous games, but each tier is now used for a
different variable set.
As always, here’s an example:
In each of the next six weeks, the students
in a college philosophy course will be
assigned to read a work by one of three
ancient philosophers—Socrates, Plato, and
Aristotle— and a work by one of five
modern philosophers—Hobbes, Descartes,
Locke, Kant, and Frege.
In this game, the students have to read two works each week. Thus, we have
to order the assigned works from the ancient philosophers and the assigned
works from the modern philosophers (which are independent variable sets).
This means you should build a tier for each one.
That’s a summary of the common features found in tiered ordering games.
The first crucial step in each game is to build the correct setup. With that in
mind, we have prepared a beautiful drill to help you hone your skills at
setting up tiered ordering games.
Good luck...
TIERED ORDERING SETUP DRILL
For each of the following situations, read over the introduction and build the
correct setup. Remember, your setup should allow you to easily organize all
of the variable sets.
1. An editor of a magazine
reviews six articles consecutively.
Each of the articles is either a news
story or an opinion piece.
2. A television talk show airs five
shows during a certain week, from
Monday through Friday. On each
show, one of five teen mothers is
interviewed and the host gives a
final thought on one of five topics.
3. Lily is sick and stays home
from school on Tuesday. She
watches six television shows
during the course of the day, each
one starting after the last one
finishes. Each show is either on
NBC, CBS, or ABC. Also, each
show is either a sitcom or a drama.
4.
There are five body builders
who compete in the Mister
Universe competition. The
competitors are each ranked on
three different criteria: biceps,
triceps, and oil application. There
are no ties in any category.
5. Mistress Laser reads the fortune
of six troubled souls over the span
of two days. She has three
consecutive appointments on the
first day and three consecutive
appointments on the second day.
Each client receives either a
positive or negative fortune from
the powerful Mistress Laser.
DOES YOURS LOOK LIKE THIS?
In this one, the editor reviews six
articles consecutively. The second tier is
built to track whether each article is a
news story or an opinion piece.
On each day, there is both a teenage
mother and a captivating final thought.
We need to build a setup with two tiers.
Poor Lily is sick. We have to order the
shows, keep track of the network on which
they air, and figure out whether each show
is a sitcom or a drama. That means three
tiers. Feel better, Lily.
In this one, we get to deal with oiled up,
steroid-filled bodybuilders. This one is
different because the same variable set (buff
guys) is being ordered three times. So we
need a tier for each of the three criteria, and
the competitors will be placed into their appropriate rank on each tier.
Even though the appointments are separated
into only two days, there is a sequence of
six appointments. The second tier is for
whether each fortune is positive or negative.
NEW RULES?
No. Well, not really. Most of the rules on tiered ordering games are
identical to those already covered (dashes, blocks, divisions, arches,
options). However, mixed in with the usual suspects will be some rules that
might appear to be new. These are really just slight alterations to our good
old rules. For instance, you might get some intriguing new blocks due to the
increased size of our setup.
Luke must sit in between two
girls.
Suppose we were doing an ordering
game with two tiers, and we are
confronted with this rule. It gives us a
concrete spatial relationship, so we
symbolize it with a block.
The goal is to create a block
that looks exactly as it will
when inserted into your
setup. Here, you would have
Luke flanked by a girl on
both sides (lucky guy).
When you represent these “new” blocks in a tiered ordering game, you might
have flashbacks to a certain video game called Tetris. Once again, your
childhood experiences pay off big time in games.1
Final Pre-Game Comments
As you have probably noticed, the setups for tiered ordering games can be
pretty intimidating. You will commonly have 10, 12, or more slots in your
setup. On less complex games, working through a number of hypotheticals
can be an effective way to attack a game. But that’s a dangerous approach on
tiered ordering games. Drawing hypotheticals with 12 slots could easily take
you deep into your golden years.
This means that it is very important to search out deductions. We know we
harp on deductions in all types of games, but they are even more crucial on
tiered ordering games. There will always be big deductions to simplify
these games. Yes, always. So let’s give it a shot. We will work through the
first game using the Blueprint Building BlocksTM technique, and then you
can try a few on your own.
1 The piece shown on this page was unequivocally one of the best pieces in Tetris,
only rivaled by the high- scoring long and straight piece. And the award for most
annoying piece goes to the little guys, the ones where you could never tell whether the
piece was pointing right or left. Still brings back bad memories.
DECEMBER 2007: GAME 3 (12-17)
This is our first tiered ordering game, and it is coming at you courtesy of the
December 2007 LSAT. This was the third game, and there are deductions
galore (as expected). We’ll set it up together, and then you will be challenged
to spot the deductions. Say hello to some criminals...
1. Setup
Detectives investigating a citywide increase in
burglaries questioned seven suspects- S, T, V, W,
X, Y, and Z- each on a different one of seven
consecutive days. Each suspect was questioned
exactly once. Any suspect who confessed did so
while being questioned. The investigation
conformed to the following:
This is pretty
edgy stuff for
the LSAT.
Some of this should sound very familiar to you. There are seven suspects
who must be questioned over a span of seven days. Since the days have an
inherent order to them, we will use them as the base of our setup.
The seven suspects are the players
in the game. But then there is a
twist. Some of the suspects cracked
under the pressure and confessed.
This introduces another variable
into the mix: whether or not each of
the suspects confessed. Thus, it’s
necessary to add a second tier.
I just made one tier and
circled the ones that confessed.
Ditz McGee:
Be careful with such a strategy. While that might get you through this game,
you want to be consistent with your setup. Most variable sets you encounter
will present more options than just “yes” or “no,” so you always want to
build a new tier.
2. Rules
As we work through the rules, make sure to visualize how each one is going
to fit in the setup. Some of the rules in this game are hard to represent, so we
are going to work slow and make sure we understand all of them. Slow and
steady wins the race. Even on the LSAT.
The first two rules start out very nicely. First, you are told that T was
questioned on day 3. Got it - just plug that into the setup. It’s not clear
whether T confessed or not. Second, you know that whoever is on day 4
didn’t confess. At this point, any suspect except T could be fourth. However,
our second tier lets us plug this rule right into the setup. And we move on.
The third and fourth rules give you basic ordering principles. There’s no
suspect in common between the rules, so there’s no way to combine them at
this point. Moving right along...
This one can be confusing. To represent this rule, diagram that there can be
no confessions after W is questioned.
As you might expect, this rule is going to be very important. Think about it
for a second. Some suspects confessed, but others didn’t. If no suspect
confessed after W was questioned, do you think W is going to be questioned
early or late in the sequence?
Since no one can confess after W, he or she will
likely be one of the last suspects questioned. And W just
became very important.
Ninja Note:
As an example, if we find out that the fifth suspect
confessed, then W could be questioned no earlier than
fifth. We will revisit this W character. Last rule...
This one can go right into
our setup. We already
know that T is questioned on day 3, and we know the suspect on day 4 did
not confess. This rule is huge.
You can see that this rule will be very
restrictive on the game. T is questioned
on day 3 and the suspect on day 4
doesn’t confess. Since exactly two
suspects confess after T is questioned,
two of the final three suspects have to
confess. Big time.
Phew. That is a slew of rules. But, luckily, we were able to represent many of
them directly in our setup. That should make our next step easier.
3. Mighty Deductions
Ordinarily, we would start out the deductions phase by making a number of
restrictions. There are a few ordering rules that outlaw certain players from
certain slots, and those deductions are always helpful. But screw that. Let’s
think big.
If you were the detective on this case, which suspect would you investigate
further? Hopefully, you answered W. He seems to be a shady character, and
we need to get to the bottom of this. There are three helpful facts about W:
S is questioned after W is questioned.
No suspect can confess after W is questioned.
Exactly two of the final three suspects must confess.
Challenge : On the next page, try to narrow down the
days on which W could be questioned. Also, try to
determine if any suspects confess or not.
Since W must be questioned before S, W cannot be questioned last.
Two of the last three suspects confess. If W is questioned on day 5, at least
one of the suspects that is questioned after W would have to confess.
However, no suspects can confess after W is questioned. Therefore, W
cannot be questioned fifth (or anytime before then).
The only day left to question W is day 6. Bam. First big deduction.
Once you find a huge deduction, there’s likely more to follow. Let’s review
the other rules in the game:
S must be questioned after W is questioned,
so S must be the last suspect questioned.
No one can confess after W is questioned,
so the seventh suspect (S) does not confess.
Two of the final three suspects have to
confess, so the suspect on day 5 and W (on day 6) must confess.
It’s easy to miss this
deduction if you move through the
rules too quickly. Without it, the next
10 minutes of your life get very
complicated. It’s what we call a game
changer.
BP Minotaur:
That is some solid work, and you should be feeling pretty good. But never let
a game off easy. Hit it when it’s down. And then hit it again. Show no mercy.
It’s time to review the other rules to see if we can find more deductions.
Challenge : Make deductions about the remaining
suspects.
Before we get to the deductions, you might have noticed that we removed the
other rules from the setup above. Mentally, you want to perform the same
maneuver. Once we place W and S and determine which of the final three
suspects confessed, there is no reason to worry about those rules any longer.
Just cross them out and move on with your life.
Here’s what we were hoping you would find:
Y is not involved in any rule, so Y is
random.
Since Z must be questioned before
both X and V, neither one can be
questioned on day 1.
Z and Y are the only possibilities left for day 1, so we can make an option.
Z must be questioned before both X and V, so Z can’t be questioned on
day 4 or day 5. In fact, Z must be questioned on day 1 or day 2. That’s nice
to know.
Those are some serious deductions. This is a great example of a tiered
ordering game and the types of deductions you can expect. We have
dramatically simplified this game, and now it’s time for the big payoff - the
questions.
4. Questions
Question #12
12. Which one of the following could be
true?
No elimination question?!?
In a slight twist, there is no elimination question on this game. Rather, we
start things off with an absolute question. Considering the breadth of the
deductions in this game, it shouldn’t come as a huge surprise. In this one, you
need to find something that could be true.
Challenge: Use
the setup and
deductions to
locate the
answer that
could be true.
(A) X was questioned on day one.
(B) V was questioned on day two.
(C) Z was questioned on day four.
(D) W was questioned on day five.
(E) S was questioned on day six.
If you are confronted with an absolute question
this early in a game, it’s a test for your deductions. If you
can’t easily identify the answer, it’s time to rewind and go
Ninja Note:
look for more.
(A) X cannot be questioned on day one since X must come after Z.
(B) As long as Z is questioned on day one, then V could be
questioned on day two. (B) could be true and is the correct
answer.
(C) Z must be questioned on either day one or day two.
(D) The big early deduction places W on day six.
(E) S must be questioned on day seven, so this is the final loser.
Question #13
13. If Z was the second suspect to confess,
then each of the following statements
could be true EXCEPT:
This is the first conditional question for this game. The new condition is that
Z is the second suspect to confess.
This one is easy. Z is the
second suspect, so you just go ahead
and plug that Z into the second slot.
Cleetus Comment:
Careful, Cleetus. The question actually says that Z is the second suspect to
confess. That doesn’t necessarily mean that Z is the second suspect to be
questioned. If some of the other suspects don’t confess, Z could be
questioned later than second and still be the second suspect to confess. It’s
your turn again. Time to figure out how this one works. Turn the page for
another challenge.
Challenge : Build a hypothetical where Z is the second
suspect to confess. Then, try to find the correct answer
(must be false).
(A) T confessed.
(B) T did not confess.
(C) V did not confess.
(D) X confessed.
(E) Y did not confess.
Since Z is the second suspect to confess, Z can’t be questioned on day 1.
Z also can’t be questioned on day 4 or day 5. Thus, Z must be
questioned on day 2.
Z is the second suspect to confess, so the suspect who is questioned on
day 1 must also confess. Two more bad guys off the streets.
Both X and V are questioned after Z, so there should be linked options
for them in slots 4 and 5.
Y is the only suspect left, so Y must be first.
(A) Even with these
deductions, it’s not clear
whether T confessed or
not.
(B) Ditto.
(C) Since V could be questioned on day 4, it could be true that V did
not confess.
(D) Since X could be questioned on day 5, X could confess.
(E) Y must be the first suspect questioned, and Y must confess.
Bam.
Question #14
14. If Y was questioned after V but before
X, then which one of the following could
be true?
On this question, we are given an additional ordering restriction. Y must be
questioned after V but before X. The first objective is to incorporate this
information with the earlier ordering rules.
Z must be questioned before V, so Z can also
be tacked onto the front of the new ordering
chain.
Next up, apply this fancy new ordering chain to your setup.
Challenge : Fill in the remaining slots, and find the
answer that could be true.
(A) V did not confess.
(B) Y confessed.
(C) X did not confess.
(D) X was questioned on day four.
(E) Z was questioned on day two.
To satisfy the new ordering rule, Z must be questioned on day 1, V must
be questioned on day 2, Y must be questioned on day 4, and X must be
questioned on day 5.
(A) V must be questioned
on day two. However, it
is not clear whether V
confessed or not. Thus,
it could be true that V
did not confess, and (A) is the big winner.
(B) Since Y must be questioned on day four, Y cannot confess.
(C) X is now questioned on day five, so X must confess.
(D) X must be questioned on day five.
(E) Z must be questioned on day one. He learned quickly you can’t run
from the law.
Question #15
15. Which one of the following suspects
must have been questioned before T was
questioned?
(A) V
(B) W
(C) X
(D) Y
(E) Z
This absolute question is a big test of our deductions. Luckily, our deductions
always pass the test. T is questioned on day 3, and you need to find a suspect
that is questioned before T. Well, Detective, it’s your job. Identify the suspect.
Challenge: Capture the
suspect that must be
questioned before T
(you have five seconds
to complete this task).
T must be questioned on day three. Since Z must be questioned before
both X and V, Z must be questioned on either day one or day two.
Z must be questioned before T, so the correct answer is (E).
Question #16
16. If X and Y both confessed, then each
of the following could be true EXCEPT:
Here’s another conditional question. In this one, both X and Y confess. As
with previous questions, the four open spots will be the emphasis here. One
thing to notice is that Y was a random. Now that you know Y confessed, your
job is to see how that interacts with the other deductions. This one is a bit of
challenge, but we think you’re up for it.
Challenge : Now that both X and Y confess, try to
figure out when they (and any other suspects) must be
questioned. There’s a setup waiting for you on the next
page.
In this situation, the big deduction stems from noticing what can’t happen.
Since X and Y both confess, and the fourth suspect doesn’t confess, X and
Y can’t be the fourth suspect questioned.
Z also cannot be questioned on day four, so V is the only option left for
day four.
Z could be questioned on day one or day two (as long as Y is questioned
on day one).
The suspect that was questioned and confessed on day five is either X or
Y.
The only suspect that gets nailed down is V. Each of the three other suspects
(X, Y, and Z) have a couple options. Since this question requires an answer
choice that can’t be true, it’s likely we have gone far enough. Time for the
answers. We are looking for something that must be false :
(A) V confessed.
(B) X was questioned on day
five.
(C) Y was questioned on day
one.
(D) Z was questioned on day
one.
(E) Z did not confess.
(A) Now that V is questioned on day four, V cannot confess. Looks
like we found our answer.
(B) X could be questioned (and confess) on day five.
(C) Y could be questioned (and confess) on day one.
(D) As long as X and Y confess on days two and five, Z still could be
questioned on day one.
(E) Z could be questioned on day one or day two. On either day, Z
could still get out without a confession.
Question #17
17. If neither X nor V confessed, then which one
of the following must be true?
Last question!
The last question is conditional. Now, neither X nor V confessed. (It’s sure
hard to get a read on these suspects - they keep flip-flopping.)
The deductions on this question are similar to the last one. Now that you
know neither X nor V confessed, use that new information to eliminate
possibilities and fill in other players.
Challenge: Try to place some of the remaining players
(V, X, Y, and Z). Also, it would be great if you could
figure out whether they confessed or not.
Cleetus Comment:
Boy, the questions sure seem to get easier as
you go.
That’s the idea, Cleetus, and we are very encouraged (although slightly
surprised) to hear you say that. There are only so many deductions in any
game, so they have to keep going back to the same well time after time.
The suspect questioned on day five must confess. Since X and V now
don’t confess, X and V can’t be questioned on day five.
Y is the only option remaining for day five, so Y is questioned on day five.
Z is questioned before both X and V, so Z is questioned on day one.
There should be linked options for X and V on days 2 and 4.
Since neither X nor V confess, the suspect that is questioned on day two
doesn’t confess.
With all of those deductions in hand,
it shouldn’t be hard to find an answer
that must be true.
(A) T confessed.
(B) V was questioned on day two.
(C) X was questioned on day four.
(D) Y confessed.
(E) Z did not confess.
(A) We still do not know whether T confessed or not, so (A) could be
false.
(B) V could be questioned on day two or day four, so this one also
could be false.
(C) Just like V, suspect X could be questioned on day two or day four.
(D) Y must be questioned on day five, and the suspect on day five
must confess. Thus, Y must confess, and (D) wins in a landslide.
(E) Z is questioned on day one, but it is not clear whether Z confesses
or not.
Woohoo! We made it through our first tiered ordering game. Here are some
keys to this game:
1. An effective setup is always the first step. You need a tier
for each variable set. Here, we had to order the suspects
and the confessions, so we had two tiers.
2. Some of the rules were similar to previous ordering rules,
but a few new rules dealt with the relationship between the
tiers.
3. Always watch for constrained players. W was involved in two huge
rules. As soon as we did a little digging, we found a huge deduction
about W.
4. When you find a big deduction, that normally leads to more. Milk it
for all you can.
5. The setups for tiered ordering games can be big and scary, but you
will always find big deductions. Just keep looking.
Now, it’s time to try one on your own. Once you are mentally prepared, turn
the page and try the next game from start to finish.
DECEMBER 2002: GAME 1 (1-5)
Questions 1-5
Eight files will be ordered from first to eighth. Each file falls into exactly one
of three categories: red files (H, M, O), green files (P, V, X), or yellow files
(T, Z). The files must be ordered according to the following conditions:
H must be placed into some position before O, but H cannot immediately
precede O.
X must be placed into some position before V.
X and V must be separated by the same number of files as separate H and
O.
Z must immediately precede M.
The first file cannot be a red file.
1. Which one of the following is an acceptable ordering of the files
from first to eighth?
1
2
3
4
5
6
7
8
(A)
H
X
O
V
Z
M
P
T
(B)
P
M
Z
H
Z
O
V
T
(C)
P
Z
M
H
O
T
X
V
(D)
X
Z
M
V
H
T
P
O
(E)
Z
M
H
P
O
X
V
T
2. The largest possible number of files that can separate Z from H is
(A)
(B)
(C)
(D)
(E)
two
three
four
five
six
3. If each of the three red files is immediately followed by a green
file, which one of the following must be a yellow file?
(A)
(B)
(C)
(D)
(E)
the first
the second
the third
the fourth
the fifth
4. The largest possible number of files that can separate X from V is
(A)
(B)
(C)
(D)
(E)
three
four
five
six
seven
5. If Z is placed in the fifth position, then which one of the following
is a complete and accurate list of the positions, any one of which
could be H’s position?
(A)
(B)
(C)
(D)
(E)
first, third, fourth
first, second, third
second, third, fourth
second, third, fourth, sixth
third, fourth, sixth, seventh
FILLING IN THE FILES
This one started off the Games section in December 2002. There are some
rules that are difficult to visualize, but hopefully you made it through in one
piece.
1. Setup
This is an example of a tiered ordering game in which the players come from
different categories.
The intro lays out the game in a
relatively straightforward
fashion. There are eight files
that must be ordered from first
through eighth. There’s a nice
1:1 correspondence between
the files and slots. However,
the files are different colors,
and this adds a second tier to
our setup.
Important deductions will be
made about the individual files, as well
as the colors, so the second tier is
crucial.
BP Minotaur:
2. Rules
The rules on this game throw you some curve balls, but it is important to
know how to respond. If you represented some of the rules in a slightly
different manner than we do, that’s not the end of the world. But make sure
you understood the implications of each one.
There are a variety of ways to represent this first rule, but this ordering chain
is the best way. The rule starts out by giving you a basic ordering principle H must come before O. However, there is also a division because H cannot
be immediately in front of O. It’s always better to deal with one rule than
two, so represent it like we did. H must come before O, and at least one file
must come between the two.
The second rule is easy (X is before V), but the third one surprises you a
bit. This rule builds on the first two. H must come before O, and X must
come before V. Now, there must be the same number of files between the
two pairs. Since H can’t immediately precede O, X can’t immediately
precede V. The goal is to represent all three of these rules in a manageable
way.
Many students represent these rules by forming two dashes and two
divisions. This is accurate, but complicated. When rules deal with the same
players, always search for ways to combine them.
Now that we’ve gotten past that big hurdle, on to the rest of the rules.
The fourth rule gives you a block with Z and M. As always, that will be
important.
The final rule is a restriction - the first file can’t be red. In other words, it
must be either green or yellow.
At this point, there are very strong rules about six
of the players. You have to find spots for H and O as well
as X and V. In addition, you need to save room for the ZM
block. That’s going to be the big challenge.
Ninja Note:
3. Deductions
In this game, it’s impossible to nail down exactly
where any of the files are placed. However, there are
some important restrictions and other quick notes
which end up saving lots of time.
There are no rules involving either P or T, so those two files are random.
Lots of restrictions...
At least two files must come after H, so H cannot be the seventh or
eighth file.
O must be placed after H and at least one other file. In addition, H
cannot be first. So O cannot be first, second, or third.
At least two files must come after X, so X cannot be the seventh or
eighth file.
At least two files must come before V, so V cannot be the first or second
file.
Because Z must immediately precede M, Z cannot be the eighth file.
The key deductions
in this game are the
restrictions. You
should note that the
first and last slots
are the most
restricted.
Those restrictions are going to be very helpful. But the big challenge that
remains is untangling the knot created by the first three rules. You have two
important pairs of files (H and O, X and V). The pairs must be separated by
the same number of files. For example, if X and V are in spots 3 and 7, you
must have room to place H and O with three slots between them as well.
Once you are done with that, your last task is to find a place for the block
with Z and M.
Sounds easy enough... so let’s check out the questions.
4. Questions
Question #1 (elimination, could be true)
This game starts off nicely with an elimination question. Not only should it
be an easy point, but it’s a great opportunity to double-check your rules.
In (C), H immediately precedes O; that is not allowed. So (C) is gone.
X is placed before V in all of the answers, so the second rule doesn’t
help us out. Darn.
The third rule takes a minute to check, but X and V must be separated
by the same number of files as H and O. Down in (E), X and V are right
next to each other, while H and O are separated by one file. (E) is gone.
Z must immediately precede M according to the fourth rule, and this
rule is broken in (B).
The last rule tells us that the first file cannot be red (H, M, or O). In (A),
the first file is H, so (A) is no good.
The last answer left standing is (D).
Question #2 (absolute, could be true)
This question asks for the maximum number of files that could separate H
and Z. Instead of just blindly plugging away (never a good idea), it’s
important to approach this question with a good plan.
In ordering games, it’s
common for questions to ask for the
maximum or minimum number of slots
that can separate two players.
Ninja Note:
Before we talk about this question, let’s discuss general strategy for this type
of question. Questions with conditions about the maximum or minimum
number of slots that can separate two players are often vexing for students.
For a maximum question, you want to create hypotheticals that space out
the two players as much as possible (slots 1 and 8 in this game). Often
times, that won’t work. The next step is to move the players one spot
closer to each other (either 1 and 7, or 2 and 8). If that still doesn’t work,
keep moving the players closer together until you find an acceptable
arrangement.
For a minimum question, do the opposite. Start with the two players as
close to each other as possible (holding hands or sitting on laps is
encouraged). If that doesn’t work, keep spreading them out. The players
might not like it at first, but you have to keep adding spaces in between
until you get a hypothetical that works.
Now, back to question #2.
One of the big questions is whether, in order to separate the two files by as
many other files as possible, H should be near the front with Z near the back,
or vice versa.
Since H cannot be the seventh or eighth file, we should place H close to
the front and Z closer to the back.
H can’t be the first file
(because it’s red), but H
could be the second file.
Z can’t be the eighth file,
but Z could be the seventh
file (M is the eighth file).
You still have to check out the big rule to make sure that this hypothetical
works. It’s not clear how many spots separate H and O; O could be the
fourth, fifth, or sixth file. There are a number of ways that X and V could
be separated by the same number of files as H and O. For example, if O is
the fourth file, X could be the third file and V could be the fifth file. This
satisfies the rules since both pairs would then be separated by just one file.
Since the rules all check out, you know this hypothetical works.
This hypothetical shows the maximum number of files between H and
Z is four, so the answer is (C).
Question #3 (conditional, must be true)
Here is the first conditional question. The new condition is that each of the
three red files is immediately followed by a green file.
If the three red files are each immediately
followed by a green file, then you are dealing
with the following blocks:
At this point, some students get stuck. It doesn’t
seem clear where any of these blocks will land.
Congratulations! Sometimes the trick to a question
is knowing when you don’t know much. Here, the placement of these
blocks isn’t clear. Our answer is going to have to come from something
else.
The last rule told us that the first file can’t be red. From the very
beginning of this game, the first file had to be either green or yellow.
One of the rules forbid any of the red
files from being first. The first file must
be either green or yellow. But if you
check out the green files, there’s no way
any of them are going first - they all
have to follow red files. All of a sudden, we are very low on options for
the first file. It must be yellow (either T or Z).
A yellow file must be first, so the answer is (A).
Question #4 (absolute, could be true)
This one is similar to question #2. There, we were asked to separate H and Z
by as many files as possible. Here, we are asked for the maximum number of
files that can separate X and V. Just like before, think big and keep moving
them closer until it works.
X can go in slot 1. And V
can go in slot 8. So the maximum in
between ‘em is 6. Easy.
Cleetus Comment:
Not so fast. You have to learn to avoid this mistake on questions asking for
the minimum or maximum number of slots that can separate two players.
While Cleetus is correct that X can be the first file and V can be the eighth
file, that doesn’t necessarily mean that those two things can be true at the
same time.1 You still have to check the other rules and see if everything
works out.
It might appear that X could be the first file and V the last, but this
doesn’t work. If X and V were the first and last files, there is no possible
way for H and O to be separated by the same number of files.
There’s no way to space out H and O by the same
number of files. This is
a big loser. Bummer.
The next step is to move the
two files closer. If X and V
were placed second and
eighth, respectively, there would still be no way to have the same
number of files that separate H and O (since H cannot be first).
The next possibility is first
and seventh. And here we
have a winner. If X is the
first file and V is the seventh
file, therecouldbe 12345678
the same number of slots between H and O.
This hypothetical works out just fine. The maximum number of files
that can separate X from V is five, and our answer is (C).
1 Analogies from real life abound. One can go to Las Vegas. One also can make good
life decisions. You might be aware that it very difficult to do both at the same time.
One can impress members of the opposite sex. One can also wear a fanny pack in
public. Both? Not so much.
Question #5 (conditional, could be true)
Our last question is conditional. However, the answer relies on the
restrictions formed a few steps back (they always pay off down the road).
The task is to identify all of the possible spots for H if Z is the fifth file.
Before you incorporate the new information, it’s helpful to review the
restrictions in place on H. H cannot be the first, seventh, or eighth file.
That eliminates (A), (B), and (E).
As for the new condition, M
must be the file immediately
after Z. If Z is the fifth file,
then M must be the sixth
file. Bye-bye to (D).
The only remaining
possibilities for H are
second, third, and fourth,
so (C) is the correct answer.
Note: You can also approach this question by plugging H into the three
remaining slots (second, third, and fourth). However, try to avoid any
unnecessary work by keeping track of which answer choices are still
in the running.
The last question asks for all
of the possible placements for H. That
sounds challenging. But a true ninja
realizes that you just have to count all
of the slots that don’t work for H.
Ninja Note:
That was a fun ride. By this point, you’ve probably
noticed that deductions come in many shapes and
sizes. In our first tiered ordering game, we figured
out the placement of one of the suspects, and
everything flowed nicely from there. In this game,
the restrictions reigned supreme. The moral of the
story? You don’t know what form the deductions for
a game will take, but you have to diligently search
until you uncover something.
Tiered ordering games are very common on the
LSAT, so it’s important to get lots of practice. To
that end, you will see another game on the next
page. This game is slightly different. The setup is, to
put it simply, big. It’s larger than the last two games.
Tiered ordering games will sometimes have more
than just two tiers. Remember, you have to build a
tier for each variable set that is introduced.
When you’re ready, give the next game a shot. See you
soon.
SEPTEMBER 2009: GAME 3 (13-17)
Questions 13-17
Flyhigh Airlines owns exactly two planes: P and Q. Getaway Airlines owns
exactly three planes: R, S, T. On Sunday, each plane makes exactly one
flight, according to the following conditions:
Only one plane departs at a time.
Each plane makes either a domestic or an international flight, but not both.
Plane P makes an international flight.
Planes Q and R make domestic flights.
All international flights depart before any domestic flight.
Any Getaway domestic flight departs before Flyhigh’s domestic flight.
13. Which one of the following could be the order, from first to last,
in which the five planes depart?
(A)
(B)
(C)
(D)
(E)
P, Q, R, S, T
P, Q, T, S, R
P, S, T, Q, R
P, S, T, R, Q
T, S, R, P, Q
14. The plane that departs second could be any one of exactly how
many of the planes?
(A)
(B)
(C)
(D)
(E)
one
two
three
four
five
15. If plane S departs sometime before plane P, then which one of the
following must be false?
(A)
(B)
(C)
(D)
(E)
Plane S departs first.
Plane S departs third.
Plane T departs second.
Plane T departs third.
Plane T departs fourth.
16. Which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
Plane P departs first.
Plane Q departs last.
Plane R departs second.
Plane S departs first.
Plane T departs fourth.
17. If plane S departs third, then each of the following can be true
EXCEPT:
(A) Plane R departs sometime before plane S and sometime
before plane T.
(B) Plane S departs sometime before plane Q and sometime
before plane T.
(C) Plane S departs sometime before plane R and sometime
before plane T.
(D) Plane T departs sometime before plane P and sometime
before plane S.
(E) Plane T departs sometime before plane R and sometime
before plane S.
GETAWAY GETS GROUNDED
This game has a larger setup than you
have seen before. However, an effective
setup and a few key deductions once again
make the difference. There are two
airlines with very interesting names: the
infamous Flyhigh Airlines1, and the more
appropriately named Getaway Airlines.
1. Setup
You have to do quite a bit of reading before you can truly visualize the setup
for this game. At the onset, all you are told is that there are two different
airlines. Flyhigh has two planes, and Getaway has three planes. No wonder
you’ve never heard of them. They all have to take off on a certain Sunday. It
sounds like they are hinting at an ordering game, but it’s still pretty
ambiguous.
If a game doesn’t give you
much information in the introduction,
just wait until you read a couple rules
to visualize the best setup.
Ninja Note:
At this point, we think we have an ordering game, but the number of slots
isn’t quite clear. Hypothetically, planes
could take off at the same time (unlikely
and dangerous, but possible ). It’s
important to note that the planes do come
from two different airlines, so this game
features players from different categories.
We can see tiers in our future. But for now,
the setup is pretty lame.
2. Rules
As expected, the first
couple rules help to
define the nature of
this game. At this
point, you need to
build a setup to help
visualize all of the
variable sets in this
game.
The first rule states that only one
plane departs at a time. Perfect we’ve got a 1:1 game. There
should be two tiers and five
slots. The first tier is for the plane
and the second tier is for the
airline.
The second rule introduces another variable set. Some of the flights are
international, and others are domestic. (Apparently, Flyhigh does a lot with
those two planes.)
This is truly a momentous
occasion. It’s our first game with three
tiers. The first tier is for the plane, the
second is for the airline, and the third
tier will track whether each flight is
domestic or international.
BP Minotaur:
In this game, the real setup isn’t revealed until you get through the first two
rules. That’s not very common, but don’t let it throw you. Now we know we
have a tiered ordering game with three tiers. That’s a lot of slots. Don’t be
intimidated by the size; we should be able to find lots of deductions to
simplify things.
Back to the rules.
These two rules provide you with three great blocks. Make sure to include the
airline in the block. This gives you less to think about and makes it easier to
visualize exactly how the block will fit into your setup.
P and Q are the only two Flyhigh planes. So
Flyhigh makes one international flight (P) and one
domestic flight (Q). Keep this in mind as we search for
Ninja Note:
deductions.
It’s crucial to combine these two rules. Any and all international flights
take off before the first domestic flight. Then, once the international flights
are set, all Getaway domestic flights depart before the first Flyhigh
domestic flight.
See how the final rule refers to
Flyhigh’s domestic “flight”? If you
missed it before, here’s another hint
that there’s only one.
Ninja Note:
That’s the end of the rules. However, we’ve already seen some hints that
there will be big deductions, so it’s time to investigate further.
3. Deductions
There is one humongous, game-changing deduction based on the last two
rules.
The last two rules establish an order for the types of flights. There are
two airlines, and they both could make international and domestic
flights. But the Flyhigh domestic flight has to depart later than any
international flights (by either airline) and any domestic Getaway
flights. So any Flyhigh domestic flights have to depart last.
Flyhigh only has two planes: P and
Q. P makes an international flight,
and Q makes a domestic flight. The
last flight to depart must be a
Flyhigh domestic flight. Since Q is
the only Flyhigh domestic flight, Q
must be the last plane to depart.
When you find a huge deduction like this one, it’s time to see if the
remaining rules interact with it in any way.
Since all international flights have to
depart before any Getaway domestic
flights, plane P (international flight) must
depart before plane R (Getaway domestic
flight).
P cannot be the fourth plane to depart
(remember Q is fifth), and R cannot be
the first plane to depart.
Planes S and T are not involved in any of the rules; they are random.
Don’t stop there. Here are a couple more helpful points to note:
The only planes that could
depart before P are S or T, or
both. If either of these Getaway
flights depart before P, they
must be international flights. If
they depart after P, they could
be either international or
domestic.
The first plane must be an
international flight, and the fourth plane must be a domestic Getaway
flight.
With all of that information at our disposal, bring on the questions.
1 Is this some sort of joke on the part of the LSAT writers? Fly “high”? Did a gangster
rapper name this airline? Hopefully the hobbies of the pilots don’t coincide with the
name of the airline.
4. Questions
Question #13 (elimination, could be true)
In a not-so-shocking development, the first question here is an elimination
question.
The first thing to check is our big deduction. Q must be the last plane to
depart. Say good night to (A), (B), and (C).
Our secondary deduction states that P must depart before R. In (E), P
departs after R, so (E) is gone.
That was quick. The correct answer is (D).
Question #14 (absolute, could be true)
Next up, an absolute question. The challenge is to identify how many of the
five planes could depart second.
Q must be the last plane to depart, so Q definitely can’t be second.
P could easily be the second plane to depart if either S or T is first.
If P is the first plane to depart, then the Getaway flights are pretty
flexible. In that situation, any of the Getaway flights (R, S, and T) could
be the second to depart.
That makes a total of four planes (P, R, S, and T) that could depart
second. The correct answer is (D).
Question #15 (conditional, must be false)
This conditional question stems from our earlier deductions. The new
condition is that S departs before P.
Since plane P must depart before plane R, if
plane S departs before P, then plane S must
depart either first or second.
If plane S departs before plane P, then S must
make an international flight.
I drew out three hypothingys
and it took me like 12 minutes!
Ditz McGee:
Well, that’s definitely not the goal. This is always a judgment call. If you can
visualize the ordering restrictions, then don’t take the time to build
hypotheticals. Here, as long as you can see that the planes are very restricted
(S must be first or second, P must be second or third, R must be third or
fourth), and you notice that T is still a big wildcard, then you should be fine
to test the answers. The ordering chain itself should be sufficient.
(A) S could depart first, and plane P could still depart second or third.
(B) Plane S always has to depart before Q. For this question, plane
S must also depart before both P and R. Therefore, the latest S
could depart is second. If S departs third, there wouldn’t be
room for both P and R. (B) must be false, and it’s the correct
answer.
(C) Plane T could depart second as long as plane S departs first and
plane P departs third.
(D) Plane T could depart third as long as plane S departs first and
plane P departs second.
(E) Plane T also could depart fourth (talk about freedom) as long as S
is first, P is second, and R is third.
Question #16 (absolute, must be true)
Big deductions will always result in a huge payoff. Here, we need to find an
answer that must be true. This one comes from the one major deduction we
were able to make.
Since Q is the only Flyhigh domestic flight, Q must be the last plane to
depart. (B) is the winner. That feels nice.
Question #17 (conditional, must be false)
The last question of this game is challenging. The new condition is that plane
S departs third.
If plane S departs third, plane P could still depart either first or second.
Since this question is looking for an answer that must be false, it’s now
advisable to draw out both possibilities.
You don’t have a nice
ordering chain to help you out, so just
take a minute and jot out the two
hypotheticals.
Ninja Note:
If plane P departs first, then you
can make linked options for planes
R and T for the second and fourth
departures.
The second and third flights could
still be either international or
domestic.
If plane P departs second, T must
depart first and be an international
flight.
Plane R must depart fourth.
(A) In the first situation, plane R could
depart before both planes S and T if
plane R is the second to depart.
(B) In the first situation, plane S could
depart before both planes Q and T, as
long as T is the fourth plane to depart.
(C) Plane S can’t depart before both planes R and T in either
situation. If S is the third plane to depart and Q must be the
fifth plane to depart, then there is only room for one more
plane to depart after plane S. (C) must be false, and (C) is the
answer.
(D) In the second situation, plane T is the first to depart. So plane T
could depart before all of the other planes, including both planes P
and S.
(E) See the explanation for (D), but sub in planes R and S at the end.
Well, that was quite a ride.
Luckily, we landed safely (as
opposed to many Flyhigh
passengers, we can only
assume). On the September
2009 LSAT, that was a big
swing game. Some students
identified the huge deduction
and left plenty of time for the
fourth and final game. Other students stumbled and burned valuable time in
the not-so-friendly skies.
Here are some important lessons learned from this game:
1. You might read the introduction to a game and feel very
underwhelmed. In a few games, you won’t be able to build a setup
or discern the basic task in a game until you check out a few rules.
2. This is the first time you had three tiers. Make sure to build a tier
for every variable set in a game so that you can visualize the
information well.
3. Some games feature a make-or-break deduction. Here, it was
spotting that plane Q had to be the final flight.
At this point, we should also discuss the end of our journey through tiered
ordering games. They are very common games, so it’s important to practice.
Make sure to review this chapter, but here is a quick overview of the strategy
for tiered ordering:
1. Some ordering games involve more than two variable sets. When
this occurs, continue to identify the variable set with an inherent
order and use that as the base of your setup. Then, build a tier for
each additional variable set.
2. Tiered ordering games come in three varieties: (1) players from
different categories, (2) players with characteristics, and (3)
independent variable sets. The setups will look similar, but the
games will function in slightly different ways.
3. There aren’t many new types of rules. However, you can expect to
see some big, oddly- shaped blocks and conditions that deal with the
relationship between tiers.
4. Tiered ordering setups are ginormous.2 Don’t panic. There will
always be big deductions to simplify the game.
Finally, and most importantly, this is a huge moment for us. We have reached
the end of the world of ordering. As we mentioned earlier, there are only two
basic processes that rule all games on the LSAT - ordering and grouping.
Ordering is now behind us. Congrats. But there is more work to be done.
Up next: The wide, wide world of grouping. Cue
dramatic music...
2 Gigantic + enormous = ginormous. It’s just math.
15/helloGROUPING
PART DEUX
Welcome to the second act, also known as the world of grouping games. Just
when you were getting comfortable with ordering games, we go and switch it
up on you. Remember, the basic Blueprint strategy relies on classifying
games according to two basic processes: ordering and grouping.
Grouping games are neither
more challenging nor less challenging
as a whole than ordering games.
Students are generally split on which
game type they find to be more
difficult.
BP Minotaur:
Grouping games present an entirely new challenge. Ordering games are
centrally concerned with spatial relationships. Who goes in slot 5? How early
can Shelly appear? Can Bruce and Wayne be next to each other? Your mind
has been filled with thoughts about the relative positions of the players in a
game for some time now. As of this chapter, all of those concerns fade into
the background (they will be back later). Grouping games challenge you to
assign players to different teams. The important questions will now center
around whether players can or cannot be selected together.
Can Charles be assigned to the green team?
Which ingredients can’t be included in the same
recipe?
If Polly is in the ceramics class, can Hank be in the
painting class?
How many squirrels can be selected?
In ordering games, your job was to line things up.
Now, your focus is on making teams.
Our strategy for grouping games will be very similar to ordering games. First,
we will learn to represent and interpret all of the rules. Then we will work
through the different permutations of grouping games. And our focus the
whole time will be on spotting common and powerful deductions.
Grouping games are nearly as
common as ordering games, and you
can bet you will see at least one on
your LSAT.
Ninja Note:
PICKING TEAMS
Every grouping game involves selecting players for a group or assigning
players to different groups. You might be selecting friends to invite to a
party, or fish for an aquarium. You might be picking squads for a
cheerleading competition, or assigning plumbers to different work teams. But
the basic process remains the same: grouping.
Each of the following is an example of a grouping game:
Each of two boats, boat 1 and boat 2, will
be assigned exactly four people. Exactly
eight people, three adults—F, G, and H—
and five children—V, W, X, Y, and Z—
must be assigned to the boats according to
the following conditions:
A panel of five scientists will be formed.
The panelists will be selected from among
three botanists—F, G, and H—three
chemists—K, L, and M—and three
zoologists—P, Q, and R. Selection is
governed by the following conditions:
Each of five students—Hubert, Lori, Paul,
Regina, and Sharon— will visit exactly one
of three cities—Montreal, Toronto, or
Vancouver—for the month of March,
according to the following conditions:
Five children—F, G, H, J, and K—and
four adults—Q, R S, and T—are planning
a canoeing trip. The canoeists will be
divided into three groups—groups 1, 2,
and 3—of three canoeists each, according
to the following conditions:
To prepare for fieldwork, exactly four
different researchers—a geologist, a
historian, a linguist, and a paleontologist
—will learn at least one and at most three
of four languages—Rundi, Swahili,
Tigrinya, and Yoruba. They must learn the
languages according to the following
specifications:
Note the wording that
introduces grouping games. Rather
than referring to “consecutive” or
“adjacent” positions, these games
“assign” or “divide” the players.
BP Minotaur:
Just like ordering games, the actual subject matter of grouping games doesn’t
matter at all.1 Now that you know what a grouping game will sound like, it’s
time to talk strategy.
1 No need to book a trip to Canada, buy a canoe, or find a Yoruban dictionary. None
of that will help you much on the LSAT (or in life).
THE SETUP
Building a consistent setup is a key step in approaching any game. If you
build a different setup for each game, it will be hard to recognize their
repetitive nature. In every grouping game, you will have a set of players
(students, chairs, toys, power tools) that is being assigned to groups (classes,
rooms, cabinets, hardware stores). There could be one group, two groups,
three groups, or more. However, there will always be one variable set that is
being assigned to another.
The groups should always form the base of your setup.
Arrange the groups vertically and then insert players
into the groups.
Consider the following game:
Five children—F, G, H, J, and K—and
four adults—Q, R S, and T—are planning
a canoeing trip. The canoeists will be
divided into three groups—groups 1, 2,
and 3—of three canoeists each, according
to the following conditions:
In this game, it’s time for an amazing canoe trip! (Excitement greatly
exaggerated.) There are a total of nine people (five children and four adults).
These lovely folks are being assigned to three different groups. The groups
(1, 2, and 3) should form the base of your setup. An appropriate setup for this
game would look very similar to the one drawn to the left. Now, it will be
easy to visualize which people are together and how many people are in each
group.
Don’t overlook the setup phase of any game. When you have the correct
setup, visualizing the game will be easy, and deductions will come early and
often. A bad setup can doom you to a painful section and LSAT mediocrity.
Students commonly try to
build a grid to organize grouping
games. Grids have numerous flaws and
should always be avoided. Attacking a
game with a grid setup is like going to
war with a squirt gun.
Ninja Note:
There are some elements that are important to note in grouping games. When
you first recognize a grouping game, noticing some salient features will help
you determine your strategy. Turn the page and let’s discuss the three big
ones.
To investigate the subtleties of grouping games, we will use a set of eight
students. Let’s call them Alf, Babak, Chuckford, Delta, Ebenezer, Farrah,
Gary, and Harry (in proper alphabetical order). That’s quite a crew.
In each grouping game, your basic approach will be informed by the answer
to each of the following three questions. These features will also foreshadow
the types of deductions you are likely to uncover.
ONE GROUP OR MULTIPLE GROUPS?
The first distinction that must be made is based on the number of groups
being formed. This distinction can drastically change how you build your
setup and how the rules in a game function.
In some games, you will only be selecting one group. Certain players will be
chosen and others will be left behind.
Eight students are trying out for a high
school softball team. There are exactly five
spots available on the team, so a coach
must select five of the eight students to fill
the roster spots.
In this game, the task is to select five of the eight students. There is only one
group being selected: the students that make the softball team. Many
grouping games will ask you to select some members of a larger group. You
might have to select some friends to be part of a wedding, or some types of
cookies for a cookie jar, or some actors to be on stage during a play, but the
common feature is that you are just selecting one group.
At Blueprint, we call these In and Out grouping games. The name is
designed to remind you of the basic task in the game. Some players are
being selected and others aren’t. The rules and deductions will focus on the
players that are selected, but it’s also important to track those that are not
selected.
To track all of the players, we will build a setup with two groups: an In
group (for the players that are selected) and an Out group (for the players
that don’t make the cut). Our setup for the game above would look like
this:
The majority of questions in
a game like this one will focus on the
players that are selected. But important
deductions will arise from knowing
who isn’t selected.
BP Minotaur:
As you can imagine, the rules for In
and Out games will place
restrictions on which players can,
must, or cannot be selected
together. Here are some rules you
should expect to see:
Chuckford and Ebenezer
cannot both make the team. If
Farrah makes the team, then
Gary also makes the team.
In other grouping games, you will be challenged to separate players into
multiple groups. Red team or blue team? Good or evil? Tall, grande, or
venti? In the previous game, only some of the players made the team. In
these games, everyone makes a team - your job is to figure out which
one.
Here’s an example. In this game, the students are being assigned to three
distinct groups (team X, team Y, and team Z). You will run into games that
include two, three, four, or even more groups.
Eight students are separated into three
different teams to compete in a trivia
challenge. The teams will be named team
X, team Y, and team Z. Each student must
be assigned to one of the teams, and each
team must be assigned at least two
students.
These are called multiple-group grouping games.
The first big change is your setup.
Rather than having an In group and an
Out group, your setup now must
reflect the different groups. And just
because we like to remind you about
things over and over and over again,
always use the groups as the base of
your setup.
The rules in a game like this will also be slightly different. Rather than
principles about which players can and cannot be selected, the rules will
focus on the players that can, must, or cannot be assigned to each team.
Some rules will be general principles that are applicable to every group,
and other rules will focus on just one of the groups. Here are a few
examples:
If Alf is assigned to team Z, then Babak
must be assigned to team Y.
Ebenezer and Harry cannot be assigned to
the same team.
The distinction between In
and Out grouping games and those
with multiple groups is so fundamental
that we will cover these games in
separate chapters.
BP Minotaur:
STABLE OR UNSTABLE GROUPS?
In all grouping games, it’s important to watch for restrictions placed on the
sizes of the groups. Some games will define exactly how many players are in
each group, and other games will place little or no restrictions on the sizes
of the groups. To put it simply, sometimes you will know that the red team
has four members, and other times you won’t know how many members it
has.
Here is an example taken from the high pressure world of competitive
spelling.1 In this game, it is established that each group has four players.
Two four-person teams, the Polygons and
the Quadrilaterals, will be selected for a
spelling bee. The teams will be selected
from a group of eight students. Each
student will be assigned to only one team.
A game in which we know the exact size of each group is called a stable
grouping game. The sizes of the groups remain stable throughout the game
and do not fluctuate. With all of the moving pieces in games, it’s nice to
know that at least one thing isn’t going to change.
Your setup should always reflect
any information that you are given
about the groups. In this one, we
can insert four slots for each of the
two teams. Here is an effective setup for this game:
Groups can be the same size or different sizes,
but you can reflect either in your setup. In a stable
Ninja Note:
grouping game, important deductions
will occur when you run out of spots in
one of the groups.
A stable grouping game specifies how many players
are in each group.
There will, of course, be other grouping games in which you will not be
furnished with this information. Some games will only put partial
restrictions (at least two, at most four) on the sizes of the groups. Other,
even more terrifying games will place no restrictions on the group sizes
at all. This doesn’t mean the game is more difficult. We just need to use a
slightly different approach. Check out the example on the next page.
A chef is creating a new main dish for a
special dinner party. She must include
some of the following eight ingredients in
her new dish.
First, notice that this is an In and Out grouping game. A chef must select
some ingredients and not others. It is not clear, however, how many
ingredients are included. There could be just one ingredient (a relatively
unimpressive dish), three ingredients, or possibly all eight ingredients. Any
and all restrictions should be represented in your setup. In this case, however,
there’s not much to work with. All we know is that there must be at least one
ingredient in the In group.
But you know that not all of
the ingredients are in the dish, so you
can put some slots in the Out group,
too.
Ditz McGee:
Actually, that’s not the case. The setup to this game states that “some” of the
ingredients are included in the dish. The chef might decide to just throw all of
the ingredients together and see what happens.2 The rules in this game will
likely rule out that possibility (by stating, for example, that two ingredients
cannot both be included), but there are grouping games in which some groups
can be empty.
This is called an unstable grouping game. This does not refer to your
mindset while taking the LSAT; rather, it refers to the fact that the group
sizes can change during the game. There might be four ingredients included
in the dish on one question and six ingredients on the next. It’s important to
think about the range of possibilities for each group.
This game has almost no restrictions, but other
games will place partial restrictions on the sizes of the
groups. For instance, the chef could use “at least three” or
BP Minotaur:
“at most five” ingredients. Your setup
should always adjust to these
restrictions by placing the correct
number of slots in each group.
An unstable grouping game does not specify exactly
how many players are in each group.
Two questions down, one to go...
1 If you haven’t had the pleasure of watching the national spelling bee on ESPN8, we
highly recommend it.
2 The word “some” is consistent with “all” on the LSAT. If you’re told that some of
the people at a party drank mojitos (not an example you are likely to deal with on the
LSAT), it could be true that everyone enjoyed the minty goodness. This comes up
much more often in Logical Reasoning, but it’s worth noting here.
CATEGORIES OR NO CATEGORIES?
The last distinction which will prove to be important in grouping games
relates to the actual players in the game. In most games, you will be
presented with a set of interchangeable players that don’t come from different
categories.
In this one, we see our familiar eight students. This game has four distinct
groups, and each one is assigned two players. There is nothing that
distinguishes one player from the next (other than their fascinating names).
Thus, you can just write out the variables.
Eight students—Alf, Babak, Chuckford,
Delta, Ebenezer, Farrah, Gary, and Harry
—will be assigned to four different teams
for the upcoming sack race. Each team will
be assigned two of the eight students and
each student will be assigned to only one
team.
In other grouping games, the players will actually come from different
categories. Males and females. Tall and short. Spicy, tart, and sweet. Irish,
German, and French. When this occurs, it is important to note this early on
because it adds a level of complexity to any game.
Here is an example of a game in which the players come from different
categories. This is an In and Out grouping game. It’s a little unstable
because you are selecting either five or six of the eight students. The new
characteristic is that the players come from different categories (freshmen,
sophomores, and juniors). This needs to be part of your setup when you are
symbolizing the variable sets.
There are eight students campaigning for
positions in student government. Three of
the students—Alf, Babak, and Chuckford—
are freshmen, three students—Delta,
Ebenezer, and Farrah—are sophomores,
and two students—Gary and Harry—are
juniors. Either five or six of the eight
students will be elected.
When a game introduces different categories, you can expect different types
of rules to pop up. There will be the normal rules about players that can or
cannot be selected together, but there will also be restrictions about how
many players can be selected from each category.
In this game, you will have to track (1) which players are elected, and (2)
how many freshmen, sophomores, and juniors are elected. Here are a few
of the different rules that you could encounter:
If Farrah is not elected, then Delta must be
elected. At least one student from each
class must be elected. There must be more
sophomores elected than either freshmen
or juniors.
Next up is a drill to make sure you got it.
Grouping Game Drill
This drill is designed to reinforce the grouping distinctions we just covered.
For each one of the following games, circle (1) whether the game is selecting
just one group or multiple groups, (2) whether the group sizes are stable or
unstable, and (3) whether the players come from different categories or not.
Then, use that information to build the appropriate setup.
1. A college student is packing for a semester abroad. Her
suitcase is filling up quickly and she can fit no more than
five of the following seven items—camera, deodorant,
fanny pack, mascara, passport, toothbrush, and visor. Her
selection must accord with the following:
2. Seven employees of Stigmata Corp—Abigail, Benji,
Candace, Duane, Ernie, Fergie, and G—are each assigned
to one new project. One employee is assigned to the Mars
project, three are assigned to the Pluto project, and three
are assigned to the Venus project. The following must
obtain:
3. Five males—Javier, Kyle, Laird, Matt, and Nin—and
four females—Ophelia, Paula, Robin, and Sally—are
divided according to their reading abilities. Each person is
classified as remedial, intermediate, or advanced,
according to the following:
4. A circus act will include exactly five animals. The
animals must be chosen from among three lions—F, G,
and H, three tigers—J, K, and L, and four bears—W, X,
Y, and Z. The selection must be consistent with the follow
restrictions:
Grouping Game Drill Answer Key
Here’s what your setups should look like, along with an explanation. If you
missed anything, make sure to look back and review the important features of
each game.
1. Pack the Fanny Pack?
This is an In and Out grouping
game. Our student must pick some
items to be packed and must leave
others behind (hopefully not the
deodorant).
The group sizes are unstable, although you should put two slots in the
Out group since you know she can only take a maximum of five items.
There are no categories.
2. Men are from Mars...
There are three separate groups
in this one - Mars, Pluto, and
Venus.
The group sizes are stable, even
though the groups are different
sizes.
There are no categories.
3. Center for Reading Good
This game also has three distinct groups which are the three different
reading levels.
The group sizes are completely
undetermined, so this game is
unstable.
There are two different
categories in this one - males
and females.
4. Lions, Tigers, and Bears. Oh my!
This is an In and Out game
– you are selecting one set
of animals for the act.
Exactly five animals are
selected, so this is a stable
grouping game.
There are three different
categories.
That wraps up our introduction to grouping games. This might seem to be a
lot to think about when you are attacking a game. That’s because it is. But
these distinctions will become automatic with some good practice. Speaking
of good practice, let’s keep moving.
16/theBIGfour
GROUPING RELATIONSHIPS
Before we actually attack grouping games, it’s important to check out the
rules. In ordering games, there were some rules we kept seeing over and over
again. The same will be true in grouping games. So we have to make sure to
understand (1) how to represent the rules and (2) how to use them to spot
deductions.
The rules in grouping games
are even less original than they were in
ordering games. There are only so
many ways that you can say two
players can’t be on the same team.
Ninja Note:
In ordering games, you will run into a conditional rule every now and then (if
J is in spot 3, then K must be in spot 5). However, they are more of an
afterthought to the blocks, arches and options. In grouping games,
conditional rules play a much more prominent role. In fact, grouping
games are based entirely on rules that restrict which players can be selected
or assigned together. These restrictions are generally introduced with
conditional rules.1
Here is a medley of rules you might see in a grouping game. These are all
conditional rules. Some games will present you with five or six of these
bad boys, so it’s crucial to understand how they work.
If A is selected, then B is also selected.
C and D cannot both be on team X.
E cannot be in the first class unless F is in
the second class.
If G is uncovered at the site, then neither H
nor J is uncovered at the site.
Here’s the great news: Each of these rules (and all the others just like them)
can be simplified. It might seem that grouping games introduce a huge
number of different relationships. But that’s not the case. Not even close. In
fact, every single conditional rule you encounter in a grouping game can be
broken down into one of four basic grouping relationships.
Four distinct grouping relationships dominate the rules
in grouping games.
Now it’s time to take a look at the big four. Through years of teaching the
LSAT, we at Blueprint came to notice that the big four grouping relationships
parallel the four stages of a romantic relationship. Weird. But you have likely
experienced some, most, or even all of these stages in your own relationships,
so the analogy can be helpful. Bet you never realized your ugly breakup
would help you understand grouping games on the LSAT, huh?
Warning: This is not the classic love story. We are not
promising happily ever after. Just like real life, there
are some bumps in the road.
We need two characters for our little love story. A boy and a girl - let’s go
with Bobby and Suzy.
Stage 1: Stalker
A number of grouping rules will establish that one player always follows
another player. This is appropriately called the stalker phase of the
relationship.
Time for our analogy. You might deny it, but we’ve all
been there. You have a little crush, and that turns into
some light stalking. Bobby hears that Suzy is going to be
at a certain party, and he changes his plans so he can
attend as well. Bobby stands close by (but not too close),
hoping to strike up that “impromptu” conversation.
Nothing too creepy, but he doesn’t want to miss his
chance. Our little love story starts out in the traditional
mold. Boy chases girl. That’s how every great relationship
begins. However, does that mean Suzy goes everywhere
that Bobby goes? Not so much. She doesn’t even know he
exists at this point.
Here’s the basic form of this rule in a grouping game: If one player is
assigned to a certain group, then another player must also be assigned to
that group. We call this the stalker phase to emphasize the fact that one
player follows another, but it does not go both ways. No one ever stalks their
stalker. Here are examples of grouping rules that introduce this relationship:
If Suzy performs at the concert, then
Bobby must also perform.
This rule would be part of an In and Out grouping game. If Suzy performs at
the concert, then Bobby must also perform. But it could be true that Bobby
performs and Suzy does not. Remember to always write out the
contrapositive. If Bobby doesn’t perform, then neither does Suzy.
Here is another example. This one would originate from a game with multiple
groups.
If Suzy is assigned to the traffic
committee, then Bobby is also assigned to
the traffic committee.
This would be a different type of game, but a very similar rule. This game
would involve assigning people to different committees.
Once again, we want to diagram the conditional
rule and its contrapositive. If Suzy is assigned to
traffic, then Bobby follows along. And if Bobby
is not assigned to traffic, then neither is Suzy.
That sure looks like they
have to be together.
Cleetus Comment:
We didn’t take you for a romantic, Cleetus. That is a very common mistake,
and one that’s important to avoid. If Bobby is assigned to the traffic
committee, we have no information about Suzy. She could be assigned to the
traffic committee, but she could also be assigned to another committee (like,
say, the crime committee that focuses on arresting stalkers). With a simple
conditional relationship, don’t assume that the two players must be
together.
It’s also possible for one person to be very popular and have two stalkers. But
even when a rule involves more than two players, it’s important to recognize
the basic relationship.
If Suzy attends the review session, then
both Bobby and Ralph will also attend.
Uh-oh. Suzy just picked up another stalker. This rule includes three players,
but the principles are very similar to the previous rules. If Suzy heads to the
review session, then both Bobby and Ralph are sure to follow. If either
Bobby or Ralph does not attend the review session, then Suzy must not be
there.
Remember those flaws we
discussed earlier. If Bobby or Ralph
attends, that doesn’t mean Suzy must
attend. Even if both Bobby and Ralph
attend, Suzy might choose to skip it.
And if Suzy doesn’t attend, that
doesn’t help us either.
BP Minotaur:
That wraps up the first stage of our relationship. You’re probably wondering
where we could go from here. Restraining order? Jail time? Stalking doesn’t
always turn into something special. But think positive. Here comes the good
part...
Stage 2: Love (Must Be Together)
Grouping rules will also commonly state that two players must always be
assigned to the same group. Sometimes, you can’t have one without the
other.2
Woohoo! Here’s the happy part of our story. Bobby
finally gets his chance. He wins her over, and they fall
head over heels in love. Now they’re attached at the hip.
They quickly become the annoying couple that holds
hands in public and has nicknames like “honey bear” and
“cuddle bum.” They’re always together.
In grouping games, many rules will establish that two players must always be
together. This introduces a reciprocal relationship in which the players
can never be separated. Here are some examples of the love relationship.
Suzy reviews the movie if, but only if,
Bobby also reviews the movie.
This rule could be part of a game in which some people review a movie (at
this stage in the relationship, most likely a romantic comedy). As soon as you
know that either Suzy or Bobby reviews the movie, it is safe to conclude that
they both do so. This relationship should be diagrammed with the
reciprocal (double-sided) arrow.
Just like every other
conditional rule, this one has a
contrapositive. If either Suzy or Bobby
reviews the movie, the other one must.
Also, if either Suzy or Bobby does not
review the movie, then the other one
does not. Here’s the simplest way to
interpret a rule like this one: You either get both of them,
or you get neither one.
Ninja Note:
This type of rule is much more common in games where players are being
segregated into multiple groups. Here comes an example.
Suzy and Bobby attend the same monster
truck rally.
In this one, Bobby and Suzy are part of a classy group choosing which
monster truck rally to attend (always a tough choice). Suzy and Bobby likely
bought matching outfits for the occasion, so we have to keep them together.
The rule should again be symbolized with the reciprocal arrow.
These fools in love will help you with deductions.
You will often find that there are not many groups that
have two open spots.
Ninja Note:
Did you ever think that games could be so romantic? Make sure to never
confuse the stalker relationship with love (in life and on the LSAT). Stalker
goes one way. Love is reciprocal.
That brings an end to our second grouping relationship. But here is where
things go terribly wrong. As with most relationships, there are some rocky
patches. And, well, let’s just say that a storm is brewing...
Stage 3: Hate (Cannot Be Together)
The third type of rule in grouping games will specify that two players cannot
be selected or assigned to a group together. You must keep them apart.
At this point, things start to fall apart. Rumors circulate
about Suzy spending too much time with Antonio. Bobby
starts to suspect that Antonio is more than just a friend,
and the fighting begins. Eventually, the fights begin to
look more like cage matches. Yelling, crying, and
unflattering videos on the internet quickly follow. This
couple is on an express train to Break-up-ville. Just like
that, our happy couple falls into the hate phase of the
relationship. To avoid any violent outbreaks, they must be
kept apart.
In grouping games, rules will very commonly give you combinations of
players that cannot be together. This type of rule will pop up when you are
selecting one group or assigning players to multiple groups. Below are a few
examples.
The hate relationship is the
most common one in grouping games.
So much hate, such a shame.
BP Minotaur:
Suzy and Bobby cannot both attend the
ceremony.
This would be an In and Out game in which some people attend a fancy
ceremony. Suzy and Bobby are not getting along at this point, so they cannot
both attend the ceremony. To represent this relationship, we will negate the
reciprocal arrow used for the love relationship.
Now... here comes a little twist. This rule states that Suzy and Bobby cannot
be together, which we correctly recognize as the hate relationship. However,
grouping games can introduce the same exact relationship in a couple of
different ways. It’s crucial to always simplify each rule and recognize the
basic relationship. Let’s say they gave you this rule instead:
If Suzy attends the ceremony, then Bobby
does not.
According to the original, if Suzy attends, then Bobby does not. The
contrapositive tells you if Bobby attends, then Suzy does not. What does that
mean? They can’t both attend. This rule is synonymous with the first one,
so you should represent it in the same way.
With some practice, you will
learn to spot different versions of the
same relationship automatically.
Ninja Note:
Just like you can sometimes have more than one stalker, you can also have
more than one hater. You know how that works. The breakup goes down, and
it’s not long before her friends and family are giving you dirty looks. Ouch.
Grouping games will commonly express that one person must stay away
from two others. Check it out:
If Bobby appears on stage, then neither
Suzy nor Kelly appears on stage.
In the original diagram, if Bobby appears, then Suzy does not appear and
Kelly does not appear. The contrapositive tells us that if either Suzy appears
or Kelly appears, then Bobby cannot appear on stage. This complicated
conditional rule can also be simplified. In this situation, Bobby is now hated
by both Suzy and Kelly. It will be much easier to deal with two hate
relationships than the original, ugly conditional rule.
All of the previous examples would stem from games in which some players
were being selected and others were left behind (In and Out grouping). The
hate relationship also occurs frequently in games with multiple groups.
Consider this rule from a game in which players are assigned to different
committees:
Bobby and Suzy cannot be assigned to the
same committee.
It’s a different type of game, but this is definitely the same stage of our
relationship. Whether there are two, three, or seven committees in this game,
Bobby and Suzy hate each other, and we must keep them apart.
There are other permutations of the hate relationship that are more complex,
but we will cover those when we get into the specific game types. For now,
just know that there’s going to be a lot of hate flying around, and you have to
be able to spot it.
It would seem that this is clearly the end of the road for our couple. A little
stalking, some puppy love, and an ugly breakup - what could come next?
Well, this is where it gets really complicated... Bobby
receives an unwelcome phone call from Suzy about
nine months later and...
Stage 4: Baby/Custody (At Least One)
Oh crap. Well, we told you that this was not the classic love story. Bobby is
now informed that he is the baby’s daddy, and dirty diapers are now a staple
of his weekends. How does this relate to grouping games? You see, babies
are helpless, so someone must watch them at all times. Someone always has
to be there. The same relationship pops up in grouping games. It will
commonly be asserted that at least one of two players must always be
selected.
In the conclusion of our analogy, the unbelievable occurs.
Bobby moved on assuming Suzy was just a part of his
past. But not so fast. Bobby and Suzy should have been
more careful during that love phase, and now they are
sharing custody of a little one (interestingly, Suzy insisted
on naming the little guy Antonio). The baby needs help at
all times, so either Bobby or Suzy must always be with
him. Welcome to parenthood.
The baby relationship is almost always introduced indirectly. It is your job to
simplify the rules and get to the basic relationship, just like we did with the
hate relationship.
If Suzy does not receive a promotion, then
Bobby does receive a promotion.
This rule would be part of an In and Out grouping game in which you must
determine which lucky employees receive promotions. If Suzy does not
receive a promotion, then Bobby must. Also, if Bobby does not receive a
promotion, then Suzy must. Thus, if either one of them does not receive a
promotion, then the other one does. This rule can be simplified to state that
either Suzy or Bobby must always receive a promotion.
Remember that “or”
statements are always inclusive. Either
Suzy or Bobby must get a promotion,
but it’s also possible that they both do.
Ninja Note:
In games that involve multiple groups, you can also be confronted with rules
that introduce a similar relationship. Check this one out, from a hypothetical
game about a fashion show.
If Mandy does not wear green, then Nina
must wear green.
Nina has to rock some green if Mandy does not. Also, if Nina does not wear
green, then Mandy must wear green. Note that the structure of this rule is
similar to the first example. If one condition is not met, then another
condition must be. This rule can be simplified to state that at least one of
Mandy and Nina must wear green.
The baby phase of the relationship is less common than the others. However,
make sure to pay close attention when it pops up (just like an actual baby).
Since this type of rule establishes that one of two options must always be
met, it will be important in nearly every question that follows.
It’s been quite a ride, but that concludes our modern day love story. Best of
luck to Bobby and Suzy. We loved, we cried, we reproduced. What more
could you ask for? Here is a quick overview of the big four relationships.
This shows you the possibilities that are allowed and those that are outlawed.
Believe it or not, these four basic relationships will be the basis for all
grouping games. If you understand these relationships, you will have a great
foundation to help you conquer any grouping game they throw at you.
Remember, this is just an introduction. There are numerous drills and games
waiting for you in the following chapters.
Now that we have covered the basics of grouping games, it’s time to jump in.
The next four chapters will be dedicated to the four common types of
grouping games:
1. In and Out Grouping 2. Two Groups 3. More Groups
4. Profiling
1 If you are already getting nervous at the mere mention of conditional rules, or if you
start to feel nervous about these rules at any time, go back and review the chapter on
conditional statements (Chapter 11).
2 Please refrain from making any “awww” noises. Feelings have no place on the
LSAT.
17/INandOUT
WHAT IS IT?
In and Out grouping games will present you with a variety of challenges. You
might have to select which students get elected for student government, or
which animals get to ride on the ark, or which appliances are purchased.
However, these games all have one defining characteristic in common:
They require you to select some members from a larger group.
Since you are only selecting
members for one group, In and Out
games are the most basic form of
grouping. However, basic does not
entail easy.
BP Minotaur:
In and Out games only require the formation of one group, namely the
players that are selected. However, it’s also vital to track the players that are
not selected. Thus, our setup will always include a spot for both the players
that are In and those that are Out.
Here is a quick example of how to build the appropriate setup for an In and
Out game:
The vice president of Synergy Corporation
reviews seven different budget proposals.
She will approve exactly four of the seven
proposals, in accordance with the
following restrictions:
In this game, your basic task is to determine which of the budget proposals
are approved. So we are dealing with an In and Out grouping game. In the
setup, you want to build an In group for the four proposals that are selected
and an Out group for the three that don’t make the cut.
Let’s review some of the other grouping elements that we covered in the last
two chapters.
Are the group sizes stable or unstable?
It is very important to note whether an In and Out game is stable or unstable.
When you read the introduction to the game, make sure to notice any
restrictions given. How many players are selected?
Some games will define exactly how many players are selected, other games
will give you loose restrictions, and some evil games will say next to nothing.
Here are some examples:
A new scene for a
television show is being
filmed. It will feature five
of the following eight
actors.
This is a stable grouping game. The scene must feature exactly five actors,
and three out: actors are not selected. This should be reflected in your setup.1
A new scene for a television show is
being filmed. It will feature some of
the following eight actors.
This game is different because it’s not clear how many actors are featured in
the scene. It could be one, two, or all eight at this point. This is an unstable
grouping game. All you get is one slot in the In group. (At least it’s not a
weird French film about an inanimate object.)
This distinction will help
inform your approach. With a stable
game, you want to watch for situations
that would violate the group sizes.
With an unstable game, it’s all about
the rules since the setup is so open.
Ninja Note:
So now you know what to watch out for when you are building the setup for
an In and Out game. However, a large part of your success depends on your
ability to interpret the rules.
Rules (Stalker/Love/Hate/Baby)
In and Out grouping games will present you with a combination of
conditional rules. It will boil down to your understanding of our big four
grouping relationships. The subtle distinctions between the rules will spell the
difference between success and failure. That’s right, your future relies on
being able to distinguish whether Anna and Bubba must always be together,
or whether you can have Bubba without Anna.2 With that in mind, it’s time to
review.
Here’s the plan: We are going to outline each of the relationships again and
then insert a quick quiz to make sure you have a deep understanding of each
one. For this exercise, we are throwing a party. It’s a pretty cool party, so
some people are invited and others aren’t.
Stage 1: Stalker
In the first relationship, if one player is selected, then another player must be
selected. Here are some examples and the correct diagrams:
If Alfonso is invited, then Buddha is
also invited.
Cable is invited only if Dennis is
invited.
If Edward is not invited, Foulad is not
invited.
Note: On the last example, you have to form the contrapositive to see that this
rule introduces the same relationship. The rule could have stated, “If
Foulad is invited, then Edward is invited,” and it would have been
identical in meaning.
Note how important it is to
diagram the rules correctly (“if” versus
“only if”). Make sure to review the
correct way to diagram each claim.
BP Minotaur:
When a rule introduces the stalker relationship, it’s important to recognize
this doesn’t imply the players must be together. One player brings along
another, but the second player could be selected alone.
Now it’s your turn. Diagram the following rule and then answer the
questions that follow.
If George is invited to the party, then
Hercules is also invited.
1. Can George be invited without Hercules?
2. Can Hercules be invited without George?
3. If George is not invited, does Hercules get
invited?
Here’s how the rule should be diagrammed:
1. Nope. George cannot be invited without
Hercules. Anytime you invite George, Hercules
is always sure to follow.
2. Yep. You definitely could have Hercules without
George. If Hercules is invited, George may or
may not be invited.
3. Maybe. If George isn’t invited, we don’t know
anything about our boy Hercules.
Stage 2: Love
In and Out games are not big on love. It is very rare for them to tell you that
two players must be selected together. Thus, be careful when concluding that
two players must be selected together if it isn’t explicitly stated.
The most common way for a love relationship to be
introduced in an In and Out grouping game is with the
phrase if and only if, or if but only if.
Here’s a quick example:
Ivan is invited to the party if and only if
Jerome is invited to the party.
This wording tells you that Ivan is invited if Jerome is invited. And Ivan is
invited only if Jerome is invited. Therefore, Jerome being invited is both
sufficient and necessary for Ivan to be invited. This should be represented
with a reciprocal arrow.
There’s a basic and more helpful interpretation of this rule. Only two
possibilities exist for these players: Either (1) they are both selected or (2)
neither one is selected. According to this rule, Ivan and Jerome are both In,
or they are both Out.
Stage 3: Hate
The final two relationships are undoubtedly the most important to understand
in these games. First up is the hate relationship. This is the most common rule
in all grouping games, including In and Out games. It establishes that two
players cannot both be selected.
Kendra and Leonardo cannot both be
invited to the party.
This is the most basic version of the hate relationship. They just flat out tell
you that Kendra and Leonardo cannot both be invited. We heard it was an
ugly breakup, something to do with his past as a Teenage Mutant Ninja
Turtle.
However, the same relationship can be introduced in a sneakier manner.
Check this one:
If Kendra is invited, then Leonardo is not
invited.
If Kendra is invited, then Leonardo is not. If Leonardo is invited, then Kendra
is not. So they can’t both be invited. Wait just a second - that’s exactly the
same as the last rule! And since they mean the same thing, you want to
represent them in the same exact way.
So, basically, one of them is
invited and one isn’t?
Ditz McGee:
Actually, no. This rule restricts both of the players from being selected at the
same time, but there’s no problem with selecting neither one. At least one of
them must be Out, but both of them also could be Out. According to this rule,
three possibilities exist:
(1) Kendra is invited and
Leonardo is not.
(2) Leonardo is invited and
Kendra is not.
(3) Neither Kendra nor
Leonardo is invited.
Don’t forget that both
of them could be Out.
Visually, it’s easy to see this type of rule and think that two players cannot be
together anywhere (they can’t be In together and they can’t be Out together).
But that would be a mistake. Since an In and Out game really only involves
selecting one group, and most (if not all) of the questions will relate to the In
group, our rules focus on the In group. You can’t select both players, but
it’s acceptable to select neither.
When dealing with an In and Out game, the hate
relationship means that both players cannot be In, but
they both can certainly be Out.
Time for another drill. Same idea - diagram the rule and answer the
questions.
If Marco is invited, then Nala is not
invited.
1. Can both Marco and Nala be invited?
2. Can neither Marco nor Nala be invited?
3. If Marco is not invited, is Nala invited?
We’re sure you aced it, but just in case:
1. Nope. This rule implies that Marco and Nala
can never both be invited.
2. Yep. There’s no problem at all with them both
getting the cold shoulder. They could both be
Out.
3. Maybe. If Nala is Out, then Marco could be In
or Out.
The feeling of hate commonly spreads in these games. It’s important to
simplify rules so you can identify the basic relationship. Rules will
commonly assert that selecting one player means that neither one of two
other players can be selected.
If Oprah is invited, then neither Phil nor
Quentin is invited.
Note the form of this rule - if one person is selected, then someone else
cannot be. But now there are two haters. If Oprah is invited, then Phil cannot
be invited. In addition, if Phil is invited, then Oprah is not. So Oprah and Phil
hate each other. But wait, there’s more! If Oprah is invited, then Quentin
cannot be. Also, if Quentin is invited, then Oprah is not. So Oprah and
Quentin also hate each other. After all of that work, you can simplify the
rule to state that Oprah and Phil cannot both be invited, and Oprah and
Quentin cannot both be invited.3
Once you split this rule into two separate hate relationships, they function
exactly the same as any other hate relationship. Remember, they both cannot
be selected, but the two participants (Oprah and Quentin, for example) can
both be Out.
And yet another drill...
If Suzette is invited, then neither T-Bone
nor Unicorn is invited.
1. Can Suzette and Unicorn be invited?
2. Can all three of them not be invited?
3. Can T-Bone and Unicorn both be invited?
One more time:
1. Nope. Suzette and Unicorn cannot both be
invited. For the record, Suzette and T-Bone
also cannot both be invited.
2. Yep. The rule only prevents people from being
invited together. Inviting no one is an easy way
to avoid problems.
3. Yep. T-Bone and Unicorn can both be invited
as long as Suzette is not. This rule doesn’t give
you any relationship between T-Bone and
Unicorn.
Stage 4: Baby (or Custody, if you prefer)
The last relationship is another tricky one, and it must be simplified. Rules
will commonly claim if one player is not selected, then another player must
be. There’s a much easier way to interpret these rules: At least one of the
two players must be selected.
If Violin is not invited, then Wilbert is
invited.
If Violin is not invited, then Wilbert is. If Wilbert is not invited, then Violin
is. In other words, if either one of them is Out, the other one must be In.
There is no chance of getting by without either Violin or Wilbert. So we
simplify this rule to state that either Violin or Wilbert must be invited.
I gets it - so one of them
is In and the other is Out.
Cleetus Comment:
No, no, and no. You have to avoid that mistake. The baby relationship asserts
that at least one of two players must be selected. But it’s totally acceptable
for both of them to be selected. Just like the hate relationship, there are
three possibilities for this rule:
(1) Violin is not invited and
Wilbert is invited.
(2) Wilbert is not invited and
Violin is invited.
(3) Both Violin and Wilbert are
invited.
“Or” always allows for
both.
When you deal with one of these rules, picture the words “or both” at the end.
When dealing with an In and Out game, the baby
relationship means that at least one of the players must
be selected, and it’s acceptable for both to be selected.
One last drill for you...
If Xan is not invited, then Yolanda is
invited.
1. Can Xan be invited without Yolanda?
2. Can both Xan and Yolanda be invited?
3. If Yolanda is invited, is Xan invited?
While you are still a white
belt, you can write each rule out and
then simplify it. However, as you
progress to black belt status, you will
want to immediately simplify each
rule.
Ninja Note:
And the winners are:
1. Yep. At least one of them must be invited, but Xan
could be invited without Yolanda, and vice versa.
2. Yep. At least one must be invited, but both could
be.
3. Maybe. As soon as Yolanda is invited, then the rule
is satisfied. Xan could then go either way.
We have one last hint with the baby relationship. Since this rule establishes
that at least one of two players must always be selected, it’s generally helpful
to insert this into your setup. You don’t want to forget that one of these
players always has to be part of the In group. Here’s how this would work for
the last rule:
If Xan is not invited, then Yolanda is
invited.
As soon as you know that either Xan or Yolanda must be selected, throw an
option into the In group. When later questions ask for a complete and
accurate list of those people that are invited, or for the minimum number of
people that must be invited, this option is an easy way to remember this
important rule.
Alrighty, now you should have a basic understanding of the four relationships
and how they function with an In and Out grouping game. But basic
understanding is not our goal here. It’s time for a drill that will really
challenge you. Yep, it’s that thing staring at you from the other page. Get in
there.
1 Watch the distinction between “exactly” and “at least” on the LSAT. When a person
claims that they had exactly two drinks before driving, that is entirely different than if
they claim to have had at least two drinks. Confusing those two claims can ruin an
entire game (or city block). Don’t confuse the two, and don’t drink and drive.
2 To be fair, his name’s Bubba. So the love relationship is a long shot.
3 Is it a coincidence that we used a person named Oprah to illustrate a complex hate
relationship? We’ll let you answer that question yourself.
IN AND OUT RULES DRILL
This drill is designed to test your ability to understand and simplify
conditional rules. Check out the following game. Then, diagram each rule
(and its contrapositive). Whenever possible (almost always), try to simplify
the rule into one of the big four grouping relationships.
When making dishes, a famous chef includes
some of the following ingredients in each dish—
alfalfa sprouts, beets, carrots, duck, eggs, feta
cheese, garlic, horseradish, iceberg lettuce,
jicama, kalamata olives, lentils, mango, nutella,
oregano, parsley, and quail—according to the
following:
This is a classic In and Out game - some ingredients
are included and others are not.
1.
If a dish includes jicama, then it also includes oregano.
2. Any dish that includes horseradish does not include parsley.
3. Feta cheese is included in any recipe that includes alfalfa
sprouts.
4. Mango is included in a recipe if, but only if, nutella is included.
5. If garlic is in a recipe, then both duck and eggs must be
included.
6. Beets are not included in a recipe if both carrots and lentils are
included.
7. If quail is in a recipe, then neither oregano nor kalamata olives
is included.
8. If garlic is not included in a recipe, then beets must be included.
9. If a recipe does not include both horseradish and iceberg
lettuce, then it must include both beets and carrots.
10. If parsley is included in a recipe, then lentils cannot be
included.
11. If alfalfa sprouts are not in a recipe, then feta cheese must be
included but garlic is not.
12. Nutella is included in every recipe that mango is not.
ANSWER KEY
1)
j[o
O[J
If jicama is included, then oregano must be included. And if oregano is
not, then jicama cannot be included. However, you could select oregano
without jicama. This is stalker, not love.
2)
h]p
If horseradish is selected, then parsley cannot be. If parsley is included,
then there is no horseradish. This is the hate relationship. You can never
include both horseradish and parsley. Remember, a recipe could include
neither ingredient.
3)
a[f
F[A
The word “any” introduces a sufficient condition. So alfalfa sprouts is
sufficient to guarantee feta cheese. However, this does not mean that you
must select both. Feta cheese could be in a recipe that lacks alfalfa sprouts.
4)
m\n
“If, but only if” introduces a reciprocal relationship. Either both mango and
nutella are included in a recipe, or neither one is included.
5)
g[d+e
D or E [ G
Garlic, duck, and eggs? Tasty. If garlic is included, then both duck and
eggs must be in the recipe. Thus, if either duck or eggs is not included,
then there will be no garlic.
6)
c+l[B
b [ C or L
“If” introduces the sufficient condition, even when it is buried in the end of
a rule. If both carrots and lentils are included, then there are no beets. So if
beets are in a recipe, then the chef cannot include both carrots and lentils
(at least one is left out).
7)
q]o
q]k
This rule gives you a lot of hate. If quail is in a recipe, then oregano is out
and kalamata olives are out. Thus, if either one is included, then quail
cannot be included. This can be simplified to two hate rules: quail and
oregano cannot both be selected; and quail and kalamata olives cannot both
be selected.
8)
g or b
If there is no garlic, you must have beets; if there are no beets, then there
better be some garlic. This is the baby phase of the relationship. Either
garlic or beets (or both) must always be included.
9)
H or I [ b + c
B or C [ h + i
This one is tough. Remember that “not both” translates to an “or”
statement, but with both terms negated. If either horseradish or iceberg
lettuce is not included, then both beets and carrots must be included; and if
either beets or carrots are not included, then both horseradish and iceberg
lettuce must be in a recipe. This rule could also be simplified: Every recipe
must have either both beets and carrots or both horseradish and iceberg
lettuce (or three of the ingredients, or even all four).
10)
p]l
Back to hate. If parsley is included in a recipe, then lentils cannot be
included. And if lentils are included, then parsley must be left out. So no
recipe can include both parsley and lentils. You could have parsley, or you
could have lentils, or you could have neither one, but it’s impossible to
have both.
11)
A[f+G
F or g [ a
This rule is pretty complex. If alfalfa sprouts are not in a recipe, then feta
cheese must be included but garlic is not. To form the contrapositive, just
flip the sufficient and necessary conditions, negate all conditions, and swap
the “and” to an “or.” Thus, if feta cheese is not in a recipe or if garlic is in
a recipe, then alfalfa sprouts must be in the recipe.
12)
n or m
If mango is not included in a recipe, then nutella must be included. And if
the chef chooses not to include nutella in a recipe, then mango must be
included. This is the baby relationship yet again. If either ingredient is not
included, then the other one must be. Thus, either nutella or mango (or
both) must be included in every recipe.
Sweet. If that drill went well, it’s time to move on. If you struggled with
some of those examples, make sure to go back and examine Chapter 4 and
the beginning of this chapter.
So that takes care of individual rules. However, from our study of ordering
games, you already know that the powerful deductions come from combining
rules.
Next up: Let’s see how these ugly rules play together
to form powerful deductions.
FOLLOW THE ARROWS
When you are doing an In and Out grouping game, you will be given a
number of conditional rules. It’s very important to look for transitive
conclusions.1 These deductions will save lots of time and make the questions
much easier. Let’s return to our raging party and check out a few examples.
If Leon is invited, then Mathias is invited.
If Mathias is invited, then Nena is not
invited.
Always diagram the original conditional rule first. If Leon is invited, then
Mathias is invited. And if Mathias is invited, then Nena is not. Thus, if Leon
is invited, then Nena is not. Also, if Nena is invited, then Leon is not. Look at
that, we just discovered a hate relationship. Leon and Nena can’t both be
invited. As you move through the game, you can immediately eliminate any
answer choice that has both Leon and Nena on the invite list.
If Pattie is invited, then Santa is also
invited. Tania is not invited only if Pattie is
invited.
In this one, anytime that Pattie is invited, Santa must be invited (must be an
ugly Christmas sweater party). Also, if Tania is not invited, then Pattie must
be invited. If you put the rules together, you can see that if Tania is not
invited, then Santa must be invited. And Tania is invited if Santa is not. By
gosh, we just found a baby. Either Santa or Tania (or both) must always
be invited. If an answer choice includes neither one, it’s out.
In the second example, you
can see that the order of the rules
doesn’t matter. Just make sure to never
flip around an arrow.
Ninja Note:
On the next page, you will see a drill that’s designed to help you spot these
helpful deductions.
1 We discussed the transitive property briefly at the end of Chapter 11. It might be a
good idea to go back and review. It’s also a good idea to use the term “transitive
property” in front of anyone you want to impress.
IN AND OUT DEDUCTION DRILL
For each of the following games, use the introduction to build the correct
setup. Work slowly through the rules. Represent each rule correctly, and
simplify whenever possible. Finally, try to combine the rules to find those
big-time deductions.
Game 1: Buff You Up
A bodybuilder does four of the following types of exercises during
one day’s workout—bench press, crunches, dead lifts, jumping
jacks, lunges, push-ups, and squats. She chooses the exercises
according to the following:
If she doesn’t do lunges, then she does squats.
If she does push-ups, then she does bench press.
If she does bench press, then she does crunches.
If she does dead lifts, then she doesn’t do lunges.
1. If she does dead lifts, what other exercises must she do?
2. If she does push-ups, then what exercises cannot be done?
Game 2: Splash of Color
An interior designer is selecting colors for the interior of a house.
He must choose from the following colors: aqua, beige, chartreuse,
devilish red, eggplant, and fuschia. The selection of colors must
conform to the following:
If he chooses eggplant, then he does not choose chartreuse.
If he chooses either fuschia or devilish red, then he chooses
eggplant.
If he does not choose aqua, then he does choose beige.
He does not choose aqua unless he chooses chartreuse.
1. If he chooses fuschia, what other colors must be chosen?
2. If he chooses aqua, then what colors cannot be chosen?
ANSWER KEY
Game 1: Buff You Up
1. Squats, Crunches
If she does dead lifts, then she
cannot do lunges.
If she does not do lunges, she must
do squats.
If she didn’t do crunches, she could
do neither bench press nor pushups.
2. Dead Lifts, Jumping Jacks
If she does push-ups, she must do bench press and crunches.
The fourth exercise must be lunges or squats, so dead lifts and jumping
jacks cannot be included.
Game 2: Splash of Color
1. Eggplant, Beige
If fuschia is chosen, then so is eggplant.
If eggplant is chosen, then chartreuse is not.
If there’s no chartreuse, there’s no aqua.
No aqua? Say yes to beige.
2. Eggplant, Fuschcia, Devilish
Red
If aqua is chosen, then
chartreuse is chosen.
If chartreuse is chosen, then
eggplant, fuschia, and devilish
red are not.
Wow. That was quite a ride. At this point, we’ve introduced grouping games.
We have zoomed in and looked at In and Out games. Shoot, we even
discussed the rules and deductions that you can expect to see. What do you
say we try an actual grouping game?
DECEMBER 2004: GAME 3 (13-17)
This game marks a number of different firsts. It’s our first In and Out game.
More importantly, it’s our first grouping game. Big time. As with all games,
we will stress the impact of each rule and the quest for deductions. Since this
is new, we are going to use our Blueprint Building BlocksTM approach.
This game is about a group of friends. They aren’t named Chandler, Joey,
Ross, Monica, Phoebe, and Rachel, but they are a pretty good group
nonetheless.
1. Setup
An album contains photographs picturing seven
friends: Raimundo, Selma, Ty, Umiko, Wendy,
Yakira, Zack. The friends appear either alone or
in groups with one another, in accordance with
the following:
What’s a
group of friends
without a
Yakira?
Your first task is to identify the type of game. There is clearly no ordering
involved in this one. Rather, your task is to determine the group of friends
that are contained in a set of photographs. So this is a grouping game. More
specifically, the questions are going to center around which friends are in
each photograph and which friends are out of each photograph. In other
words, we are selecting some of the friends and leaving others behind, which
makes this an In and Out grouping game.
Next, look for restrictions on the sizes of the groups. Do they tell you how
many friends are in each photograph? Nope. This game is unstable.
Since the friends appear “alone or in
groups,” there must be at least one
friend in each photograph. At this
point, there could be one, two, or all
seven friends in a photograph. The
setup looks pretty boring, but
sometimes boring is effective.
As soon as you accurately represent the variables and build your setup, it’s
time for the rules.
When an In and Out has an
unstable setup and no crazy tricks in
the intro, you know it’s going to be all
about the rules.
Ninja Note:
2. Rules
Our next challenge is to diagram the rules. Since it’s so important to diagram
these rules correctly, we are going to let you try each one. In the space
provided, diagram the rule and its contrapositive. Remember, always try to
simplify the relationship. Do they hate each other? Stalker? At least one?
Go ahead and try the first one.
Hint - the word “every” indicates a sufficient
condition. Wendy appears in every photograph that Selma
BP Minotaur:
appears in.
It is very common for this rule to be diagrammed incorrectly. When that
happens, this game becomes less fun than a root canal.
Ditz McGee:
Cleetus Comment:
I just said they have to be
together.
Naw. If you got Wendy, then you got a
Selma.
You are both wrong, actually. Whenever you diagram a conditional rule, look
for words that indicate sufficient and necessary conditions. In this one,
“every” is the key. “Every” always introduces a sufficient condition - it’s
synonymous with “if.” We could rephrase that rule to say, “If Selma appears
in a photograph, then Wendy appears in that photograph.”
The correct diagram is shown to the right. If Selma is
in a photograph, then Wendy must be in that
photograph. You can’t have Selma without Wendy.
But remember, you can have Wendy without Selma.
As you will see in some of the remaining rules, the LSAT loves to go back to
the same old tricks time and time again. But that’s why memorizing these
rules really pays off. Go ahead and diagram the next rule.
Fool me once, shame on you. Fool me twice, the LSAT isn’t going to go very
well. “Every” is once again used to introduce the sufficient condition.
If Umiko appears in a photograph, then Selma appears in
that photograph. According to the contrapositive, if
Selma does not appear in a photograph, Umiko also does
not appear. As always, avoid those fallacies - Selma
could appear in a photograph without Umiko.
Did you notice that Selma is
involved in both of the first two rules?
This will spell deductions
BP Minotaur:
Go ahead and diagram the third rule.
“Every” again? We get it. This rule includes some of the same wording, but
actually introduces a radically different relationship than the first two rules.
Ditz McGee:
I figured that means you can’t have both of them.
You get one or the other.
You are trying to simplify, but you’re slightly off. This
is our first baby relationship. If Yakira doesn’t appear,
then Raimundo must, and if Raimundo doesn’t appear,
then Yakira must. So at least one of them must be in
every photograph.
One more thing: Since this rule will be important to
remember on every question, we should insert an option
into the setup. Either Raimundo or Yakira must always
take a spot in the In group.
Ninja Note:
One more rule...
Remember, it’s also fine to
have both R and Y.
In this one, they swap “every” with “any.” However, both words are used to
introduce sufficient conditions. So this rule can be restated as, “If Wendy
appears in a photograph, then neither Ty nor Raimundo appears.”
This rule can be simplified even further. If
Wendy appears in a photograph, then neither
Ty nor Raimundo appears in that photograph.
And if Ty or Raimundo appears in a
photograph, then Wendy cannot. (Wendy is
apparently not very popular.) We have two
hate relationships. Wendy cannot appear in
the same photograph as Ty, and Wendy also cannot appear in the same
photograph as Raimundo. But don’t forget that these players could be Out
together. In fact, Raimundo, Ty, and Wendy could all be Out.
Simplifying these complex
relationships can be difficult. You will
survive as long as you diagram
correctly, but keep trying to identify
hate relationships when you can.
BP Minotaur:
Diagramming the rules for an In and Out game can be challenging. But focus
on the details: Swapping around any one of these rules will spell disaster.
That brings us to the end of the rules in this game. But, as you know by now,
we are far from ready to hit the questions. Take a look back at the rules. Our
old rule applies: When a player is mentioned in more than one rule, always
look for deductions. Well, we have Selma, Wendy, and Raimundo all
mentioned in multiple rules. Time to get to the bottom of this.
3. Land of Deductions
In this game, you are faced with only conditional rules. This is common when
dealing with an In and Out grouping game. It’s unlikely that you will be able
to determine where any of the players must be assigned. But there are other
types of deductions that are just as important. You should be able to combine
some of the rules and find transitive conclusions.
Up next, your task is to see if you can spot deductions by combining the rules
in this game. We will give you a small prompt, and then you have to apply as
many rules as possible.
First challenge: If Umiko appears in a photograph,
what else can you conclude?
Wow, Umiko provides a wealth of knowledge. Hopefully, you were able to
find a host of deductions if Umiko is in a photograph. Later, you’ll have to
use the rules to identify the most helpful players on your own. Here’s an
outline that starts with Umiko:
If Umiko is in a photograph, then Selma must appear in that photograph.
And if Selma appears in a photograph, then Wendy must appear in that
photograph. So you can combine these rules.
If Wendy appears in a photo, then both Ty and Raimundo cannot appear
in that photo, so this condition can be combined with the previous rules.
Either Raimundo or Yakira must
always be in the photo, so if
Raimundo is not in a photo, Yakira must be in that photo. This can also
be added into the growing chain.
This chain is going to be super helpful. As you work through the questions,
you can refer back to this chain to find quick deductions. Umiko is in a
photograph? No problem. We immediately know that Wendy and Yakira are
in the photograph, and Ty and Raimundo are not.
But don’t stop there. We know that the rules work nicely together, so let’s
investigate further. In the next one, we are going to discuss a photograph that
doesn’t feature Yakira.
Second challenge: If Yakira doesn’t appear in a
photograph, what else do you know?
Getting rid of Yakira also leads to a ton of deductions. Note that our last
chain ended with Yakira in the photograph. By putting Yakira out, we can
test the contrapositives of these relationships.
If Yakira does not appear in a photograph, then Raimundo must be in the
photograph.
If Raimundo appears in a photograph, then Wendy cannot. This can be
added to the previous rule.
If Wendy does not appear in a
photograph, Selma also does not.
If Selma does not appear in a photograph, Umiko also does not.
Note that Ty is involved in the first chain, but not the second. Be careful
when you have conjunctions (“and”) or disjunctions (“or”). In the first chain,
Ty is Out because Wendy is included. But in the second chain, the fact that
Raimundo appears in the photograph doesn’t determine anything about Ty.
There is one more deduction that is important to notice.
Our man Zack is not involved in any of the rules, so Zack is totally
random. As we learned in Saved by the Bell, Zack plays by his own
rules. This is important to note, so go ahead and circle Zack in your
setup.
In a game like this, a random
will play a big role. For example, if a
question asks you for the maximum
number of friends that could be in a
photograph, we gotta remember Zack.
Ninja Note:
As always, we have spent a good amount of time diagramming the rules
and making deductions. But that will save us time in the long run. Below,
you can see where we are. At this point, we are ready to attack the
questions.
4. Questions
In a game like this, you can expect a number of conditional questions (“If so
and so appears in a photograph, then...”). The makers of the LSAT want to
determine whether you were able to comprehend the implications of each
rule. Our deductions will get us through these questions quickly and
accurately.
Question #13
13. Which one of the following could be a
complete and accurate list of the friends
who appear together in a photograph?
In a shocking development, the first challenge here is an elimination
question. Just like ordering games, almost all grouping games will give you
one of these questions first. You can use the same strategy. Simply use the
rules to eliminate the four incorrect answers.
Challenge: Use the elimination approach to find the
correct answer.
Wendy appears in every photograph that
Selma appears in.
Selma appears in every photograph that
Umiko appears in.
Raimundo appears in every photograph
that Yakira does not appear in.
Neither Ty nor Raimundo appears in any
photograph that Wendy appears in.
(A) Raimundo, Selma, Ty, Wendy
(B) Raimundo, Ty, Yakira, Zack
(C) Raimundo, Wendy, Yakira, Zack
(D) Selma, Ty, Umiko, Yakira
(E) Selma, Ty, Umiko,
The first rule states that Wendy must appear in every photo that Selma
appears in. In (D), our photo features Selma but no Wendy. In (E), we get
Selma but no Wendy again. So both of those answer choices are knocked
out. Two birds with one stone, as they say.
The second rule does not eliminate any of the remaining answer choices.
The third rule tells you that either Raimundo or Yakira must appear in
every photograph. Unfortunately, Raimundo is right at the front of the
remaining answer choices. That means that this rule also does not help at
all. Annoying, but no need to stress.
Finally, we get some help here. The fourth and final rule states that neither
Ty nor Raimundo appears in any photo that Wendy appears in. (A)
features Wendy in a photo with both Ty and Raimundo, so that is wrong.
In (C), Wendy is featured in another photo with Raimundo, so that one
goes bye-bye as well.
It took some work, but we eliminated the four incorrect answer
choices and we’re left with the correct answer, (B).
Question #14
14. If Ty and Zack appear together in a
photograph, then which one of the
following must be true?
This is the first conditional question for this game. Our deductions will
start to pay off big time. The new condition is that both Ty and Zack appear
together in a photograph. Remember, Zack is totally random. Ty, however, is
going to be very helpful. Let’s create a hypothetical where Ty and Zack are
both In. Next, use the deductions to fill out as many other spots as you can.
Challenge: Work out a hypothetical and identify the
correct answer (must be true).
(A) Selma also appears in the photograph.
(B) Yakira also appears in the photograph.
(C) Wendy also appears in the photograph.
(D) Raimundo does not appear in the
photograph.
(E) Umiko does not appear in the
photograph.
If Ty appears in the photograph, then Wendy cannot appear in the
photograph.
If Wendy does not appear in the photograph, then Selma does not
appear in the photograph.
If Selma does not appear in the photograph, then Umiko does not appear
in the photograph.
You can see how those previous deductions helped guide you through this
question. At this point, five of the friends are determined, and the only wild
cards are Raimundo and Yakira. At least one of them must appear in the
photograph, or they could both appear. However, this is a must be true
question, so you should look for one of your solid deductions in the answer
choices.
The last deduction was that Umiko does
not appear in the photograph, so (E) is
going to be our winner.
Question #15
15. What is the maximum number of
friends who could appear in a photograph
that Yakira does not appear in?
This is a conditional
question masquerading as an absolute
question. Note the new information
that’s introduced at the end (Yakira
does not appear in the photograph).
BP Minotaur:
On this conditional question, we have to plug Yakira into the Out group and
see what happens. The first deduction is a quick one:
We simplified the third rule to state that either Raimundo or Yakira
must appear in every photograph. As soon as Yakira is Out, Raimundo
must be In.
Challenge: With Yakira and Raimundo in place, try to
determine the other friends.
You aren’t fooling us, silly
LSAT. These are the exact same
deductions we made on the last
question.
Ninja Note:
Since Raimundo is in the photograph and he
hates Wendy, Wendy cannot be in the
photograph.
If Wendy is Out, then both Selma and Umeko must also be Out.
Most of the work is now done. However, the question asks for the maximum
number of friends that could be in the photograph. Both Ty and Zack are
missing from our hypothetical, so we have to see if we could throw that
combo into the photograph.
Zack is a random, so Zack can always be added to the photograph.
Can Ty be in the photograph? Sure. Ty hates Wendy, but Wendy is
already Out.
If Yakira does not appear in a
photograph, the maximum
number of friends that could
be in the photograph is three
(Raimundo, Ty, and Zack).
This leads us very nicely to our
answer, (D).
(A) six
(B) five
(C) four
(D) three
(E) two
Question #16
16. If Umiko and Zack appear together in a
photograph, then exactly how many of the
other friends must also appear in that
photograph?
As anticipated, this game is chock-full of conditional questions. Our
deductions have not let us down thus far, so you should be feeling confident.
The new condition is that Umiko and Zack appear together in a photograph
(how cute). We already noted that Zack is random, so we want to look to the
powerful Umiko for deductions.
Questions can sometimes be
used to guide your strategy. Since this
question asks for the exact number of
friends in the photograph, you know
you are going to be able to figure out
all seven friends.
Ninja Note:
This one is all up to you. Turn the page and get to it.
Challenge: We threw Zack and Umiko into the setup.
You take care of the rest. Determine the exact number
of other friends in the photograph.
(A) four
(B) three
(C) two
(D) one
(E) zero
Wow, talk about a big payoff from our early deductions. Based on the
inclusion of Umiko in the photograph, you can quickly figure out exactly
which friends are in the photograph.
This is a trick question. There
are five friends in the photo, but that’s
not an answer.
Ditz McGee:
That’s a common mistake on this question. While you probably got the
correct deductions, Ms. McGee, you have to read the questions very
carefully. This one asks for the exact number of friends other than Zack and
Umiko that appear in the photograph.
Here’s an outline of the deductions:
If Umiko appears in the photograph, then Selma must appear in the
photograph.
If Selma appears in the photograph, then Wendy also appears in the
photograph.
Wendy hates Raimundo and Ty, so both of them cannot appear in the
photograph.
Either Raimundo or Yakira must
always appear in the photograph, so
Yakira is the final piece to our
puzzle.
There are three friends other than Zack and Umiko, so (B) is the
answer.
Question #17
17. If exactly three friends appear together
in a photograph, then each of the following
could be true EXCEPT:
We have reached the final question. Yet again, it’s a conditional question.
This one has a bit of a twist. Rather than just relying on the conditional rules,
this question sets a strict limit on the sizes of the groups. The new restriction
is that exactly three friends must appear together in a photograph.
There are seven friends in total. If there are only three friends together
in the photograph, then there must be four friends who do not appear in
the photograph.
The option with Raimundo or Yakira will be helpful in this situation.
Remember, at least one of the three spots must be taken by Raimundo or
Yakira.
The approach on this question is a little different. There are a number of
combinations that could work in the photograph. It’s more helpful to look
for any friends who can’t be in the photo. If you do so, there’s one very
helpful deduction.
Challenge: See if you can identify the one friend who
cannot be in the photograph.
Hint: There are only three spots (one of which must be
taken by R or Y). Look for a player who, if selected,
would force too many other friends into the
photograph.
Since the In group only has three slots total (and only two slots after you
account for either Raimundo or Yakira), you want to search for players that
would force too many additional friends into the photograph.
If Umiko is in the photograph, then both Selma and Wendy must be in
the photograph. So if Umiko is In, at least four friends appear in the
photograph, which isn’t possible on this question. Thus, Umiko cannot
be in the photograph.
After you make that deduction, it’s time to hit the answer choices. For the
other questions, we were able to anticipate the correct answers. Here, there’s
no option but to jump in and search for it.
(A) Selma and Zack both appear in
the photograph.
(B) Ty and Yakira both appear in
the photograph.
(C) Wendy and Selma both appear
in the photograph.
(D) Yakira and Zack both appear in
the photograph.
(E) Zack and Raimundo both appear
in the photograph.
(A) If Selma appears in the photograph, then Wendy must appear
in the photograph. If Zack also appears in the photograph, our
list becomes Selma, Wendy, and Zack. This list might seem to
be acceptable, until you remember that you always have to
include either Raimundo or Yakira. This would actually give us
a minimum of four friends who must appear in the photograph.
This violates the new condition for this question, so (A) must be
false. We have a winner.
Note: Even though Umiko doesn’t actually appear in the correct answer
choice, going through the search for that deduction gave you a better
idea of what to look for.
(B) If Ty appears in the photograph, then Wendy, Selma, and Umiko
cannot be in the photograph. However, we could have Ty appear
with Yakira and either Raimundo or Zack, so (B) could be true.
(C) If Wendy appears in the photograph, then Raimundo and Ty
cannot appear in the photograph. This implies that Yakira must
appear in the photograph, making our three friends Wendy, Selma,
and Yakira, which could be true.
(D) Yakira could appear in the photograph with a variety of other
friends, and Zack is random, so (D) could also be true.
(E) If Raimundo appears in a photograph, then Wendy, Selma, and
Umiko cannot appear in the photograph. However, Raimundo
could appear with Zack and either Ty or Yakira.
That concludes our first In and Out grouping game. Before we move on, let’s
recap the lessons that we’ve learned and how they were illustrated in this
game.
1. Grouping games are an entirely new challenge.
You might have enjoyed that, or you might have hated it. Either way, you are
going to have to deal with with it. Grouping represents the other huge
obstacle in our quest to conquer Logic Games. Rest assured, this is our very
first grouping game, and they will get easier as you get more practice.
However, note that the challenge, from the setup to the rules to the
deductions, was entirely different than ordering games.
2. Conditional rules rule the day.
As you just experienced, diagramming the rules
correctly in grouping games is essential to your
success. If we had flipped around one of those
rules, we would still be banging our head
against the wall trying to get through the first
question. Make sure to review those conditional
statements over and over again.
3. The same old strategy leads to powerful new
deductions.
The deductions in this game arose from combining the rules. Wait, Wendy is
mentioned in the second rule and the fourth rule - let’s check that out. While
the powerful deductions in grouping games are based on combining
conditionals rather than ordering chains and blocks, a number of the same
basic principles apply. Work slowly through the rules, and look for instances
where certain players are mentioned more than once.
Now it’s your turn. On the next page, you will see a game from the
September 2009 LSAT. Build an effective setup, and then focus on
diagramming and simplifying the rules. When you finish, check out the
explanations to make sure you took all of the appropriate steps.
SEPTEMBER 2009: GAME 2 (7-12)
Questions 7-12
A company organizing on-site day care consults with a group of parents
composed exclusively of volunteers from among the seven employees—
Felicia, Leah, Masatomo, Rochelle, Salman, Terry, and Veena—who have
become parents this year. The composition of the volunteer group must be
consistent with the following:
If Rochelle volunteers, then so does Masatomo.
If Masatomo volunteers, then so does Terry.
If Salman does not volunteer, then Veena volunteers.
If Rochelle does not volunteer, then Leah volunteers.
If Terry volunteers, then neither Felicia nor Veena volunteers.
7. Which one of the following could be a complete and accurate list
of the volunteers?
(A)
(B)
(C)
(D)
Felicia, Salman
Masatomo, Rochelle
Leah, Salman, Terry
Salman, Rochelle, Veena (E) Leah, Salman, Terry, Veena
8. If Veena volunteers, then which one of the following could be true?
(A) Felicia and Rochelle also volunteer.
(B)
(C)
(D)
(E)
Felicia and Salman also volunteer.
Leah and Masatomo also volunteer.
Leah and Terry also volunteer.
Salman and Terry also volunteer.
9. If Terry does not volunteer, then which one of the following
CANNOT be true?
(A)
(B)
(C)
(D)
(E)
Felicia volunteers.
Leah volunteers.
Rochelle volunteers.
Salman volunteers.
Veena volunteers.
10. If Masatomo volunteers, then which one of the following could be
true?
(A)
(B)
(C)
(D)
(E)
Felicia volunteers.
Leah volunteers.
Veena volunteers.
Salman does not volunteer.
Terry does not volunteer.
11. If Felicia volunteers, then which one of the following must be
true?
(A)
(B)
(C)
(D)
(E)
Leah volunteers.
Salman volunteers.
Veena does not volunteer.
Exactly three of the employees volunteer.
Exactly four of the employees volunteer.
12. Which one of the following pairs of employees is such that at
least one member of the pair volunteers?
(A)
(B)
(C)
(D)
(E)
Felicia and Terry
Leah and Masatomo
Leah and Veena
Rochelle and Salman
Salman and Terry
MASATOMO OR VEENA?
And we’re back. Hopefully you were able to work your way through that
game and answer some burning questions that will have a profound effect on
your legal career - such as, can Rochelle volunteer if Salman doesn’t?
As we work through this game, remember the bigger picture. Your focus
should not be solely on right and wrong answers. The process is just as
important.
1. Setup
In the intro paragraph, you are given a set of seven brand-spanking-new
parents. Very cute. Your job is to determine the makeup of a volunteer group
for on-site day care. Who volunteers and who doesn’t? Since we have to
select the parents that volunteer, this game should quickly be identified
as an In and Out grouping game.
The next thing that you have to hunt for are
restrictions on the sizes of the groups. Do
we have any idea how many parents
volunteer? In this game, we literally have
no idea. This grouping game is completely
unstable. Not only do they not specify how
many parents volunteer, they don’t even
tell you that some of them do volunteer. At
this point, it’s possible for no one to volunteer, or for any number of parents
(between one and seven) to volunteer. Even though this leaves things wide
open, it’s important to notice.
2. Rules
It’s all conditional rules in this one,
as expected. You need to focus on
diagramming the rules correctly
and simplifying the relationships
whenever possible.
This one starts out with a stalker.
If Rochelle is In, then Masatomo
must be In. Don’t confuse this with love. You could have Masatomo
without Rochelle.
Next rule:
This rule is similar to the first one.
However, you should note the
transitive structure of the first two
rules. If Rochelle volunteers, then so
does Masatomo. And if Masatomo
volunteers, then so does Terry .
These two rules should be
combined. As always, make sure to write out the contrapositive.
This one is different. If Salman
doesn’t volunteer, then you must
have Veena. If you don’t have
Veena, you better have some
Salman. What does that mean? If
you don’t have one, you have to
have the other. So either Salman or
Veena must volunteer. Or you
could have both.
The next rule sounds pretty similar to this one.
See, we told you these rules are
repetitive. If you don’t get Rochelle,
better have some Leah. No Leah?
Say hello to Rochelle. You should
simplify this rule just like the last
one. Either Rochelle or Leah (or
both) must volunteer.
It will also be helpful to remember that Rochelle was involved in the first
rule.
When you have baby relationships (at
least one), make sure to always represent
them in your setup. The last two rules
should be visualized as options for the In
group.
Moving on to the last rule...
With the last rule, some hate pops up
in this game. If Terry volunteers,
then neither Felicia nor Veena
volunteers. Thus, if either of them
volunteers, then Terry cannot. Terry
is a hated man (or woman). Terry
cannot volunteer with Felicia, and
Terry cannot volunteer with Veena.
That takes care of our first challenge - diagramming the rules correctly.
3. Deductions
Since you are dealing exclusively with conditional rules, it’s time to look for
transitive conclusions. There are a number of ways to arrive at the same
deductions, but here is a general overview of how the rules play together:
If Rochelle volunteers, then
Masatomo volunteers. If
Masatomo volunteers, then
Terry must volunteer. This is a
great place to start.
If Leah doesn’t volunteer, then
Rochelle must volunteer. This
can be added to the front of our growing chain.
If Terry volunteers, then neither Felicia nor Veena volunteers.
If Veena doesn’t volunteer, then Salman does. This is the last piece to
our puzzle.
Ditz McGee:
How do I know where to
start?
That’s a very common question from students. The answer, surprisingly, is
that it shouldn’t matter much. As long as you go through every rule to see if it
can be added to the chain, and you never flip around an arrow, the result will
be the same.
After combining all of the rules into a chain, it’s time to examine the
contrapositives. Since we already found how all of the rules can be
combined, writing out the contrapositives shouldn’t be a huge challenge. Just
remember the rules: (1) reverse sufficient and necessary conditions, (2)
negate everything, and (3) switch ands and ors. Here it is:
If Salman doesn’t volunteer, then Veena must volunteer.
If Veena volunteers, then Terry does not volunteer.
If Terry does not volunteer, then Masatomo does not volunteer.
If Masatomo does not volunteer, then Rochelle does not volunteer.
If Rochelle does not volunteer,
then Leah must volunteer.
What about Felicia?
Don’t she have to be In too?
Cleetus Comment:
Actually, no. The original rule told us if Terry volunteers, then neither Felicia
nor Veena volunteers. We only know that Veena volunteers. That is enough
to conclude that Terry doesn’t volunteer, but it doesn’t tell us anything about
Felicia. Felicia could volunteer along with Veena, or she could be Out.
When you make conditional chains, remember
to avoid those ugly fallacies. For instance, if Veena
volunteers, you can draw conclusions about Terry,
Rochelle, and others. But it doesn’t tell us anything about
BP Minotaur:
Salman. If you think it does, you’re
committing the fallacy of the converse.
At this point, it’s on to the questions. Bring it on.
4. Questions
Question #7 (elimination, could be true)
The questions start off with a nice elimination challenge. Just use the rules.
The first rule has Masatomo stalking Rochelle. In (D), we have Rochelle
without Masatomo. Can’t happen; get it out of here.
The second rule requires Terry to volunteer if Masatomo volunteers. (B)
goes away.
According to the third rule, either Salman or Veena must volunteer. All
of the remaining answer choices are fine.
The fourth rule kicks out (A) since you always have to include either
Felicia or Leah.
The last rule establishes that Terry hates Felicia and Veena. In (E), you
see both Terry and Veena. No bueno.
All that’s left is (C).
Question #8 (conditional, could be true)
The deductions make this one quick and easy. The new condition is that
Veena volunteers.
If Veena volunteers, then Terry does not.
If Terry does not volunteer,
then Masatomo also does not.
If Masatomo does not volunteer, then Rochelle does not.
You gotta have Rochelle or Leah, so now Leah volunteers.
The wild cards at this point are Salman and Felicia. They could go either
way.
(B) says that Felicia and Salman also volunteer. No problem there.
They could both volunteer, so (B) is the champ.
Question #9 (conditional, must be false)
Even though this in a new question, it’s asking about the exact same
deductions.
If Terry does not volunteer,
then Masatomo does not
volunteer.
If Masatomo does not volunteer, then Rochelle does not.
You gotta have Rochelle or Leah, so now Leah volunteers.
Veena, Salman, and Felicia are still undetermined.
Rochelle doesn’t volunteer. So (C) must be false.
Question #10 (conditional, could be true)
Here is yet another conditional question. Now we know that Masatomo
volunteers. We always knew he sounded like a nice guy.
If Masatomo volunteers,
then so does Terry.
If Terry volunteers, then
neither Felicia nor Veena volunteers.
You gotta have either Veena or Salman. So here we have some Salman.
Those deductions rule out four of the answer choices. At this point,
we don’t know if Leah volunteers. So it could be true that Leah
volunteers. Say hello to (B).
When you have a could be
true question, think about the players
that are up in the air (on this one, Leah
and Rochelle). Then just go look for an
answer about one of them.
Ninja Note:
Question #11 (conditional, must be true)
Shocking - another conditional question. Of course, there were no huge
deductions about the volunteers, so these are the questions you should expect.
On this one, Felicia volunteers.
If Felicia volunteers, then
Terry does not.
Now it’s time for the chain.
If Terry does not volunteer,
Masatomo also does not.
If Masatomo does not volunteer, then Rochelle does not.
Rochelle or Leah must be selected, so now Leah volunteers.
Veena and Salman are still up in the air. At least one of them must
volunteer.
Leah must volunteer, so (A) is the correct answer.
The deductions on this
question are very similar to #9.
BP Minotaur:
Question #12 (absolute, must be true)
This is the first and only absolute question on this game. The key here is to
know how to approach a question like this. It asks for a pair of employees
such that at least one of them must volunteer. In other words, you need to
find a pair that can’t both be Out.
If they ask a question like this,
check the answers by putting both of
the players in the Out group. If it
doesn’t work, then you know at least
one of the two must be In, and you
found the answer.
Ninja Note:
To make this question easier, just think about the grouping relationships.
The baby relationship is the one in which you have to have at least one of
two players. So this question is asking for you to identify a baby
relationship. If you check out the deductions, we actually have a bunch of
them.
(A) Terry and Felicia hate each other, so they cannot both volunteer.
But they can both be Out.
(B) If Leah doesn’t volunteer, then
Rochelle must volunteer. If
Rochelle volunteers, then
Masatomo volunteers. Thus, if
Leah doesn’t volunteer, then
Masatomo volunteers. And if
Masatomo doesn’t volunteer, then Leah does. You should
recognize that as our baby relationship. Either Leah or
Masatomo (or both) must volunteer. Bam.
(C) If Leah doesn’t volunteer, then Veena also doesn’t volunteer. So
they could both be Out.
(D) If Salman doesn’t volunteer, then Rochelle doesn’t volunteer. This
one is similar to (C) - they could both be Out.
(E) Just like the last two answers, if Salman doesn’t volunteer, then
Terry doesn’t volunteer. So they could both be Out.
That takes care of the volunteering game. Many of the same challenges were
present. There were a variety of conditional rules, and it was important to
simplify the rules whenever possible and combine them to form deductions.
The next game will present some new challenges. We are going to walk
through it together using the Blueprint Building BlocksTM technique, but you
will do most of the work on your own.
JUNE 2003: GAME 4 (18-23)
The first two grouping games that we covered had a number of similarities.
First, they both involved In and Out grouping. In addition, they were (1)
unstable grouping games and (2) the players did not come from different
categories.
In this game, you get to see something different. And you get to see monkeys,
pandas, and raccoons. We know you’re excited. Let’s go.
1. Setup
For a behavioral study, a researcher will
select exactly six individual animals from
among three monkeys—F, G, and H—three
pandas—K, L, and N—and three raccoons
—T, V, and Z. The selection of animals for
the study must meet the following
conditions:
This game would be much more entertaining if
they gave names to the pandas.
First, you must identify the type of game. The basic task here is to select the
animals that are included in the behavioral study. Some of them are included
and others are not. That spells In and Out grouping.
Next, you should search for restrictions placed on the sizes of the groups.
Well, look at that. For the first time, they do define the sizes of the groups.
Exactly six of the nine animals must be selected.1 This a stable grouping
game, and the group sizes must be incorporated into your setup.
But you aren’t done yet. The
third thing to notice is that the
players in this game come
from different categories. It’s
not just nine random animals.
There are monkeys, pandas,
and raccoons. This introduces
in: another level of complexity
to the game. Not only will you
out: have to track which
animals are selected, you will
also have to track how many animals are selected from each category.
(e.g. How many pandas could be selected?)
2. Rules
At this point, you should feel more comfortable with the grouping
relationships. So we are going to turn this into a challenge. It’s important to
diagram the rules correctly, but we also want you to attempt to combine rules
if possible.
Challenge: Diagram the rules and look for helpful
combinations.
F and H are not both selected.
N and T are not both selected.
If H is selected, K is also selected.
If K is selected, N is also selected.
Here we have a little bit of hate combined with some light stalking. Each of
these rules fits nicely into one of the common grouping relationships.
F and H cannot both be selected, so they hate each other. At least one of
them must be Out.
N and T also hate each other. You can’t select both of them.
If H is selected, then K is also selected. This means that H cannot be
selected if K is not selected. But you could have K without H.
If K is selected, then N is also selected. This is the stalker relationship,
just like the last rule.
Here’s an overview of the setup and the rules at this point:
3. Deductions
There are a number of helpful deductions in this game. First, it’s helpful to
notice that there are a bunch of animals not involved in any of the rules.
G (monkey), L (panda), V (raccoon), and Z (raccoon) are all randoms
by virtue of not being mentioned in any rules.
But now it’s time to focus on the animals that are involved in the rules. The
chain that you build in this game isn’t quite as impressive as in the last game,
but it’s going to be very helpful nonetheless. The best place to start is with H.
If H is selected, then K is
selected.
If K is selected, then N is
selected.
If N is selected, then T is not selected.
Just like in the previous games, it’s helpful to work through the
contrapositives as well.
If T is selected, then N is not
selected.
If N is not selected, then K is
not selected.
If K is not selected, then H is not selected.
Ninja Note:
If H is selected, you also know that F can’t be
selected.
At this point, we have a good grasp on the game and the rules. However, we
are not quite done. When you are working through a
grouping game with players from different categories,
you want to search for rules that involve two players
from the same category. For instance, if there is a rule
about two pandas, you want to check that out.
F and H cannot both be selected. F and H are
both monkeys. Put those two things together and
you get a nice deduction: You cannot select all
three monkeys. In other words, the most monkeys you can have is two
(very disappointing).
If two monkeys are selected, the combo can’t be F and H. Thus, G must
be one, and F or H must be the other.
Always search for
deductions about how many players
can be selected from each category.
BP Minotaur:
With all of that, we are ready for the questions. But don’t get too
comfortable. You still have a fair amount of work in front of you.
1 If math is not your strong suit, feel free to use your fingers to count the total number
of players. Also, make sure to wear open-toed shoes to the LSAT in case the total
goes over 10.
4. Questions
There are two open issues that will form most of the questions in this game:
(1) Which animals are selected?; and (2) How many animals can be selected
from each category?
We are going to go about this a bit differently. You are now going to attempt
all of the questions on your own. We will give a slight bit of assistance along
the way, but we won’t discuss the questions and answers until you have
completed them all.
Challenge: Work through the questions on your own.
Question #18
18. Which one of the following is an
acceptable selection of animals for the
study?
F and H are not both selected.
N and T are not both selected.
If H is selected, K is also selected.
If K is selected, N is also selected.
(A) F, G, K, N, T, V
(B) F, H, K, N, V, Z
(C) G, H, K, L, V, Z
(D) G, H, K, N, V, Z
(E) G, H, L, N, V, Z
No big shock here. We start off with an elimination question.
Question #19
19. If H and L are
among the animals
selected, which one of
the following could be
true?
L is random, but H is a
big one.
(A) F is selected.
(B) T is selected.
(C) Z is selected.
(D) Exactly one panda is selected.
(E) Exactly two pandas are selected.
Question #20
Next up is an absolute question. Time to check our deductions. Remember
when we combined all of those conditional rules? Turns out that was a good
idea.
20. Each of the following is a pair of animals that
could be selected together EXCEPT
(A) F and G
(B) H and K
(C) K and T
(D) L and N
(E) T and V
Question #21
21. If all three of the raccoons are
selected, which one of the following must
be true?
Plug in the
three raccoons
(T, V, and Z)
and go!
(A) K is selected.
(B) L is selected.
(C) Exactly one monkey is
selected.
(D) Exactly two pandas are selected.
(E) All three of the monkeys are selected.
Question #22
22. If T is selected, which one of the
following is a pair of animals that must be
among the animals selected?
(A) F and G
(B) G and H
(C) K and L
(D) K and Z
(E) L and N
Even though you’re doing these questions on your own, we aren’t going to let
you waste your precious time. On the last question, selecting T created lots of
deductions. Now, you get T again. Don’t recreate your work - just check
out the hypothetical for #21.
Question #23
Last question!
23. The selection of animals must include
(A) at most two of each kind of animal
(B) at least one of each kind of animal
(C) at least two pandas
(D) exactly two monkeys
(E) exactly two raccoons
The last one here is an absolute question. And it’s a tough one. Feel free to
use work from previous questions to help you eliminate answer choices.
At this point, you have to stop playing with the monkeys and pandas. With
deductions in hand, we are hoping you felt pretty comfortable running
through the questions. Here’s an overview so you can check your work.
Question #18 (elimination, could be true)
F and H cannot both be selected. That gets rid of (B).
N and T cannot both be selected. Say goodbye to (A).
If H is selected, then K must be selected. (E) doesn’t pass the test.
If K is selected, then N is also selected. This kicks out (C).
(D) is the winner.
Question #19 (conditional, could be true)
As expected, selecting H leads to all kinds of deductions. They are outlined
on the next page.
If H is selected, then F
cannot be selected.
If H is selected, then K
must be selected.
If K is selected, then N is also selected.
If N is selected, then T cannot be selected.
(A) Nope. F can’t be selected.
(B) Ditto. T can’t be selected.
(C) Sure. Z is totally random and we have two open spots. Z could
be selected. There’s our answer.
(D) All three of the pandas (K, L, and N) are already selected.
(E) See (D).
Question #20 (absolute, must be false)
If K is selected, then N is also
selected. If N is selected, then T
cannot be selected. Thus, if K is
selected, then T is not selected. And if
T is selected, then K is not. So K and
T cannot both be selected.
Since K and T cannot both be selected, (C) is the answer.
On a question like this, it’s
unlikely that a random will be correct
since there are no restrictions on that
player. (A), (D), and (E) give you G, L,
and V, respectively, which are all
randoms. Kill those answers quickly.
Ninja Note:
Question #21 (conditional, must be true)
If T is selected, then N is out.
If N is out, then K is out.
If K is out, then H is out.
All of a sudden, the Out group is full. The remaining animals (F, G, and
L) must all be selected.
Even though L is totally random in this game, L must be selected
because there are no spots left in the Out group. Thus (B) is the
answer.
In stable grouping games,
big deductions follow when you can
fill up one of the groups.
BP Minotaur:
Question #22 (conditional, must be true)
In question #21, the selection of T told us exactly which three animals
weren’t selected (N, K, and H), which completed the entire hypothetical. This
question relies on the exact same deductions.
N, K, and H are the three
animals that aren’t selected, so
the other animals all must be In.
To find an answer, just eliminate
anything with N, K, or H in it. There goes (B), (C), (D), and (E). (A)
gives us the combo of F and G.
Question #23
(absolute, must be true)
The tricky part about this question is that we don’t know any animals that
must be selected. So it seems odd that they would ask a question about
animals that must be included. But this question relies on the categories and
not the individual animals.
(A) This one definitely doesn’t work. On question #21, we had to
answer a question in which all three raccoons were selected. The
three pandas could also all be selected.
(B) It’s very important to know how to test this answer choice. The
answer says that you must always have at least one of each kind
of animal. To test whether that is true, you should test to see if
it’s possible to put all three of any category in the Out group.
The nice thing about this game is that you only have three
animals that aren’t selected. So as soon as you don’t select one
category of animal, the other animals must all be selected.
If none of the monkeys (F, G, and H) were selected, then both N
and T would have to be selected. That’s not allowed.
If none of the pandas (K, L, and N) were selected, then both F
and H would have to be selected. So that’s no good.
If none of the raccoons (T, V, and Z) were selected, then both F
and H would have to be selected. Strike three.
When you put all of that together, you can conclude that you must
always have at least one monkey, at least one panda, and at least
one raccoon. So (B) is the final answer. This is not a deduction you
should expect yourself to spot at the beginning of the game.
(C) In both questions #21 and #22, L is the only panda selected. So
you don’t have to select at least two pandas.
(D) This one can be tempting. As we just discussed in (B), at least one
monkey must be selected. In addition, we deduced in the beginning
of the game that at most two monkeys can be selected. However, it
is possible to select only one monkey. For instance, in question
#19, H could be the only monkey selected.
(E) This one goes out just like (A). In question #21, all three raccoons
were selected.
That takes care of our monkeys and raccoons. It was a wild ride. There were
two important differences in that game. First, the players (animals) came
from different categories. This informed our deductions, and there were a
couple tough questions related to how many players could be selected from
each category. Second, the group sizes were stable. This had a big effect on
the deductions in the game since we had to always keep track of how many
free spots remained in each group.
Say goodbye to In and Out grouping games.
As you learned over the past few games, it’s essential to understand and
simplify the big four grouping relationships. The most helpful deductions are
found when combining the conditional rules. Make sure to review the
beginning of this chapter before moving on.
The defining characteristic of the games in this chapter was the selection of
one group. Some of the players were selected, and others were left behind. In
the next chapter, we evolve. We are still going to be looking at grouping
games, but now there will be more groups from which to choose.
18/TWOgroups
THIS WAY OR THAT WAY
From this point on, we are going to be attacking grouping games that include
two or more groups. Rather than selecting who makes the team as we did in
the last chapter, now we must deduce to which team players are assigned.
This is one of the basic distinctions that we introduced when we first
discussed grouping games.
We are going to start off by discussing games that have two distinct
groups. Turns out, they are pretty similar to In and Out grouping games.
Rather than forming an In group and an Out group, now you might have a red
team and a blue team, or classroom 1 and classroom 2, or gold stars and
silver stars. But many of the rules and deductions will function in a similar
fashion.
Other games will assign
players to three, four, or even five
groups, but such games will be covered
in later chapters.
BP Minotaur:
Let’s walk through the best way to set these games up and
approach the rules.
THE SETUP
The setup for games with two distinct groups is going to look nearly identical
to In and Out games. Make sure to label the groups with the appropriate
group names. Here’s a quick example:
Eight local parents will be assigned to two
four-person neighborhood watch
committees—the nighthawks and the
watchbirds. Each parent is assigned to one
of the committees. The assignment must be
consistent with the following conditions:
Stable or Unstable?
Just like we did in the last chapter, it’s important to track whether these
games are stable or unstable. Both types are common.
The game above specifies the size of each group (four), so it’s stable. But a
game could just tell you that each committee has at least two members, or
even give you no restrictions at all.
THE RULES
It’s all about the big four, just like before. (That was not an intentional
rhyming scheme.) Games with two distinct groups continue to center around
conditional rules. It’s crucial to simplify the rules and understand the
basic relationship between the players. When you do so, you will find that
the relationships are very similar to In and Out grouping games. You get a
little love, a little hate, maybe even a baby now and then. However, there are
slight differences in how the rules function (due to the different nature of the
game).
Before we look at the specific rules, there’s one principle to cover. When you
are assigning players to one of two groups, you can simplify a lot of rules.
When there are only two options, if you aren’t in one group, you must be in
the other.1 Check out this example:
If Cleopatra is assigned to team 1, then
Demetrius is assigned to team 2.
Assume you are doing a game in which the players are assigned to team 1 or
team 2. The original rule is easy enough to diagram. The contrapositive of
that statement is, “If Demetrius is not assigned to team 2, then Cleopatra is
not assigned to team 1.” But that complicates the issue. If Demetrius is not
assigned to team 2, then he is assigned to team 1. If Cleopatra is not assigned
to team 1, she is assigned to team 2. Thus, the contrapositive can be
simplified to state, “If Demetrius is assigned to team 1, then Cleopatra is
assigned to team 2.” Much better.
Simplifying the contrapositive
doesn’t change anything at all. You are
writing the exact same relationship in a
more helpful way.
Ninja Note:
Let’s walk through the different relationships in a game with two groups. For
these rules, assume that you are doing a game with just two groups: team 1
and team 2.
Stalker (Follows Along)
The stalking continues in these games. If one player is assigned to a team,
then someone else follows along. Kinda creepy. Check out the example on
the next page.
If Kilowatt is assigned to team 2, then
Lynly is assigned to team 2.
If Kilowatt is assigned to team 2, then Lynly is assigned to team 2. So Lynly
is the stalker. Zenas and Yavar are assigned to the same team. So Kilowatt
also does some stalking. As always, stay away from the conditional fallacies.
What do we know if Lynly is on team 2? Nada.
I gots it - those two is in
love. Gotta be together.
Cleetus Comment:
Nope. It can be tempting to draw that conclusion, but it’s important to avoid
that mistake. If Kilowatt is on team 2, then Lynly is on team 2. And if Lynly
is on team 1, then Kilowatt is on team 1. So there is a good chance that they
will be together on some questions, but Lynly could be on team 2 if Kilowatt
is on team 1. It’s possible for them to be apart (sad, but true).
Love (Must Be Together)
For all of you romantics out there, we are still going to see some love. Many
rules will assert that two players must be assigned to the same team.
Zenas and Yavar are assigned to the same
team.
Now that’s a cute couple - Zenas and Yavar. According to this rule, they
must be together, so they are in love. They must both be assigned to team 1
or both be assigned to team 2.
Since there are only two
options for these players, this is a good
place to search for deductions.
Ninja Note:
Hate (Cannot Be Together)
Then, it all goes south. A very common rule in these games will establish that
two players cannot be assigned to the same team. We’ve entered a world of
hate.
Jacqueline and Henry cannot be assigned
to the same team.
Jacqueline and Henry can never be assigned to the same team due to their
intense hatred for each other. We represent this rule in a similar fashion to In
and Out grouping games.
When we were working
through In and Out grouping games,
two players in a hate relationship
couldn’t both be selected. But it was
acceptable for neither one to be
selected. In a game with multiple
groups, this rule should be interpreted
to mean that the two players cannot be
together in any group.
BP Minotaur:
That’s an important point. Even though the rules look identical, they work in
a slightly different fashion depending on what type of game you are doing.
Here’s the cool part. When you are doing a grouping game with two
groups, the hate relationship is pure gold. You have two groups, and
now there are two players that cannot be together. Rather than keeping this
rule off to the side, it should immediately be inserted into the setup as
linked options.
If you have two
groups and they give you
a hate relationship, make
linked options in your
setup. This will make it
much easier to visualize
this rule and to make
further deductions.
Ninja Note:
To recap, this rule asserts that two players cannot be together in any group.
It’s best to visualize this relationship by forming linked options in the two
groups. There is, however, a more limited version of the hate relationship.
It’s an important one to simplify. Take a look.
If Gertrude is assigned to team 2, then
Fonzi is assigned to team 1.
The first step is to make sure to simplify the contrapositive. If Gertrude is on
team 2, then Fonzi is on team 1. This also tells us that if Fonzi is not on
team 1 (or, more simply, if he is on team 2), then Gertrude is not on team 2
(she is on team 1). But wait, you can even go a little farther...
So they totally hate each
other - like Brad and Jen.
Ditz McGee:
Not exactly. It might appear that good old Gertrude and Fonzi can never be
together, but take a closer look at the rule. If Gertrude is on team 2, then
Fonzi can’t be on team 2. And if Fonzi is on team 2, then Gertrude is not on
team 2. But what if they were both assigned to team 1? Any problems with
that? Nope.
This rule establishes that Gertrude and
Fonzi can’t both be assigned to team 2.
Either one of them could be on team 2
without the other, or they could both be
assigned to team 1. We can represent this
with a variation on the classic hate relationship.
A complicated game could
present you with a medley of these
rules. Staring at four or five rules
coupled with contrapositives can be
overwhelming. Make sure to work
through the rules slowly so you can
simplify each relationship to its most
basic form.
BP Minotaur:
Now for some active learning. Rules such as the one above can make or
break a game, so let’s make sure you got it down. For this drill, continue to
assume that everyone is assigned to either team 1 or team 2.
Challenge: In the space provided, represent the rule
and try to simplify the diagram. When you are ready,
attempt the questions that follow.
If Rusty is assigned to team 1, then Krusty
is assigned to team 2.
1. If Krusty is assigned to team 1, which team is Rusty
assigned to?
2. Could both Rusty and Krusty be assigned to team 1?
3. Could both Rusty and Krusty be assigned to team 2?
Here’s the proper diagram for this rule:
1. 2. This one comes straight from the
contrapositive. If Krusty is on 1, Rusty is on 2.
2. Nope. The simplified version of this rule tells
you that it’s impossible for both Krusty and
Rusty to be assigned to team 1.
3. Fo-sho (translates to “yes”). There’s no problem with both Rusty and
Krusty hanging out together on team 2.
Just one more relationship...
Baby (At Least One)
The baby relationship will not show up in grouping games with two groups,
so don’t worry about babies for now.2 These rules will return, however, in
grouping games with more than two groups.
You will still confront
situations where at least one of two
players must be assigned to a team. For
example, the rule above states that
Rusty and Krusty cannot both be
assigned to team 1. That implies that
either Rusty or Krusty (or both) must
be assigned to team 2.
BP Minotaur:
That concludes our introduction to grouping games with two groups. We
have a crazy idea - let’s try one. We will attack the first game using our
Blueprint Building BlocksTM technique, and then you will tackle the second
one all by your lonesome.
1 It’s just like real life. If you aren’t unemployed, then you have a job. If you aren’t a
virgin, then you’ve had sexual relations at some point (congrats).
2 Phew... it’s like the pregnancy test came back negative.
OCTOBER 2003: GAME 3 (13-17)
Next up, we get to hobnob with some executives. It’s time to assign members
of a board of directors to different committees. Yee-haw! There are going to
be two groups. That’s no surprise. But the challenge is to simplify the rules
and spot those deductions. Let’s do it.
1. Setup
Each of the seven members of the board of
directors— Guzman, Hawking, Lepp,
Miyauchi, Upchurch, Wharton, and Zhu—
serves on exactly one of two committees—
the finance committee or the incentives
committee. Only board members serve on
these committees. Committee membership
is consistent with the following conditions:
First thing you probably noted are the powerful names. Miyauchi? Zhu?
Upchurch? Once you get past all that, it’s time to focus on the game and
build an appropriate setup.
This is clearly a grouping game, evidenced by the fact that players are being
assigned to different committees. There are two groups - the finance
committee and the incentives committee. The first question should always
be the same: Do you know how many players are in each group? The answer
here is a resounding no. At this point, the finance committee could have zero,
one, two, three, four, five, six, or seven members. We have an unstable
grouping game on our hands.
The setup at this point is relatively simple. You have seven players and two
groups. Don’t put any slots in the groups because you wouldn’t want to fool
yourself into thinking that a
minimum has been established for
either group.
But they both gotta have
at least one, right?
Cleetus Comment:
Actually, no. A lot of games will give you that restriction, but this one
doesn’t. It’s likely that the rules will forbid you from putting all seven players
in either group (it would be kinda boring if everyone served on the finance
committee), but don’t take anything for granted. There are some ugly,
unstable grouping games in which it’s possible to have no players assigned to
one of the groups. Now let’s tackle the rules.
2. Rules
Even at this early stage, you should have some idea of what to expect in the
rules. First, we want to watch for restrictions on the sizes of the groups.
Second, since this is a grouping game with two groups, we know there will
be conditional rules that we can simplify. Bring it on, LSAT. You are going to
have the chance to diagram each of the rules.
Challenge: Diagram the first two rules and attempt to
simplify the relationships.
Remember to always simplify the contrapositive since there are only two
options for each player in this game. If someone isn’t on the finance
committee, they are on the incentives committee, and vice versa.
I simplified that first one
because it looks like Hawking and
Guzman can’t be together.
Ditz McGee:
Good try, Ditz, and we’re glad you’re attempting to simplify. But that’s a
little off. Here’s an overview of these two rules:
The first step is to diagram the rule correctly. If Guzman serves on the
finance committee, then Hawking serves on the incentives committee. Also,
if Hawking serves on the finance committee (not the
incentives committee), then Guzman serves on the
incentives committee (not the finance committee).
But it gets better. If either Guzman or Hawking is
on the finance committee, the other can’t be. So
the most straightforward way of representing
this rule is to say that Guzman and Hawking
can’t both be on the finance committee. They could, however, both happily
be assigned to the incentives committee.
The second rule is one that bewilders and destroys students. First, you have to
make sure to diagram the original rule and contrapositive correctly.
If Lepp serves on the finance committee, then both
Miyauchi and Upchurch serve on the incentives
committee. Thus, if either Miyauchi or Upchurch
serves on the finance committee (not the incentives
committee), then Lepp must serve on the incentives committee (not the
finance committee).
Somehow, for some reason, diagramming this rule correctly still leaves you
with an unsatisfied feeling. Why? Because that rule is really ugly.1 It’s not
going to be fun dealing with rules like that when you are running through the
questions. Thankfully, there’s a way to simplify this further.
If Lepp is on the finance committee, then neither
Miyauchi nor Upchurch can be on the finance
committee. And if either one of them is on the finance
committee, then Lepp cannot be. Therefore, Lepp
cannot be on the finance committee with Miyauchi, and Lepp also can’t be
on the finance committee with Upchurch. This version of the rule is much
easier to handle.
The first two rules place three
hate relationships on the finance
committee. Gotta keep our eye on that.
Ninja Note:
And now, let’s keep moving.
Challenge: Diagram the third and fourth rules (Hint:
these rules should be inserted directly into your setup,
which has conveniently been included).
The third and fourth rules are similar. They both give you
hate relationships. According to the third rule, Wharton
and Zhu can’t be assigned to the same committee.
Then, the hate spreads. The fourth rules says that
Upchurch and Guzman can’t be assigned to the same committee.
Normally, you want to diagram these conditions in your big list of rules at
the bottom of the page. However, this game only features two groups.
When that happens, there is a much better way to deal with such hate
relationships. Wharton and Zhu have to be split up, so you should form
linked options for the two of them. Do the same for Upchurch and
Guzman.
Since your eyes will be locked on your setup as you work
through the questions, this is an easy way to remember to
keep these players separated. In this game, four of the
seven players are involved in linked options.
These two rules also restrict
the possible sizes of the groups. Both
committees must have at least two
members, but no more than five.
Ninja Note:
There’s just one more rule. That’s one more chance to work on your
diagramming skills! They’re looking pretty strong at this point.
Challenge: Diagram the final rule.
They mix it up a little bit for the last rule, but this is another common
grouping relationship. If Zhu serves on the finance committee, then Hawking
is also on the finance committee. Sound like a stalker to you? And if
Hawking decides to go against the grain and serve on the incentives
committee instead of the finance committee, then Zhu can’t serve on the
finance committee. Sorry, Zhu. Gonna have to settle for incentives.
Make sure you diagrammed this one correctly. Other
important points: (1) Both Zhu and Hawking were
involved in earlier rules, so that might lead to deductions,
and (2) this is yet another rule about the finance
committee.
A fair proportion of students
misinterpret this rule to imply that Zhu
and Hawking must be together. Not the
case - it’s acceptable to assign Zhu to
the incentives committee and Hawking
to the finance committee.
BP Minotaur:
That brings us to the end of the rules for this game. Here’s a quick look at our
setup and rules:
2. Deductions
Believe it or not, most of the heavy lifting is already done. The rules are a
huge obstacle in this game. If you walk through them slowly, simplify when
appropriate, and visualize the linked options in your setup, as we just did, you
are in good shape. Otherwise, well...
Sometimes, it’s helpful to check out the way
the other side lives. We simplified the rules
and ended up with the beautiful situation
above. What would a less advanced
student’s setup look like at this point? Let’s
show you. Take a look at the monstrosity to
the right. Do you think that’s gonna make it
easy to find deductions? Not so much. This
student diagrammed the rules correctly, but
he failed to simplify or incorporate anything
into the setup. You can already see the frustration and wrong answer
choices coming. Small maneuvers can make a big difference.
Now let’s get back on the productive side of things. As always, you want
to look for relationships between the rules. We have two rules about
Hawking, two rules about Guzman, and two rules about Zhu, so we know
those are going to be the big guns in this game. We are going to give you a
slight push and ask you to put all the rules together.
Challenge: Investigate two situations: (1) Guzman on
the finance committee; and (2) Wharton on the
incentives committee. In each one, try to place as many
other players as possible.
In this challenge, we had you do things a little differently. Rather than just
linking together the conditional rules, you filled out two hypotheticals.
Either way can help to uncover how the rules work together. Hopefully,
you were able to place a number of the committee members in each
situation.
First, let’s talk about Guzman on that finance committee:
If Guzman is on the finance committee, then Upchurch is on the
incentives committee.
If Guzman is on the finance committee, then Hawking is also on the
incentives committee.
If Hawking is on the incentives committee, then so is Zhu.
If Zhu is on the incentives committee, then Wharton is on the finance
committee.
And now let’s play with Wharton on the incentives committee:
If Wharton is on the
incentives committee, then
Zhu is on the finance
committee.
If Zhu is on the finance committee, then Hawking is as well.
If Hawking is on the finance committee, then Guzman is on the
incentives committee.
If Guzman is on the incentives committee, then Upchurch is on the
finance committee.
If Upchurch is on the finance committee, then Lepp is on the incentives
committee.
At this point, we are in good shape. It’s crucial to simplify the rules in this
game. We took it one step farther and found some deductions by combining
rules. All of this hard work will pay big dividends when we tackle the
questions.
1 How ugly? Like “not even after too many shots of tequila” ugly. Like “not even if
we were stranded on a deserted island” ugly. Like “not even after too many shots of
tequila on a deserted island” ugly. That ugly.
4. Questions
Question #13
13. Which one of the following could be a
complete and accurate list of the members
of the finance committee?
Before we actually solve this question, we need to talk
a little bit of strategy.
Here’s our elimination question. That’s no big surprise. However, before
we get to this question, we need to discuss a bit of strategy. In the beginning
of this chapter, we noted a few approaches you should take due to the nature
of grouping games with only two groups. If a player isn’t in one group, they
must be in the other. This is just as important in elimination questions as
well.
When you only have two groups, you have to remember that anyone not in
one group must be in the other. On an elimination question, while they are
only showing you the players assigned to one group, anyone not on that list
must be in the other group. It might seem like you only have half of the
info, but they’ve really given everything away. This question relates to the
finance committee, but anyone not mentioned in an answer choice must be
on the incentives committee. You will have to eliminate answer choices
because they don’t work for the finance committee (which you can see) or
the incentives committee (which you can’t).
Challenge: On the next page, identify the set of
directors that could work together on the finance
committee.
If Guzman serves on the finance
committee, then Hawking serves on the
incentives committee.
If Lepp serves on the finance committee,
then Miyauchi and Upchurch both serve on
the incentives committee.
Wharton serves on a different committee
from the one on which Zhu serves.
Upchurch serves on a different committee
from the one on which Guzman serves.
If Zhu serves on the finance committee, so
does Hawking.
(A) Guzman, Hawking, Miyauchi,
Wharton
(B) Guzman, Lepp, Zhu
(C) Hawking, Miyauchi, Zhu
(D) Hawking, Upchurch, Wharton, Zhu
(E) Miyauchi, Upchurch, Wharton
If someone isn’t listed here, they must be on the
incentives committee.
Watch for hate relationships on elimination
questions with only two groups. If two players can’t be
together, an answer choice must have one of the players,
but it can’t have both. Thus, you can eliminate answers
Ninja Note:
that have both players or neither one.
The first rule establishes that Guzman and Hawking cannot both be on the
finance committee. (A), get out of here.
The second rule is contingent upon Lepp being on the finance committee,
which only occurs in (B). But neither Miyauchi nor Upchurch are on the
finance committee with Lepp, so (B) looks fine. For now.
Wharton and Zhu cannot be on the same committee, so (D) is gone.
The fourth rule tells you that Upchurch and Guzman hate each other. At
first glance, it might appear that this rule doesn’t eliminate anything. But
wait. This is exactly the dangerous trap about which the Ninja just warned
you. Upchurch and Guzman don’t both serve on the finance committee in
any answer choice, but we also have to check to see if they both serve on
the incentives committee in any answer. If neither Guzman nor Upchurch
shows up in an answer choice here, then they must both be on the
incentives committee. Like in answer choice (C). No bueno.
According to the final rule, if Zhu serves on the finance committee, then
Hawking is on the finance committee. In (B), you find Zhu without
Hawking. (B) is also a loser.
After a fair deal of work, (E) is the winner.
Question #14
14. Which one of the following pairs of board
members CANNOT both serve on the incentives
committee?
Deduction
time!
Next up is an absolute question. These questions are always an early test of
our deductions. The challenge is to find two players that can’t both be
assigned to the incentives committee. In this game, there were lots of rules
about the finance committee, but not very many about the incentives
committee. That should help you know where to look.
Challenge: Find the pair that can’t serve on the
incentives committee together.
(A) Guzman and Hawking
(B) Guzman and Wharton
(C) Hawking and Wharton
(D) Miyauchi and Upchurch
(E) Miyauchi and Wharton
The only rule about the
incentives committee is related to
Hawking. You can expect Hawking to
show up in the correct answer.
Ninja Note:
If Hawking serves on the incentives
committee, then Zhu also serves on the
incentives committee.
Wharton and Zhu don’t serve on the same
committee. If Zhu serves on the
incentives committee, then Wharton
serves on the finance committee.
It also works the other way. If Wharton serves on the incentives
committee, then Zhu serves on the finance committee, and so does
Hawking.
Hawking and Wharton cannot both serve on the incentives
committee, so (C) is the credited response.
Question #15
15. What is the maximum number of members
on the finance committee?
This one
looks a bit
scary.
Unstable grouping games will
commonly ask questions about the
maximum or minimum number of
players in a group. That’s why it’s
important to watch for deductions of
this sort.
Ninja Note:
As Ninja points out, this question isn’t a huge surprise. It can be challenging
because it’s not clear how to attack it. Just start plugging people into the
finance committee and see how many you can fit? Not exactly.
The first step is to review any rules or deductions that you have which
limit the sizes of the groups. If you remember, we discussed this earlier in
this game.
The linked options that we formed in the beginning of the game helped
us deduce that each group must have at least two players (either
Wharton or Zhu and either Upchurch or Guzman).
At this point, you might be tempted to say that the maximum number of
players that could be assigned to either group is five. However, it would be
foolish to just pick that answer without checking it. Also, we have noted a
number of times that there are a bunch of restrictions on the finance
committee. Those restrictions could easily cut down the maximum size of the
finance committee. Here’s the correct way to test the maximum for the
finance committee:
We need to look at the rules
and try to figure out the player
or players that are going to
cause trouble for us here. We
want to put as many people as
possible on the finance
committee. So you should
seek out players that would
force others away from the finance committee.
If Lepp is on the finance committee, then both Miyauchi and Upchurch
have to go on the incentives committee. To maximize the members of
the finance committee, you should first throw Lepp on the incentives
committee.
Since Lepp now serves on the incentives
committee, Miyauchi is free to serve on the
finance committee.
At this point, you’re in pretty good shape. You
have six of seven slots in the setup. The question at this point is whether
you can place anyone else on the finance committee.
The only unassigned player left is
Hawking.
Hawking could be assigned to the
finance committee as long as Guzman
gets bumped down to the incentives committee.
If Guzman is on the incentives committee, then Upchurch is on the
finance committee.
We now have Upchurch,
(A) two
Miyauchi, Hawking, and
(B) three
either Wharton or Zhu on
(C) four
the finance committee. The
(D) five
maximum is four, so (C) is
(E) six
the answer.
To determine the maximum
members of a group, always look to the
hate relationships. Generally, every
hate relationship knocks the maximum
down by one. However, you should
always test to make sure that other
rules don’t complicate the issue.
BP Minotaur:
Question #16
16. If Miyauchi and Wharton both serve on the
finance committee, then which one of the
following could be true?
At long last,
it’s conditional
time
Here is the first conditional question for the
game. New condition: Miyauchi and Wharton
serve on the finance committee. The first deduction about Wharton comes
nice and quick.
Wharton and Zhu can’t serve on the same committee. Since Wharton is
on the finance committee, Zhu must be on the incentives committee.
They aren’t going to let you off that easy. So now you want to look for
further deductions. This one is up to you.
Challenge: Try to place more committee members, and
find the answer that could be true.
(A) Guzman and Lepp both serve on the
finance committee.
(B) Guzman and Upchurch both serve on
the incentives committee.
(C) Hawking and Zhu both serve on the
finance committee.
(D) Lepp and Upchurch both serve on the
incentives committee.
(E) Zhu and Upchurch both serve on the
finance committee.
There aren’t as many deductions here as you might have expected. But that
happens sometimes. Remember, this is just a could be true question, so you
are unlikely to be able to completely determine the committees.
If Zhu is on the incentives
committee, then so is Hawking.
Ditz McGee:
Nope. That is the converse. Be careful. Here are the actual deductions:
If Miyauchi serves on the finance committee,
then Lepp must serve on the incentives
committee.
At this point, there are still some things you don’t know. For example,
Hawking is totally up in the air. But you can eliminate answers that are
false.
Lepp must serve on the incentives committee, proving (A) must be false.
Zhu must also serve on the incentives committee. Get rid of (C) and (E).
Answer choice (B) breaks the original condition that Guzman and
Upchurch cannot serve on the same committee.
It could be true that both Lepp and Upchurch serve on the incentives
committee, so (D) looks damn good.
Question #17
17. If Guzman serves on the incentives
committee, then which one of the following must
be true?
Last
question!
The final question is one more conditional challenge. Back to Guzman,
who now must serve on the incentives committee. We aren’t giving you any
help on this one. You gotta bring it home.
Challenge: Find the answer that must be true.
(A) Hawking serves on the finance
committee.
(B) Lepp serves on the incentives
committee.
(C) Miyauchi serves on the finance
committee.
(D) Wharton serves on the incentives
committee.
(E) Zhu serves on the finance committee.
Guzman and Upchurch must serve on different committees. Since Guzman
has claimed a seat on the incentives committee, Upchurch must serve on
the finance committee.
Lepp and Upchurch can’t both serve on the finance committee according
to the second rule. Since Upchurch is on the finance committee, Lepp must
be shoved down to the incentives committee.
Lepp is on the incentives committee, so (B) is
the answer.
Well, folks, that’s going to bring our trip through the boardroom to an end.
Hopefully, you enjoyed that one. The game definitely didn’t enjoy getting
dominated by us.
That kinda felt like one
of ‘dem In and Out numbers.
Cleetus Comment:
That’s actually a legitimate point. Because there are just two groups in this
game, many of the deductions and questions will be similar to In and Out
grouping games.
Here’s a couple quick points to summarize this game:
1. When you have a game with two groups, there are a number of
shortcuts you can use. They all stem from the same basic idea: If a
player isn’t in one group, he or she must be in the other group.
2. Like all grouping games, combining conditional rules is a great way
to find deductions.
The next game continues a number of these themes. You’re on your own for
this one. Take it slow. Focus on the correct setup, simplify the rules, and
search for deductions. Catch you at the finish line.
DECEMBER 1994: GAME 1 (1-6)
Questions 1-6
Exactly eight consumers—F, G, H, J, K, L, M, and N—will be interviewed
by market researchers. The eight will be divided into exactly two 4-person
groups—group 1 and group 2—before interviews begin. Each person is
assigned to exactly one of the two groups according to the following
conditions:
F must be in the same group as J.
G must be in a different group from M.
If H is in group 1, then L must be in group 1. If N is in group 2, then G
must be in group 1.
1. Group 1 could consist of
(A)
(B)
(C)
(D)
(E)
F, G, H, and J
F, H, L, and M
F, J, K, and L
G, H, L, and N
G, K, M, and N
2. If K is in the same group as N, which one of the following must be
true?
(A) G is in group 1.
(B)
(C)
(D)
(E)
H is in group 2.
J is in group 1.
K is in group 2.
M is in group 1.
3. If F is in the same group as H, which one of the following must be
true?
(A)
(B)
(C)
(D)
(E)
G is in group 2.
J is in group 1.
K is in group 1.
L is in group 2.
M is in group 2.
4. If L and M are in group 2, then a person who could be assigned
either to group 1 or, alternatively, to group 2 is
(A)
(B)
(C)
(D)
(E)
F
G
H
J
K
5. Each of the following is a pair of people who could be in group 1
together EXCEPT
(A)
(B)
(C)
(D)
(E)
F and G
F and H
F and L
H and G
H and N
6. If L is in group 2, then each of the following is a pair of people
who could be in group 1 together EXCEPT
(A)
(B)
(C)
(D)
(E)
F and M
G and N
J and N
K and M
M and N
MARKET RESEARCH
Time for the Pepsi challenge. We are heading out for some market research.
The basic structure of this game is similar to the last one, but there are some
important twists and turns along the way. Let’s jump in.
1. Setup
Right up front, you are given a group of eight consumers. All creativity is
absent here - the names are F, G, H, and a bunch of other letters.
Disappointing. The eight consumers are separated into two groups for
market research. At this point, we know we have a grouping game with two
groups.
The next question you ask
yourself should always be
the same: Do I know how
big the groups are? In this
game, the answer is yes.
Each group has four players.
This is a stable grouping
game since you know exactly how many players are assigned to each
group.
Make sure to reflect these restrictions in your setup.
Since there are two groups, we want to simplify
the rules. We also want to search for deductions based on
the stable sizes of the groups.
BP Minotaur:
2. Rules
The
first
rule
gives
you a love relationship. F and J
must always be together. This
is represented with our good
old reciprocal arrow.
F and J are going to make up
half of one of the groups. Keep your
eyes on this powerful twosome.
Ninja Note:
The first step is to recognize this as the hate relationship. G and M cannot be
in the same group. However, there are only two possible groups. This is best
visualized by forming linked options for G and M in the two groups. One of
them goes in group 1, the other in group 2. This representation will be very
helpful in visualizing other deductions.
The third rule is the stalker
relationship. If H is in group 1,
then L must be in group 1. This
also implies that if L is in
group 2 (not group 1), then H
must be in group 2 (not group
1).
Does this mean that H and L
must be in the same group? Nope. H could be in group 2 while L is in group
1.
Onward to the final rule. If N is
in group 2, then G is in group 1.
Don’t mistake this to mean that
they can never be together. G
and N cannot be together in
group 2, but they could be
together in group 1. They only
partially hate each other.
3. Deductions
As in previous games, symbolizing the rules is extremely important.
However, deductions are critical to getting through the questions quickly and
correctly.
Since G and M cannot be in the same group, the other six consumers
must be separated into two groups of three.
Since F and J will take up two of the three remaining slots in one of the two
groups, it’s important to investigate how this love relationship is going to
play out.
F and J must be in the same group. Thus, F, J, and either G or M must
occupy three of the spots in one of the groups.
If H is in group 1, then L must be in
group 1. Since this only leaves one
spot, F and J would then have to be in
group 2.
If L is in group 2, then H must be in
group 2. Since this also leaves one
lonely spot, F and J gotta go in group
1.
At this point, we have simplified the rules and made some great deductions.
Once you see how the rules work together in this one, it’s time to jump in.
4. Questions
Make sure you attacked each question in the most efficient manner.
Since there are no huge
deductions in this game, you should
expect lots of conditional questions.
BP Minotaur:
Question #1 (elimination, could be true)
The fun starts out with an elimination question for group 1.
The first rule states that F and J must be in the same group. Goodbye
(B).
Rule #2 states that G and M cannot be in the same group. This quickly
eliminates (E) because both G and M are in group 1. In addition, this
gets rid of (C) because neither G nor M is included in the answer. That
means they would both be in group 2, which is equally problematic.
According to the third rule, if H is in group 1, L must be in group 1. In
(A), H is in group 1 but L is not. No good.
(D) is the only answer remaining, so it is the winner.
Question #2 (conditional, must be true)
Now, it’s time to test how all the rules work together. This is the first
conditional question for this game. For this one, K and N must be in the
same group.
K is a random in this game, so there are no rules about K to help us out.
In a perfect world, it would be clear to which group K and N must be
assigned. Unfortunately, that’s not the case. But don’t let that stop you. There
are only two groups, so we should jump in and try K and N in both.
If K and N are both in group 1, there’s
no room to add F and J. So they would
have to be in group 2.
The last two consumers are H and L. If
H is in group 1, then L must be in group
1. There’s only one spot left, so that’s
not cool. H must be in group 2, and L
must be in group 1.
If K and N are both in group 2, F and J
can’t squeeze into group 2. F and J are
now pushed into group 1.
The last deduction is just like the situation above. If H is in group 1,
then L must be in group 1. Not enough room for that, so H is in group 2
and L is in group 1.
L is always in group 1 and H is always in group 2. Since this is a
must be true question, start by searching for one of those options.
Bam. (B).
This deduction is very common in stable
grouping games. Group 1 only has one seat left, so H
Ninja Note:
(who would bring L) has to find a
different group.
Question #3 (conditional, must be true)
Let’s run it back - another conditional, must be true question. At this point,
you should feel confident with the rules. Now, F is in the same group as H.
During our deductions phase, we found a relationship between H and F
that will be very helpful here. If H is assigned to group 1, L must be
assigned to group 1. There wouldn’t be enough spots left for F and J, so
F and J have to go into group 2. The short version of the deduction is
that H and F cannot both be in group 1.
For this question, F and H must be in
the same group. That’s gotta be group 2.
Now that we have a huge deduction on our
side, it’s time to fill in the rest of our slots.
F and J must be in the same group, so J
gets to join group 2.
Group 2 is filled to the brim. So the rest of
the consumers must be in the other group.
K, L, and N - say hello to group 1.
Our slots are all determined, minus the
linked options for G and M. Finding the
answer at this point should be easy.
Boom - K must be in group 1, so our
answer is (C).
Question #4 (conditional, could be true)
Our crazy guess at the start of this game is becoming a reality - lots of
conditional questions. For this one, L and M are in group 2.
K is a random in this game.
This question asks for a player that
could be assigned to either group. If we
were in Vegas, the smart money would
be on K.
Ninja Note:
Based on our original linked options, if
M is in group 2, G must be in group 1.
If L is in group 2, then H must also be in
group 2.
F and J must be in the same group.
There’s not enough room in group
2, so they must be in group 1.
K and N are the two remaining
consumers, so we can make a
linked option with the two of them.
The two consumers who could be in either group are K and N. K is
offered in (E), so we take it.
Question #5 (absolute, must be false)
This turns out to be the only absolute question in the whole game. Absolute
questions rely on deductions, and our big deductions take care of this one
quickly. The task is to identify two people who can’t both be in group 1.
Remember the deductions we found regarding H and F? They come in
handy yet again. If H is in group 1, then L must be in group 1. Since
either G or M must always be in group 1, there’s only one spot left. F
and J must be in the same group, so they would have to be in group 2.
When combined, these rules tell us that
H cannot be in group 1 with either F or
J.
(B) gives us the dangerous combo of F and H. Our deduction proves
that those two cannot both be in group 1, so (B) is the winner.
Question #6 (conditional, must be false)
The last question is a conditional challenge. It is centered around a rule that
has been mostly forgotten during the course of this game.
This is another common trick.
They keep a rule waiting in the dark
for a few questions and hope you
forget about it. Then, bam, it comes
back to get you on the last question.
Ninja Note:
If L is in group 2, then H must be in
group 2.
As has become a pattern at this point, we
run out of spots for F and J in group 2. F
and J, please take your seats in group 1.
Those are the basic deductions, but there’s an important thing to realize about
the remaining consumers.
K and N are the last two remaining consumers, so we can make linked
options for them.
Don’t forget about the last rule! G and N can’t both be assigned to
group 2.
It might appear that any of the combinations would work for group 1.
However, if both M and K were in group 1, then G and N would be
forced into group 2. Either M or K could be in group 1, but not both.
If K is in group 1, N must be in group
2. Also, if M is in group 1, G must be
in group 2. Thus, if both K and M are
in group 1, then G and N would both
be in group 2. This is a big no- no.
Since both K and M cannot be in
group 1, (D) is our answer.
First, let’s recap this game. Despite the lack of humongous, game-changing
deductions, there are a few keys to success. First, visualizing the hate
relationship between G and M on the setup shrinks the game and makes it
more manageable. Second, combining the rules to find a few quick
deductions about F and J makes the questions much easier.
It’s been quite a ride through the finance committee and the market research
teams, but that’s going to do it. To review, here are the two big lessons to
take away from this chapter:
1. Grouping games with two
groups are very similar to In and
Out grouping games, from the
setup to the rules and even the
deductions.
2. The key to games with two groups is simplifying the rules. Since
there are only two groups, many rules can be expressed in very
basic terms. This is very helpful when you attack the questions.
There are more grouping games to follow. We just wrapped up games that
involve assigning players to two groups. What in the world could come
next...?
19/MOREgroups
LOTS OF OPTIONS
In the last chapter, you could hang out in one group, or you had to hang out in
the other. But that was all - there were only two options for each player.
Now, our setup is going to grow as we are presented with games including
three, four, or even five distinct groups. Rather than just choosing between
the red and blue teams, we might have to choose between the red, blue,
green, orange, purple, aqua, mauve, and chartreuse teams. Yikes.
There is really just one fundamental change between earlier grouping games
and the games to follow in this chapter. Our shortcuts go away. First, if you
weren’t in, you had to be out. Then, if you weren’t in group 1, you had to be
in group 2. If Miyauchi didn’t show up on the finance committee, he sure as
hell was going to be found on the incentives committee. But no longer.
Eight parakeets—Sam, Tyrell, Uptown,
Velvet, Waxter, Xena, Yanni, and Zander
—are housed in three separate cages—A,
B, and C—at an estate. Each parakeet is
housed in only one cage. Each cage must
house at least two of the parakeets,
according to the following conditions:
Take a look at this game and the appropriate setup that should be built.
Now, imagine you are working through this game and you find out that
Tyrell, the cute little parakeet, is not in cage B. What does that mean?
Well, Tyrell could be in cage A or cage C.
From this example, you can see that it will be harder to simplify rules in
games with more than two groups. We will have to search elsewhere for
deductions.
Check out this rule as an example:
If Sam is housed in cage C, then
Waxter is housed in cage A.
Sure, Waxter is housed in cage A when Sam is housed in cage C. That’s the
easy part. But the contrapositive can’t be simplified:
If Waxter is not housed in cage A (if he is in either B or C), then Sam
cannot be housed in cage C (he must be in either A or B).
Underbooked? Overbooked?
It’s important to always keep your eyes on the ratio of players to slots. Some
grouping games will present you with more players than available slots,
others with less. Check out the following examples:
Each of three cities—Grantsville,
Hollytown, and Iceburg— will be
awarded grants in exactly two of the
following areas: parks, roads,
schools, theaters, or waterways.
Each grant must be awarded to at
least one of the cities. The
following must obtain:
For this game, your setup shouldn’t change
much. The three cities should be used as the
base, and there are two slots next to each
one for the grants awarded. But wait, check
your math... That’s right, there are six slots.
And only five grants.
Say hello to an underbooked grouping game. It’s important to note at
this point that one grant will be assigned to two of the cities. Deductions
will surely follow.
Nine children—A, B, C, D,
E, F, G, H, and I—will be
separated into four twoperson teams for a sack race.
Each child can be assigned to
only one team. The following
conditions govern the
assignment of children to
teams:
Now we are dealing with four groups. But if each group only has two slots,
only eight children make the four teams. What happens to the ninth child?
Well, probably lots of crying, humiliation, and psychotherapy later in life.
out: But for the purposes of this game, it’s overbooked, so we need an
additional group to keep track of the child that has to sit out the sack
race. So very sad.
The fundamentals that you have learned over the last few chapters will
continue to be helpful when working through these games. There are more
groups, but that doesn’t change most of the basic grouping operations. Let’s
take a look at one of these beasts using our Blueprint Building BlocksTM
technique.
JUNE 2006: GAME 2 (8-12)
You know how you never get anything good in the mail? Really, they should
just scrap the whole idea of a mailbox at this point. Coupon saver? Awesome.
Another furniture catalog? Sweet. New neighborhood realtor? Great. It’s all
pretty useless. In this game, you get to arrange all of the lovely gifts
bestowed upon you by the mailman (presumably before they are discarded).
Remember, the idea here is to try out a game with more than two groups.
Let’s go.
1. Setup
There are exactly five pieces of mail in a
mailbox: a flyer, a letter, a magazine, a postcard,
and a survey. Each piece of mail is addressed to
exactly one of three housemates: Georgette, Jana,
or Rini. Each housemate has at least one of the
pieces of mail addressed to her. The following
conditions must apply:
You’ve never
seen fury quite
like Jana if you
take a postcard
that is addressed
to her. So let’s
take this game
very seriously.
The basic task is to assign the pieces of mail to the housemates, so this is
surely a grouping game. We very commonly use people as the players in a
game and move them around, but that’s not the best strategy here. Since we
know each housemate gets at least one piece of mail and the pieces of mail
are assigned to the housemates, it’s much easier to use Georgette, Jana, and
Rini1 as the base of your setup.
In many grouping games,
people are assigned to groups. But
some games, such as this one, use
people as the groups. Always read
carefully and determine which variable
set is being assigned to the other.
BP Minotaur:
The other crucial piece of information at
this point is the size of the groups. Each
housemate is addressed at least one
piece of mail, but we don’t know the
exact number. This is an unstable
grouping game with three groups.
2. Rules
There’s a lot up in the air at this point. We know the five pieces of mail must
be distributed among the three housemates. Each housemate must receive at
least one piece of mail (nobody wants to feel left out). From here, there are
two general categories of rules we should expect:
1. Common grouping relationships (stalker, love, hate,
baby) among the five pieces of mail.
2. Restrictions on how many pieces of mail each of the
three housemates can receive.
Jumping right in...
This one starts out with some basic restrictions. Georgette receives neither the
letter nor the magazine. There are two basic deductions that should pop into
your mind at this point:
The letter and the magazine must each be addressed to either Jana or
Rini.
Georgette must be addressed the flyer, the postcard, or the survey.
Moving right along...
With this rule, we move right back to conditional land. This is a version of
the stalker relationship. If the letter is addressed to Rini, then the postcard is
addressed to Jana. The contrapositive can’t be simplified much - if the
postcard isn’t addressed to Jana, then the letter is not addressed to Rini.
Ninja Note:
The letter is mentioned in both of the first two
rules. Gotta watch that letter.
Just one more rule to go...
This rule establishes that the flyer can’t be alone in one of the groups
(housemates). Students sometimes have problems representing this rule. They
will commonly write “F not alone” or “F with something else” - neither is
very effective.
If you ever find yourself
transcribing a rule, please stop. The
goal is to represent the rule visually.
BP Minotaur:
That brings us to the conclusion of the rules. There are only three rules, so
it’s a pretty short ride. You might have an itch telling you deductions are
waiting. Well, it’s time to scratch and see what we find.
3. Deductions
When this game appeared on the LSAT, most students jumped straight into
the questions. That’s dangerous. There are a few small deductions that make
a huge difference. As we noted on the previous page, the letter was
mentioned in both of the first two rules. That sounds like a great place to
start.
Since the letter cannot be addressed to Georgette, the letter must be
assigned to either Jana or Rini.
The second rule is a conditional based on the letter going to Rini. Since
that is one of only two options for the letter, it’s time to dig a little
deeper.
Challenge: On the next page, see what deductions you
can uncover if the letter is addressed to Rini. (Hint:
don’t stop after the postcard.)
The first deduction about the postcard is pretty straightforward, but
hopefully you didn’t stop there. Since there are only five pieces of mail,
you will quickly run out of options in this game. You can use small
variable sets to your advantage.
If the letter is addressed to Rini, then the postcard is addressed to Jana.
The magazine cannot be addressed to Georgette, so the only two options
left for Georgette are the flyer or the survey.
The flyer can’t be the only piece of mail
addressed to a housemate. So if the flyer is
addressed to Georgette, then the survey must
also be addressed to Georgette. So there are
two options for Georgette: (1) she receives
only the survey, or (2) she receives both the
flyer and the survey. Either way, Georgette must receive the survey.
There’s a quick deduction about Jana we’ve now uncovered.
The letter could be addressed to Jana. If the letter isn’t
addressed to Jana, then it must be assigned to Rini
(because it can’t go to the fabulous Georgette). If the
letter is addressed to Rini, then the postcard is addressed
to Jana. Thus, either the letter or the postcard must
always be addressed to Jana.
Believe it or not, that last deduction pays huge dividends. You can
immediately kick out any answer choice that doesn’t give Jana what she
deserves - either the letter or the postcard.
Boom. With all of that in hand, we are ready to attack the questions. As you
can see, the addition of a third group changed some of the rules and
deductions, but many of the same principles still guide our strategy.
1 Don’t Georgette, Jana, and Rini sound like the three characters from a TV show
about female crime fighters? Think Charlie’s Angels.
4. Questions
We didn’t find any huge deductions that allowed us to assign pieces of mail
to housemates, so you should expect lots of conditional questions. That puts
pressure on you to move quickly and not do any unnecessary work. We’re
going to put most of the work on your shoulders.
Question #8
8. Which one of the following could be a
complete and accurate matching of the
pieces of mail to the housemates to whom
they are addressed?
Challenge: Use the elimination strategy to find the
answer.
(A) Georgette: the flyer, the survey
Jana: the letter
Rini: the magazine
(B) Georgette: the flyer, the postcard
Jana: the letter, the magazine
Rini: the survey
(C) Georgette: the magazine, the survey
Jana: the flyer, the letter
Rini: the postcard
(D) Georgette: the survey
Jana: the flyer, the magazine
Rini: the letter, the postcard
(E) Georgette: the survey
Jana: the letter, the magazine, the postcard
Rini: the flyer
Neither the letter nor the magazine is
addressed to Georgette.
If the letter is addressed to Rini, then the
postcard is addressed to Jana.
The housemate to whom the flyer is
addressed has at least one of the other
pieces of mail addressed to her as well.
Hint: They try to pull a dirty little trick on you here. If you get stuck, go back
and count the pieces of mail.
Ditz McGee:
I got stuck between (A) and
(B).
Neither the letter nor the magazine is addressed to Georgette. Bye-bye to
(C).
If the letter is addressed to Rini, then the postcard is addressed to Jana. In
(D), both the letter and the postcard are addressed to Rini. No way.
The flyer must be accompanied by at least one other piece of mail. In (E),
the flyer looks pretty lonely. (E) is out.
This is where Ditz and others run into a problem. What the heck? It
appears both (A) and (B) meet the stated rules in the game. And they do.
However, there’s something even more basic going on. Count the pieces of
mail in (A). Flyer? Check. Survey? Cool. Letter? Got it. Magazine? Sure.
Postcard? Um, postcard? Where’s the darn postcard? It’s not there.
Remember, all five pieces of mail have to be included. (A) is out.
After a long battle and a dirty trick by the LSAT, (B) is the champ.
Question #9
9. Which one of the following is a complete and
accurate list of the pieces of mail, any one of
which could be the only piece of mail addressed
to Jana?
Remember that
deduction about
Jana? It’s time
for that to pay
off.
For this absolute question, you have to assume that Jana only gets one piece
of mail (poor girl). Earlier, we made a few small deductions about Jana. But
even small deductions have a habit of paying off down the road.
Challenge: If Jana only gets one piece of mail, identify
the options for that one lonely piece.
(A) the postcard
(B) the letter, the postcard
(C) the letter, the survey
(D) the magazine, the survey
(E) the letter, the magazine, the
postcard
Many students will be tempted to test all five pieces of mail on this question.
That’s a huge waste of time. This is just a test to see if you spotted the
deduction about Jana.
Jana must always be addressed either the postcard or the letter. So the
flyer, the magazine, and the survey can’t be the only piece of mail for
Jana.
If Jana receives the letter, Georgette and Rini could receive all four other
pieces of mail.
If the letter is addressed to Rini and the postcard is addressed to Jana, the
survey must go to Georgette. The magazine could be addressed to Rini and
the flyer could be addressed to Georgette or Rini.
Jana could receive only the letter or only the postcard. (B) takes home
the crown.
Question #10
10. Which one of the following CANNOT be a
complete and accurate list of the pieces of mail
addressed to Jana?
Jana again?
Shocking. New
question, same
deduction.
Next up is another absolute question. It’s a bit surprising, but they are really
testing whether you understood the original conditions. You should
remember that the last question revolved around Jana. And guess who
makes a comeback here? For this question, you need a list that doesn’t work
for Jana. But keep your eyes on the same deduction.
Jana must always be addressed either the postcard or the letter.
Challenge: Try to locate the answer choice that doesn’t
work for Jana.
(A) the flyer, the letter, the magazine
(B) the flyer, the letter, the postcard
(C) the flyer, the letter, the survey
(D) the flyer, the magazine, the postcard
(E) the flyer, the magazine, the survey
Hint: Don’t rush in and testeach answer choice. Rather, think about the rules
that an answer choice might violate.
This one follows very nicely from the same deduction about Jana.
Jana must always be addressed either the letter or the postcard. She could
also be addressed other pieces of mail as well, but she always gets the
letter or the postcard.
(E) offers you the flyer, the magazine, and the survey. Sounds like a
great mix, except it’s missing the letter and the postcard, so it doesn’t
work. Winner, winner.
We didn’t find many deductions, but we did
realize that Jana is always addressed either the letter or the
Ninja Note:
postcard. It seemed small, but that
alone answers the last two questions
and likely saves us two or three
minutes on the game. Sometimes, it’s
the small things.
Question #11
11. Which one of the following CANNOT be a
complete and accurate list of the pieces of mail
addressed to Rini?
Finally, we
get to talk about
someone other
than Jana.
Yet another absolute question. But now we leave Jana behind and deal with
Rini. The form of the question is very similar to the last one - we must find a
list of pieces of mail that can’t all be addressed to the same housemate.
At first glance, this one might seem more difficult because we don’t have
many deductions about Rini. But that might actually make it easier. There’s
only one rule mentioning good old Rini - the second rule. That’s going to be
a good place to begin your search.
Reminder: If the letter is addressed to Rini, then the postcard is
addressed to Jana.
Challenge: Try to identify the answer choice that
doesn’t work for Rini.
(A) the magazine, the postcard
(B) the letter, the survey
(C) the letter, the magazine
(D) the flyer, the magazine
(E) the flyer, the letter
This question sets up the same trap as the last one. It’s easy to let them bait
you into testing each answer choice. But that leads to lots of writing, a sore
hand, and precious time wasted. During the deductions phase of this game,
we noted the letter must be addressed to either Jana or Rini. This prompted us
to test deductions when the letter is addressed to Rini. Let’s review the
deductions that followed.
If the letter is addressed to Rini, then the postcard is addressed to Jana.
The magazine cannot be addressed to Georgette, so the only two options
left for Georgette are the flyer and the survey.
Georgette could be addressed only the survey. Or she
could be addressed the flyer, in which case she also
receives the survey because the flyer can’t be the sole
piece of mail that she receives. Thus, Georgette must
receive the survey, and she could also receive the flyer.
If the letter is addressed to Rini, then the survey
must be addressed to Georgette. Thus, Rini cannot receive both the
letter and the survey. (B) is our answer.
Rini is only mentioned in the
rules once, in connection with the
letter. If you have to test answers, it
makes sense to test (B) and (C) first
since they include the letter.
Ninja Note:
Question #12
12. If the magazine and the survey are both
addressed to the same housemate, then which one
of the following could be true?
Last
question!
We have arrived at the last challenge, and we finally get a conditional
question. This one builds on the information gathered from the previous
questions. It’s vital to attack a question like this is the proper way, so let’s
walk through some of the steps together.
The new condition is that the magazine and the survey are both
addressed to the same housemate. The first question that should pop into
your head is, “Which lucky housemate gets both the magazine and the
survey?”
The magazine cannot be addressed to Georgette, so Georgette is
definitely out.
That leaves us with the usual suspects: Jana and Rini. It’s time to review our
previous deductions and see if we can figure out which one receives both the
magazine and survey.
The letter quickly becomes an issue again (surprise,
surprise). The deduction that got us through the last
question pays big dividends again. Remember, if the
letter is addressed to Rini, then the survey must be addressed to
Georgette. But, for this question, the magazine and the survey are both
addressed to the same housemate. Since Georgette can’t be addressed
the magazine, she also cannot be addressed the survey. If the survey is
not addressed to Georgette, then the letter cannot be addressed to Rini.
If the letter is not addressed to Rini, then it
must be addressed to Jana.
That’s a good start, but we still have to figure out
the magazine and survey combo.
The first big deduction is that the letter must be addressed to Jana. But that’s
not enough. The other four pieces of mail are still up in the air. We got you
started, but you get to take it from here.
Challenge: First, try to figure out which housemate
gets the magazine and the survey. Second, try to place
the other pieces of mail. Finally, locate the correct
answer (you need one that could be true).
(A) The survey is addressed to Georgette.
(B) The postcard is addressed to Rini.
(C) The magazine is addressed to Jana.
(D) The letter is addressed to Rini.
(E) The flyer is addressed to Jana.
If both the magazine and the survey were addressed to Jana, there would
have to be linked options for
the flyer and the postcard to
Georgette and Rini.
We have a big problem. The
flyer can’t be alone, and it
looks damn lonely. So the
magazine and the survey
cannot both be addressed to Jana.
Since the magazine and survey can’t be addressed to Jana, they must be
addressed to Rini. Now it’s time to nail down the other pieces of mail.
The flyer can’t be alone. Thus, there are two options
for Georgette: (1) the flyer and the postcard or (2)
just the postcard. Thus, Georgette must be addressed
the postcard. (This is the same deduction we made
about Georgette and the survey two pages ago. It’s a
common one in grouping games.)
The flyer could be addressed to any of the three housemates.
Since the flyer could be addressed to any of the housemates and
Jana is one of the housemates, the flyer could be addressed to Jana.
Yay for (E).
Your stint as the mailman has come to an end. That’s our first grouping game
with more than two groups. Here are some key points to remember:
1. The big four grouping relationships continue to
run the show.
2. It’s harder to simplify relationships when there are
more than two options for each player. This will
limit the deductions that can be made by
simplifying rules.
3. Small deductions can help in huge ways. You might have thought the
little deduction about Jana (letter or postcard) wouldn’t be very
important, but it probably saved us five minutes of time in the long run.
Now, it’s your turn. Take a stab at the game on the next page. Don’t rush your focus should remain on simplifying the rules and isolating deductions.
DECEMBER 2008: GAME 2 (7-11)
Questions 7-11
Four people—Grace, Heather, Josh, and Maria—will help each other move
exactly three pieces of furniture—a recliner, a sofa, and a table. Each piece of
furniture will be moved by exactly two of the people, and each person will
help move at least one of the pieces of furniture, subject to the following
constraints:
Grace helps move the sofa if, but only if, Heather helps move the recliner.
If Josh helps move the table, then Maria helps move the recliner.
No piece of furniture is moved by Grace and Josh together.
7. Which one of the following could be an accurate matching of each
piece of furniture to the two people who help each other move it?
(A) recliner: Grace and Maria; sofa: Heather and Josh; table:
Grace and Heather
(B) recliner: Grace and Maria; sofa: Heather and Maria; table:
Grace and Josh recliner: Heather and Josh; sofa: Grace and
Heather; table: Josh and Maria
(C) recliner: Heather and Josh; sofa: Heather and Maria; table:
Grace and Maria
(D) recliner: Josh and Maria; sofa: Grace and Heather; table:
Grace and Maria
8. If Josh and Maria help each other move the recliner, then which
one of the following must be true?
(A)
(B)
(C)
(D)
(E)
Heather helps move the sofa.
Josh helps move the sofa.
Maria helps move the sofa.
Grace helps move the table.
Heather helps move the table.
9. If Heather helps move each of the pieces of furniture, then which
one of the following could be true?
(A)
(B)
(C)
(D)
(E)
Grace helps move the recliner.
Maria helps move the recliner.
Josh helps move the sofa.
Maria helps move the sofa.
Grace helps move the table.
10. Which one of the following could be a pair of people who help
each other move both the recliner and the table?
(A)
(B)
(C)
(D)
Grace and Josh
Grace and Maria
Heather and Josh
Heather and Maria
(E) Josh and Maria
11. If Josh and Maria help each other move the sofa, then which one
of the following could be true?
(A)
(B)
(C)
(D)
(E)
Heather and Josh help each other move the recliner.
Heather and Maria help each other move the recliner.
Grace and Josh help each other move the table.
Grace and Maria help each other move the table.
Heather and Maria help each other move the table.
MOVING DAY
There’s nothing as terrible as moving day. Well, actually, what’s worse is
when your friend asks you to help on moving day. The result is guaranteed to
be a sore back, some bloody extremities, and one less friend. That’s the topic
of this game. The only saving grace is that apparently the friend in question
has very few possessions worth moving (three). Let’s see how you fared.
1. Setup
There’s definitely no ordering going on here. We are not asked to select
whether the table is moved before or after the sofa, but rather who moves
each piece of furniture. That spells grouping.
Our task is to assign the people to the pieces of furniture, so the furniture
should form the base of your setup.
Note that you also know how
many people move each piece of
furniture (two). That is another hint
when you are deciding which variable
set to use as the base.
Ninja Note:
The setup should reflect the three different groups and show the four different
people that are available. There are a few different and interesting
characteristics in this game to discuss.
Each group is assigned exactly two people. Unlike the last game we
completed, this is a stable grouping
game.
There’s some funky math going on here.
There are only four friends, but there are
three pieces of furniture. (Clearly, no one
can move a sofa alone.) Bad news: Some
of the friends are going to have to move
more than one piece of furniture. This
game is underbooked.
Since there are only four people, either two people are
each going to move two pieces or furniture, or one
poor sap has to help move all three. We are rooting for
Josh.
2. Rules
You should expect some normal grouping relationships. In addition, you have
to keep an eye on which people can or cannot move more than one piece of
furniture. For instance, ascertaining that Josh must move at least two pieces
of furniture, or that Maria can only move one piece, would be very helpful.
There are a few very important
words in this rule1. “If but only
if” signifies a very specific
relationship. It tells us that a
condition is both sufficient
and necessary. This should
always be represented with a
reciprocal arrow. It’s just a
different version of the love relationship.
“ If but only if ” = Love
As always, think about the
contrapositive. If Grace doesn’t help
with the sofa, then Heather doesn’t
help with the recliner, and vice versa.
Easy way to sum this up - either both
conditions are met, or neither one is.
Ninja Note:
The second rule gives you a
straightforward conditional
relationship. This is old hat by
now. If Josh moves the table,
then Maria helps move the
recliner. And if Maria doesn’t
help with the recliner, then
Josh doesn’t help with the
table.
Just one rule to go.
Apparently, Grace and Josh are not big fans of each other. They can’t even
pick up opposite sides of a piece of furniture. Ugly breakup? Contrasting
political views? A shared ex? It’s not clear, but Grace and Josh must be kept
apart.
That’s it for the rules, but you know what’s next.
3. Deductions
This is going to amaze you.
Surprise you. Shock you.
Ready? In this game, there
really aren’t very many
deductions. Let that sink in for
a moment.
Grace is involved in two rules, but they don’t combine to give you much.
Josh pops up twice, but the deductions are unimpressive. Sucks.
The process of searching for
deductions will always prove fruitful,
even if there is little treasure awaiting
you. You will have a better grasp on
the game, and you know the questions
will rely heavily on the rules.
BP Minotaur:
Let’s take one last look at the setup and rules before jumping into the
questions.
1 Few games have a “do or die” rule quite like this one. If you misinterpret this rule,
the whole game is screwed. This is like a pickup line at a bar. Do or die.
4. Questions
There are a few questions in this game that take students way too long, even
if they do eventually arrive at the prized correct answer. Make sure you
attacked each question using the right tactic.
Question #7 (elimination, could be true)
A quick elimination question affords us a great opportunity to make sure the
rules are working with us at this early stage.
Grace helps move the sofa if, but only if, Heather helps move the
recliner. You want to look for answer choices where only one of these
conditions is met. In (D), Heather moves the recliner, but Grace doesn’t
move the sofa. In (E), Grace moves the sofa, but Heather doesn’t move
the recliner. Two birds with one rule.
If Josh helps move the table, then Maria helps move the recliner. In (C),
Josh is doing his best with the table. But Maria is helping him with the
table rather than moving that recliner. C’mon, Maria.
Grace and Josh must be kept far apart. In (B), they are on opposite sides
of the table. That’s not going to work.
(A) is the only one that presents us with an acceptable matching.
Question #8 (conditional, must be true)
Here’s the first conditional question of the game. In this one, Josh and Maria
work together to move the recliner.
If Josh and Maria move the recliner, then
Heather doesn’t move the recliner.
If Heather doesn’t move the recliner, then
Grace can’t help move the sofa.
Grace has to help move something, so she is
assigned to the table.
(D) is the winner since Grace must help move the table.
Question #9 (conditional, could be true)
Go Heather. In this one, Heather has morphed into some scary form of
superwoman. Listen to her roar. Heather helps move each of the three pieces
of furniture.
If Heather helps move each piece of
furniture, then the three remaining people
(Grace, Josh, and Maria) move only one
piece of furniture each.
If Heather helps move the recliner, then
Grace must help move the sofa.
You should make linked options for Josh and Maria on the recliner and
table to finish up the hypothetical.
Maria could help move the recliner, so (B) is the correct answer.
Question #10 (absolute, could be true)
Finally, an absolute question. We guessed they would be rare, seeing as we
had so few deductions. This question functions more like a conditional
question anyway. You are asked to identify a pair of people that cannot help
each other move the recliner and the table.
Remember: If these two people both move the recliner
and the table, the other two people have to help each
other move the sofa.
(A) Grace and Josh can’t move anything together, so this one goes
away quickly.
(B) If Grace and Maria help each
other move the recliner and the
table, then Heather and Josh have
to help each other move the sofa.
But that doesn’t break any
conditions in the game, so we have our winner.
(C) If Heather and Josh help each other move the recliner and the
table, then Grace and Maria must help each other move the sofa.
The problem is with our buddy Josh. If Josh helps move the table,
then Maria is supposed to pitch in with the recliner. This one is
eliminated by the second rule.
(D) If Heather and Maria help each other move the recliner and the
table, then Grace and Josh must help each other move the sofa. You
might recall that Grace and Josh have irreconcilable differences, so
this is out.
(E) If Josh and Maria help each other move the recliner and the table,
then Grace and Heather must help each other move the sofa. But if
Grace helps move the sofa, Heather is supposed to grab on to that
recliner. This one is another loser.
Question #11 (conditional, could be true)
The last question is upon us. At this point, you should have a good grasp on
how the rules interact with each other. Josh and Maria now team up on the
sofa.
If Josh and Maria help each other move the sofa, then Grace doesn’t
help move the sofa.
If Grace doesn’t help move the sofa, then Heather can’t help with the
recliner.
Heather has got to grab something, so she is headed over to the table.
The only three options left for the recliner are
Grace, Josh, and Maria. Since Grace and Josh
cannot move any furniture together, Maria
must help move the recliner with either Grace
or Josh.
(A) and (B) are eliminated because Heather
cannot help move the recliner. (C) and (D)
are out because Heather must be one of the people to help move the
table. (E) could be true because Heather must help move the table,
and Maria could help her out.
That wraps up moving day. Time for a beer and a
massage. As you just experienced, not all games
will have humongous, game-changing deductions.
However, this doesn’t diminish the importance of
searching for deductions. As a wise philosopher
once said, the search is more important than the
treasure you discover.2
The key in this game was to identify the important
aspects of the game (stable, underbooked) and to represent the rules correctly.
From there, moving day went smoothly (definitive proof that the LSAT is
nothing like real life).
At this point, we’ve covered grouping games that have one, two, and more
groups. You might think we’ve exhausted the possibilities, but there’s one
last type of grouping game to discuss.
Up Next: Profiling Games
2 No philosopher has ever actually said anything like that. But we are pretty sure
we’ve read it in a fortune cookie.
20/PROFILING
EVERYTHING IS UNSTABLE
Just when you thought there couldn’t be any other way to create a grouping
game, they throw one more twist at you. This chapter marks our last type
of grouping games: Profiling.
Profiling games look a little
scary at first, but students tend to enjoy
them if they grasp the strategy for
attacking them.
BP Minotaur:
It’s very important to spot profiling games because the strategy will diverge
somewhat from other grouping games. So the first thing to discuss is what
characteristics you should look for to determine you are up against a profiling
game.
Players and Groups are Unstable
As noted in previous chapters, it’s important to track whether variable sets
are stable or unstable. How many players are on the red team? How many
ingredients are in the appetizer recipe? How many pieces of furniture can
Maria help move? When you know the answer to one of these questions, the
game is stable. When you don’t, it’s unstable.
In a profiling game, both variable sets are unstable. So there will be two
important characteristics in these games:
1. Players will be assigned to at least one group (rather
than exactly one).
2. The number of players assigned to each group will
be unknown.
Students commonly confuse
unstable grouping games and profiling
games. The difference is that players
can now be assigned to more than one
group, rather than just one. Watch for
games in which players are assigned to
“at least one” group.
BP Minotaur:
Let’s take a look at an example.
Four different community centers—
Coneydale, Deadwood, Evergreen, and
Feildheim—each offer at least one of the
following youth programs: origami,
painting, squash, tennis, and weaving.
Each program is offered by at least one
community center. The following
conditions govern the assignment of
programs to community centers:
In this game, your job is to assign programs to the different community
centers. There are some keywords, so you have to check both variable sets to
see what restrictions are in place.
1. Each community center must offer at least one program, but they could
offer more. For example, Deadwood might offer one, two, or four
different programs.
2. Each program must be offered by at least one community center, but
they could be offered by more. For example, weaving might be available
at one, two, or all four community centers.
One of these characteristics alone does not entail a profiling game. But when
you have both of these characteristics, it’s profiling time.
Now let’s talk strategy.
Slow Down the Setup
As you know by now, we are big proponents of constructing an effective
setup early in the process so you can visualize how the rules function in a
game. However, profiling games are the exception. Because both groups are
unstable in a profiling game, it’s possible to use either variable set as the
base of your setup.
In the game above, you could use the community centers as the base and
plug in the programs that they offer (setup #1 to the right). Alternatively,
you could use the programs as the base of your setup and plug in the
community centers that offer them (setup #2).
This comes with a big caveat. While it’s possible to make it through a game
with either setup, there’s always one that will make your life much easier.
On the next page, we will walk you through the
strategy to ensure the best setup
for a profiling game.
Step 1: Identify a Profiling Game
As we discussed a few moments ago, the first
step is to recognize that you are staring at a
profiling game. How? Test the sizes of the
groups. If both groups are unstable, then you
know you have a profiling game.
Step 2: Check out the rules
Once you realize you are doing a profiling game, you know that two different
setups are possible for the game. Don’t rush and simply choose one of the
setups randomly. You want to do a little research and see which setup will be
more helpful for the next 10 stressful minutes of your life. Where might you
find clues about the best way to visualize the game? The rules, of course.
The rules in a profiling game provide helpful hints in
building the most effective setup.
The big challenge in a profiling game is that the sizes of the groups are up in
the air. You don’t know much, and that can be uncomfortable. Your task is to
take back the power (take that, LSAT). As you will see very soon, the
important deductions in these games will stem from limiting the sizes of the
groups. If you can decipher that a community center offers exactly two
programs, or that tennis is offered by a maximum of two community centers,
then you will have some very valuable pieces of information.
When you read through the rules in a profiling game,
identify which variable set faces more restrictions.
Always use that variable set as the base of your setup.
Let’s use our previous game as an
example. Say you glance down and
Deadwood offers exactly two
you see this set of rules staring back
programs.
at you.
Fieldheim offers tennis and at
most one other program.
Which variable set should you use
Coneydale offers more programs
as the base of your setup?
than Fieldheim.
Evergreen cannot offer both
Try to visualize...
squash and weaving.
The community centers! The rules
all place restrictions on how many
programs the community centers
can offer, so it will be much easier to visualize if you use the centers as the
base of your setup.
Now, pretend we are in an alternate
universe. There are flying cars,
Origami is offered by more of the
dragons with goatees, and an
community centers than weaving.
entirely different set of rules. Take
Squash is offered by only one of
a look at this alternate set of rules.
the community centers.
Which variable set do you think
Tennis is offered by at least two
should be used as the base of your
community centers.
setup?
Hopefully, you are now visualizing
the programs as your base. Working from there will be much easier since the
rules place restrictions on the programs. It’s easier to assign the centers to the
programs in this situation.
Games will give you some rules that favor each
of the two possible setups, but the majority of rules will
point you in one direction.
Ninja Note:
Step 3: Work slowly through the rules
We always urge you to work slowly through the rules and look for
interrelationships, but this is of the utmost importance with profiling games.
The important deductions won’t arise from one all-important rule, but rather
from a compilation of the rules. Thus, it’s crucial to test the significance of
each rule thoroughly.
When you are looking for deductions in a profiling game, remember: They
start with all of the power.1 You are given very little. You don’t know how
many times the players appear, and you don’t know the sizes of the groups.
Your challenge is to take back the power. You can accomplish this by
limiting the sizes of the groups.
The important deductions in a profiling game will be
based on limiting the sizes of the groups.
The real deal!
A showroom contains exactly six new cars—T, V, W, X,
Y, and Z—each equipped with at least one of the
following three options: power windows, leather interior,
and sunroof. No car has any other options. The following
conditions must apply:
To illustrate, let’s do some excerpts from an actual profiling game. This is the
infamous “Pimp my Ride” game from October 2001. Check out the
introduction and identify the elements that make this a profiling game.
It’s clear that we are doing a grouping game with two variable sets - cars and
options.2 Let’s check both variable sets to see if we have a profiling game.
1. Do we know how many options are assigned to each car? Nope. Car V,
for instance, might have just one option or all three (big time).
2. Do we know how many cars have each option? Not so much. Leather
might be found in two cars or four cars.
Clearly, we have a profiling game. But before we build a setup, we need to
check out the rules and see which variable set to use for our base. Remember,
it’s all about visualization and limiting the sizes of the groups.
V has power windows and a sunroof.
W has power windows and leather interior. W and Y have no options
in common.
X has more options than W.
V and Z have exactly one option in common. T has fewer options
than Z.
Check out the rules to this game. Which variable set should be used as the
base of your setup?
It seems like the cars are the clear winner in this one. Nearly every rule
limits the options assigned to each car.
Now that you have your setup, it’s time to actually jump into the rules. We
are going to highlight a couple rules to illustrate the types of deductions that
will be key in such a game. Take a look at these three rules together. Think
for a moment about the impact they have on the cars involved. Remember,
the key is to limit the number of options that are possible for each car.
Here’s an overview of the deductions that can be drawn from this
combination of rules:
W has power windows and leather interior, but note that the rule doesn’t
tell us if these are the only options in W. Plug in the two options, but we
still want to leave open the possibility that W could also have a sunroof.
The next rule states that W and Y have no
options in common. Since W already has power
windows and leather interior, Y can’t have either
of those options.
Y has to have something, so Y must have a
sunroof.
Since W and Y have no options in common, W
can’t have a sunroof. Thus, both W and Y are
completely determined.
The third rule states that X has more options than W. Since W has
exactly two options, X must have all three options. X is totally pimped
out.
It’s important to have
conventions to track the sizes of the
groups. Only put a slot when you know
there has to be a variable there (for
example, the minimum is two) and use
a vertical line to limit the maximum
size of a group.
Ninja Note:
Isn’t that beautiful? Using just those three rules, we completely determined
three of the six cars. The deductions in a profiling game build and build into a
glorious final product. It’s common to completely determine a number of
groups before jumping into the questions.
Here’s a quick recap of the important features covered in this introduction:
1. In a profiling game, both variable sets are unstable. The players can
be assigned to more than one group, and the group sizes are
unstable. Watch for the phrase “at least one” in the introduction to
denote that you have a profiling game.
2. It’s possible to set up a profiling game using either variable set as
the base, but one is always preferable in visualizing the game. To get
some clues about the best setup, quickly read over the rules. Form
your setup using the variable set with the most restrictions on it.
3. Work through the rules very slowly on a profiling game. The
deductions will build on each other, so make sure to review rules as
you uncover deductions.
4. The most important deductions will flow from limiting the sizes of
the groups. Always test whether each rule limits how many players
could be assigned to each group.
Well, enough chat for now. Let’s give this a shot. The first game will utilize
our Blueprint Building BlocksTM technique, and then you will get to try one
on your own.
1 “They” refers to the people who write these damn games. We always picture them
with horns and a spiked collar, but feel free to make up your own mental picture.
2 The LSAT tends to date itself once in a while. Power windows? That’s a big option
on these cars? Wonder if they also offer a cassette player?
DECEMBER 2005: GAME 3 (13-17)
In this lovely game, you are going to meet a group of technicians. They are
very skilled with their hands, and they use them to repair various household
electronics. Sound like fun?
1. Setup
In a repair facility there are exactly six
technicians: Stacy, Urma, Wim, Xena, Yolanda,
and Zane. Each technician repairs machines of at
least one of the following three types—radios,
televisions, and VCRs—and no other types. The
following conditions apply:
Wim? Xena?
Yolanda? Zane?
They were
feeling very
adventurous
when choosing
the names in
this one.
Alright, we have many important things to discuss. But first you have to fight
your way through a truly eclectic set of technicians. They are really stretching
the names on this one.1 Of course, that doesn’t matter. The first thing to note
is the basic grouping element to this game - we are assigning technicians to
machines.
Now, let’s check the restrictions on the variable sets to see if we have a
profiling game.
1. How many machines does each technician repair?
No idea. You might guess that Wim is rather limited and Xena can repair
anything you throw at her. But it’s not clear how many machines are
repaired by each technician.
2. How many technicians repair each type of machine?
No clue. Radios might be repaired by one, two, or any other number of
technicians. So the number of technicians that repair each type of machine
is also unknown.
By investigating these two questions, it’s now clear that we are dealing with
a profiling game. You might be tempted to randomly select a variable set and
build a setup, but don’t jump the gun. As you know from our introduction to
profiling games, we could set this game up using the players (technicians) or
the groups (machines) as the base. But your job is to figure out which setup
will be most helpful. Time to glance through the rules and visualize the
best setup. Take a look on the next page and see if you can figure it out.
Xena and exactly three other technicians repair
radios. Yolanda repairs both televisions and
VCRs.
Stacy does not repair any type of machine that
Yolanda repairs.
Zane repairs more types of machines than
Yolanda repairs.
Wim does not repair any type of machine that
Stacy repairs.
Urma repairs exactly two types of machines.
Which
variable set has
the most
restrictions
placed on it technicians or
machines?
Hopefully, you are thinking technicians. The majority of the rules place
restrictions on the technicians, so it’s advisable to use them as the base of
your setup. For example, the fourth rule (Zane repairs more types of
machines than Yolanda repairs) and the sixth rule (Urma repairs exactly two
types of machines) will be much easier to visualize if you assign the machines
to the technicians.
The first rule favors the
machines, but it is outweighed by all of
the later rules that favor using the
technicians.
Ninja Note:
It took a little time to figure out the
best approach to this game, but now
we feel confident in building the
setup.
Check it out.
2. Rules
Now it’s time to jump into the rules. We already took a quick glance, but let’s
rewind back to the top and take a closer look.
Reminder: We have to work super slowly through the
rules to catch all of the small deductions. Watch for
deductions that restrict the sizes of the groups.
Let’s jump in and take a look. Here comes the first rule...
This is a very important rule, so you want to make sure that you correctly
represent all of the information. First, you know that Xena repairs radios.
Plug that right into your setup. In addition, the rule tells you that exactly three
other technicians repair radios. You should represent this to the side of your
setup.
It says three people, so I
wrote that three people do radios.
Cleetus Comment:
Be careful. It actually says “three other technicians” (in addition to Xena), so
there are a total of four technicians that repair radios.
As you work through the rest
of the rules, you want to keep a
running tally of the technicians who
repair radios. Four of the six people
repair radios, so deductions might be
easy to spot.
BP Minotaur:
And now we move on to the next rule...
This one seems pretty straightforward. Yolanda is a very talented lady - she
repairs both televisions and VCRs. This can be plugged right into the setup.
Good thing we chose to use the technicians as the base.
Ditz McGee:
So Yolanda doesn’t do radios, right?
Nope. Don’t assume that Yolanda doesn’t repair radios. That would be
insulting to Yolanda, and it would be an invalid conclusion to draw at this
point. The original rule would have to say “only televisions and VCRs” for us
to make this deduction. It doesn’t, so we can’t.
Next rule...
Time for deductions. Since the last
rule already outlined some of the
machines that Yolanda repairs,
this rule should catch our eye. This
one is up to you.
Challenge: See if you
can draw some
deductions about Stacy
and Yolanda.
These are very common deductions in a profiling game, so hopefully you
were able to fill in some slots or limit the size of some groups (Stacy and
Yolanda). There’s an overview on the next page.
Yolanda repairs televisions and VCRs, so Stacy
cannot repair either of those two machines. Stacy
must repair at least one type of machine, so Stacy
repairs radios.
Yolanda now can’t repair radios, so both Stacy
and Yolanda are completely determined at this
point. Use a vertical line to signify that they are
done.
It’s also helpful to note that Stacy is the second
technician to repair radios. Also, Yolanda doesn’t repair radios, so there
is only one more technician that doesn’t repair radios.
You should be feeling pretty good. Two of the six technicians are
completely determined. Sweet. But we still have lots of work to do.
Moving on...
Your turn again. Slow down and think through each rule. We have heard the
name Yolanda before, and that probably means there are more deductions.
Challenge: See if you can use this rule to draw
additional deductions.
We already know that
Yolanda repairs
televisions and VCRs, but not radios. Since Zane repairs more types of
machines than Yolanda, Zane is a very talented man. He repairs all three
types of machines.
It’s also helpful to note that Zane is yet another of our radio repairmen.
Our list now includes Xena, Stacy, and Zane. Only one more technician
repairs radios. Since Yolanda doesn’t repair radios, either Urma or
Wim (but not both) repairs radios.
We are gathering more and more deductions with each rule. Let’s see if we
can continue the trend...
Finally, Wim joins the fray. We already have rules and deductions about
Stacy, so it’s time to keep the momentum going. See what you can find.
Challenge: Take a look at Wim and see if you can spot
any deductions.
(Hint: Don’t stop
with just Wim - think
about the radios.)
Stacy repairs only radios. If Wim doesn’t
repair any types of machine that Stacy
repairs, Wim cannot repair radios.
Wim must repair either televisions or VCRs,
or both. This can be visualized by putting an
option for Wim and setting a maximum of
two machines.
But wait, there’s more...
The very first rule told us that four of the six technicians repair radios.
Neither Wim nor Yolanda is capable of repairing a radio. So the fourth
technician to repair radios must be Urma.
This is a huge deduction for the rest of
the game. It’s a huge payoff for keeping
track of the radios as we worked through
the previous rules.
Things are going well. But we have one more rule to go.
From the last rule, we deduced that Urma repairs radios. This rule states that
Urma repairs exactly two types of machines. So Urma also must repair either
televisions or VCRs, but not both. This can be visualized by plugging in an
option for Urma and setting a maximum of two types of machines.
That brings us to the end of the rules. On the next page, check out our
completed setup and rules.
Now that’s beautiful!
Look at those
amazing deductions.
Note: At this point, all of the
rules have been incorporated
into the setup. That’s
awesome. We have very little
to worry about going forward.
3. Deductions
During the introduction to profiling games, we claimed that these games are
unlikely to have big deductions from just one or two rules. Rather, there are
lots of smaller deductions to be made as you work through the rules. This
game is a perfect example. We made a plethora of deductions while
diagramming the rules, so we are in perfect shape already.
Before jumping into the
questions, check to see what is still up
in the air. Stacy, Yolanda, and Zane are
completely determined. Urma and
Wim have limited options left, and
Xena is the most up in the air.
Ninja Note:
Here’s a quick overview of the six technicians:
Stacy repairs only radios. Her future job prospects are looking dim.
Urma repairs exactly two types of machines. Radios are one, and the
other could be either televisions or VCRs.
Wim must repair either televisions or VCRs, or he could repair both.
But his maximum is two, and he can’t repair radios.
Xena is the least determined technician. She definitely repairs radios.
That could be all, or she could also repair televisions or VCRs, or both.
Yolanda repairs only televisions and VCRs.
Zane is the man. He repairs all three types of machines (and probably
any other random electronics you have around the house, as well).
1 No one under 80 is named Urma. Wim must have had a rough go in elementary
school. Xena’s career apparently didn’t take off much after her stint as the warrior
princess. Zane just gives everyone flashbacks to the Billy Zane walk off scene in
Zoolander. Quite a crew.
4. Questions
Time to get down to business. So far, we’ve made a huge number of
deductions to help us work through the questions. Profiling games generally
include a number of questions about the overlap between players. Could
Xena and Yolanda have one machine in common? Could they repair exactly
the same types of machines? Our deductions should allow us to visualize the
answers to these questions easily.
This one is going to be all on you. Since you are armed with some powerful
deductions, see if you can breeze through the questions. We will give you
minimal guidance and then review the answers at the end.
Challenge: Complete all of the questions (accurately,
of course).
Question #13
13. For exactly how many of the six technicians
is it possible to determine exactly which of the
three types of machines each repairs?
Time for all
of that hard
work to start
paying off.
There’s no elimination question in this game.
Rather, they start off by testing to see if you were
able to process all of the rules. The question asks
for the number of technicians that are completely determined. That’s easy.
(A) one
(B) two
(C) three
(D) four
(E) five
Question #14
14. Which one of the following must be
true?
Here’s an absolute question designed to test your deductions. The winner
must be true, while each of the losers could be false.
(A) Of the types of
machines repaired by
Stacy there is exactly one
type that Urma also
repairs.
(B) Of the types of
machines repaired by
Yolanda there is exactly
one type that Xena also
repairs.
(C) Of the types of
machines repaired by
Wim there is exactly one
type that Xena also
repairs.
(D) There is more than one
type of machine that both
Wim and Yolanda repair.
(E) There is more than one
type of machine that both
Urma and Wim repair.
Question #15
15. Which one of the following must be
false?
This is another absolute question, but now you are looking for something that
must be false. Eliminate the choices that could be true, and try to find the one
that breaks a rule or deduction.
(A) Exactly one of the six
technicians repairs
exactly one type of
machine.
(B) Exactly two of the six
technicians repair exactly
one type of machine
each.
(C) Exactly three of the six
technicians repair exactly
one type of machine
each.
(D) Exactly one of the six
technicians repairs
exactly two types of
machines.
(E) Exactly three of the six
technicians repair exactly
two types of machines
each.
Question #16
16. Which one of the following pairs of
technicians could repair all and only the
same types of machines as each other?
Now you are venturing into the world of things that could be true. You need
to find a pair of technicians that could repair the exact same types of
machines as each other.
(A) Stacy and
Urma
(B) Urma and
Yolanda
(C) Urma and
Xena
(D) Wim and
Xena
(E) Xena and
Yolanda
Question #17
17. Which one of the following must be true?
(A) There is exactly one
type of machine that both
Urma and Wim repair.
(B) There is exactly one
type of machine that both
Urma and Xena repair.
(C) There is exactly one
type of machine that both
Urma and Yolanda
repair.
(D) There is exactly one
type of machine that both
Wim and Yolanda repair.
(E) There is exactly one
type of machine that both
Xena and Yolanda repair.
The grand
finale!
Hopefully you enjoyed that experience. But before you celebrate, let’s review
the questions. Make sure you were able to translate the deductions into
correct answers.
Question #13 (absolute, must be true)
The first question asks you for the number of technicians that are completely
determined. In profiling games, such deductions are common, so this is not a
surprising question.
By processing the rules, we were able to completely figure out the
machines for Stacy, Yolanda, and Zane. That equals three, so (C) is
the answer.
Question #14 (absolute, must be true)
This one is an absolute question designed to test our deductions. Many of
the answer choices could be true, but you must seek out the one that must be
true.
(A) Stacy repairs only radios. Urma repairs radios and either
televisions or VCRs, but not both. Thus, there is exactly one
type of machine repaired by both Stacy and Urma - radios. (A)
must be true and is the answer.
(B) Yolanda repairs only televisions and VCRs. Xena definitely
repairs radios, and she could also repair other types of machines.
Thus, Yolanda and Xena could repair no types of machine in
common, or they could repair one type (televisions or VCRs) or
two types (televisions and VCRs) of machines in common.
(C) Wim must repair either televisions or VCRs, but he could also
repair both. Xena repairs radios and she could repair other types of
machines. Just like (B), Wim and Xena could repair no types of
machines in common, or they could repair one type (televisions or
VCRs) or two types (televisions and VCRs) of machines in
common.
(D) If Wim only repairs one type of machine, then Wim and Yolanda
would only both repair one type of machine (televisions or VCRs).
(E) Urma and Wim can both repair a maximum of one type of machine
- either televisions or VCRs, but not both.
Question #15 (absolute, must be false)
This one heads in the other direction and asks you for an answer that must be
false. To get through this one, you have to quickly diagnose the maximum
and minimum number of machines each technician could repair.
(A) Stacy is the only technician that must repair only one type of
machine, so it could be true that exactly one technician repairs one
type of machine.
(B) Stacy repairs exactly one type of machine, and Wim or Xena or
both could repair only one type of machine. Thus, it could be true
that two technicians repair one type of machine.
(C) This one also could be true - Stacy, Wim, and Xena could all
repair one type of machine.
(D) Urma and Yolanda both repair two types of machines. Thus, it
must be false that only one technician repairs two types of
machines. There it is.
(E) Urma and Yolanda repair two types of machines. Wim or Xena
could also repair two types of machines, so there could be three
technicians that repair two types of machines.
Question #16 (absolute, could be true)
This question asks for a pair of technicians that could repair all and only the
same types of machines. That’s just a complicated way of saying that they
could repair exactly the same types of machines. It’s tough to anticipate the
correct combination, so it’s acceptable to jump in and test the answer choices.
(A) Stacy repairs one type of machine, and Urma repairs two. So there
is no way they can repair the same types of machines.
(B) Urma repairs radios and Yolanda doesn’t repair radios, so this one
is out as well.
(C) Urma repairs radios and either televisions or VCRs, but not
both. Xena repairs radios and she can repair other types of
machines. So Urma and Xena could repair the same types of
machines - either radios and televisions, or radios and VCRs.
(D) Radios are a deal breaker again. Win can’t repair radios, while
Xena does repair radios.
(E) Xena is a master of radio repair, but Yolanda doesn’t know a
frequency modulator from a flux capacitor.
Question #17 (absolute, must be true)
By now, the questions should feel like a walk in the park. This game is
running scared; time to finish it off. For the last question, we need something
that must be true.
(A) Urma repairs radios and either televisions or VCRs, but not both.
Wim repairs either televisions or VCRs, or both. So Urma and Wim
could repair no machines in common.
(B) Urma and Xena must both repair radios. But Urma and Xena could
both also repair either televisions or VCRs.
(C) Urma must repair either televisions or VCRs, but not both.
Yolanda must repair both televisions and VCRs. Regardless of
whether Urma repairs televisions or VCRs, they must have one
type of machine in common. (C) is the winner.
(D) Yolanda repairs televisions and VCRs. Wim could repair both
televisions and VCRs, so they could share two types of machines.
(E) Xena could repair only radios, in which case there are no machines
that both Xena and Yolanda repair.
Take that, Xena. The warrior princess just got taken down. That was our first
profiling game. As you can see, there are some elements similar to other
grouping games, but the approach and the deductions have some unique
aspects.
To recap, let’s discuss how the strategy for profiling games played out in this
game.
1. Identify a profiling game. After reading the introduction, it was clear
that both variable sets (technicians and machines) were unstable. We
also noted the key phrase at least one.
2. Glance over the rules to determine the best setup. The majority of the
rules placed restrictions on the technicians, so we decided to use
Yolanda, Zane, and the rest of the eclectic crew as the base of our setup.
3. Work slowly through the rules with a focus on limiting the sizes of
the groups. By testing the implications of each rule, three of the six
technicians were completely determined by the rules, and the others
were severely limited.
4. Sit back and enjoy. After all that hard work, the questions were a
breeze.
Now it’s time for you to try it on your own. On the next page, it’s lunchtime.
Don’t get distracted by the delicious food options. Focus on strategy.
Remember: Work slowly. Deductions are king.
JUNE 2004: GAME 4 (18-22)
Questions 18-22
Each of exactly six lunch trucks sells a different one of six kinds of food:
falafel, hot dogs, ice cream, pitas, salad, or tacos. Each truck serves one or
more of exactly three office buildings: X, Y, or Z. The following conditions
apply:
The falafel truck, the hot dog truck, and exactly one other truck each serve
Y.
The falafel truck serves exactly two of the office buildings.
The ice cream truck serves more of the office buildings than the salad
truck.
The taco truck does not serve Y.
The falafel truck does not serve any office building that the pita truck
serves.
The taco truck serves two office buildings that are also served by the ice
cream truck.
18. Which one of the following could be a complete and accurate list
of each of the office buildings that the falafel truck serves?
(A)
(B)
(C)
(D)
(E)
X
X, Z
X, Y, Z
Y, Z
Z
19. For which one of the following pairs of trucks must it be the case
that at least one of the office buildings is served by both of the
trucks?
(A)
(B)
(C)
(D)
(E)
the hot dog truck and the pita truck
the hot dog truck and the taco truck
the ice cream truck and the pita truck
the ice cream truck and the salad truck
the salad truck and the taco truck
20. If the ice cream truck serves fewer of the office buildings than the
hot dog truck, then which one of the following is a pair of lunch
trucks that must serve exactly the same buildings as each other?
(A)
(B)
(C)
(D)
(E)
the falafel truck and the hot dog truck
the falafel truck and the salad truck
the ice cream truck and the pita truck
the ice cream truck and the salad truck
the ice cream truck and the taco truck
21. Which one of the following could be a complete and accurate list
of the lunch trucks, each of which serves all three of the office
buildings?
(A)
(B)
(C)
(D)
(E)
the hot dog truck, the ice cream truck
the hot dog truck, the salad truck
the ice cream truck, the taco truck
the hot dog truck, the ice cream truck, the pita truck
the ice cream truck, the pita truck, the salad truck
22. Which one of the following lunch trucks CANNOT serve both X
and Z?
(A)
(B)
(C)
(D)
(E)
the hot dog truck
the ice cream truck
the pita truck
the salad truck
the taco truck
FALAFEL AND ICE CREAM
Choosing a lunch truck can be one of the most vexing decisions one must
make during the day. So you can imagine why they chose this dilemma to be
the subject of a tough logic game. Lots to discuss, so let’s get to it.
1. Setup
As soon as you are assigning different kinds of food trucks to office
buildings, you know you are messing around with a grouping game.
However, the important question is, what type of grouping game? Here are
the key features to note:
There are six lunch trucks: falafel, hot dogs, ice cream, pitas, salad, and
tacos. That’s quite an assortment. Each truck serves at least one of three
office buildings. The mighty pita truck might go to one, two, or all three
office buildings. The trucks are unstable.
There are three office buildings: X, Y, and Z. How many lunch trucks
serve each of these fine office buildings? It’s not clear, so the buildings
are also unstable.
Since both variable sets are unstable, it’s a
profiling game.
Upon recognizing this is a profiling game, you should check out some of the
rules to visualize the best setup.
This game is a tough one to determine - there
are rules that would favor using the food trucks as the
base, and others that favor the office buildings. Don’t feel
BP Minotaur:
bad if you went the other way.
If you felt a little torn when looking over the rules, you’re not alone. This
game can be conquered by working with either variable set. There are rules
that seem easier to visualize using the trucks, and there are rules that favor
the buildings. However, there are a couple rules that should catch your eye.
The falafel truck, the
hot dog truck, and
exactly one other
truck each serve Y.
The taco truck does
not serve Y.
These rules should
make you think about using
the buildings as your base.
We are going to use the buildings as the base of our setup and plug in the
lunch trucks that serve those buildings. This is the best setup. But again,
don’t stress if you went with the trucks.
You should be able to arrive at the same deductions using either setup. Since
so many students are split on this one, we will also show you the setup and
deductions using the trucks at the end of our deductions phase.
Here’s our setup at this point:
2. Rules
Next up are the rules. Just like in the previous
game, working slowly through the rules leads
to tons of deductions.
Make sure to plug the falafel truck and the hot dog truck into Y. Then, cut it
off at three trucks total. Even at this early stage, we know it’s going to be
important to keep track of which food trucks could be the third one going to
Y. (This is similar to the rule about radios in the last game.)
The falafel truck already serves Y, so we just gotta
figure out whether it also serves X or Z.
We can use this rule to place limits on the ice cream and the salad trucks: I
serves at least two buildings, and S serves at most two buildings.
This is a big moment. The first rule told us that the falafel truck serves Y, and
the second rule stated that the falafel truck serves exactly two buildings.
If the falafel truck does not serve any office building that the pita trucks
serves, the pita truck can’t serve Y, and it can only serve one building.
The pita truck must serve either X or Z, and the falafel truck must serve
the other building, so you should build linked options to picture this
relationship.
There’s one more cool deduction
at this point. Since neither the
taco truck nor the pita truck can
serve Y, the third truck serving Y
must be either the ice cream truck
or the salad truck.
And the last rule is a big one...
Make sure you correctly
interpreted this rule: Two
buildings are served by both
the taco truck and the ice
cream truck.
Your first thought should be,
which buildings might those
be?
Check out the setup. There’s
definitely no room for both
the taco truck and the ice
cream truck to serve Y.
The taco truck and the ice cream truck must serve both X and Z.
Now that’s a lot of stuff. Shoot, look at that setup. It’s practically full. That
covers the rules, but there are a few more deductions to note before hitting
the questions.
3. Deductions
First, let’s run down the list of food trucks so we know what we are dealing
with at this point.
The taco truck is completely determined. It serves X and Z, but not Y.
The falafel truck (Y and either X or Z, but not both) and the pita truck
(only one building, either X or Z) are nearly done. The linked option is
the only lingering question.
The hot dog truck serves Y, and it could also serve one or both of the
other buildings.
The ice cream truck serves X and Z, and it could also serve Y.
The salad truck could serve any of the buildings at this point, but it must
serve fewer buildings than the ice cream truck.
Buildings X and Z aren’t very restricted. Both buildings could be served by
as many as five food trucks. However, there are a few additional deductions
that can help.
The ice cream truck must serve
more buildings than the salad
truck. If the salad truck serves Y,
then the ice cream truck can only
serve two buildings. Thus, if the
salad truck serves Y, that is the
only building served by the salad
truck.
If the ice cream truck serves Y, then the salad truck must serve either X
or Z, or both.
We are now ready to attack. But first, here’s a look at the setup, rules and
deductions we have at our disposal. On the right, you can see the alternative
setup with the deductions filled in.
In the final analysis, we think
using the buildings (X, Y, and Z) as the
base is superior. The linked options
(falafel truck and pita truck) and the
deduction about the third truck serving
Y are much harder to spot using the
trucks. Of course, hindsight is 20/20.
Ninja Note:
4. Questions
Even armed with our all-powerful deductions, there are a few pretty
challenging questions in this game. Let’s go.
Question #18 (absolute, could be true)
The test throws you off a bit by not offering you a nice elimination question
to get things started. But we will be just fine. Here, you are asked to find an
acceptable list of buildings served by the falafel truck.
This one was a cinch.
Falafel could do X, Y, or Z - straight to
(C). Also, what’s falafel?
Cleetus Comment:
Not so fast, Cleetus. You just made a common mistake. While the falafel
truck could serve each of the buildings, that’s not what the question is asking.
This one is actually looking for a list of the buildings the falafel truck could
serve at one time - not at any time. Check out the difference between the
following two questions:
Which one of the following is a complete
and accurate list of each of the office
buildings that the falafel truck could serve?
Which one of the following could be a
complete and accurate list of each of the
office buildings that the falafel truck
serves?
Most students think that they are being asked the question on the left, in
which case (C) would be the correct answer. However, the real question is the
one on the right.
The falafel truck must serve Y and either X or Z, but not both. (D)
is the answer since it offers you Y and Z.
If a question asks you which
answer “is” a complete and accurate
list, that means at any time. If it asks
which one “could be” a complete and
accurate list, that means at one time. A
small yet important distinction.
BP Minotaur:
Always read the question very carefully. It sucks to drop a point simply
because you are answering the wrong question.
Question #19 (absolute, must be true)
The deductions put you in a prime spot to answer this question quickly. The
challenge is to identify two trucks that must serve at least one building in
common. Using the buildings for our setup definitely makes this easier to
visualize.
(A) The hot dog truck might only serve Y, in which case it wouldn’t
serve any of the same buildings as the pita truck.
(B) Read answer choice (A) again, but sub in the taco truck at the end.
(C) The ice cream truck serves both X and
Z. The pita truck must serve either X or
Z. Either way, the ice cream truck and
the pita truck will both show up to one of
the buildings. Worst. Waffle. Cone. Ever.
(D) The salad truck could just serve Y and steer
clear of any and all ice cream trucks.
(E) Read answer choice (D) again, but sub in the taco truck at the end.
Question #20 (conditional, must be true)
This is the first conditional question for the game, so it’s time to create a
hypothetical.
If the hot dog truck serves more buildings than the ice cream truck, the
hot dog truck must serve all three buildings.
If the ice cream truck serves fewer buildings
than the hot dog truck, then the ice cream
truck can’t serve all three buildings. Since
the ice cream truck must serve both X and
Z, it can’t serve Y.
If the ice cream truck doesn’t serve Y, the salad truck must be the third
truck that serves Y.
We’ve made our deductions, so it’s time for answers.
(A) The hot dog truck serves all three buildings and the falafel truck
only serves two buildings, so there’s no chance they can serve the
same buildings.
(B) Since it’s not clear exactly which buildings are served by the
falafel truck or the salad truck, there’s no way to know whether
they serve the same buildings.
(C) The ice cream truck serves two buildings and the pita truck only
serves one. No good.
(D) The salad truck serves Y and the ice cream truck does not. That’s
four strikes.
(E) The ice cream truck serves X and Z. The taco truck serves X
and Z. Those would be the same buildings. That’s what we
were looking for.
Question #21 (absolute, could be true)
Here’s an absolute question that flows nicely from the deductions. The task is
to identify every truck that could serve all three office buildings at one time.
The falafel truck only serves two buildings.
The pita truck only serves one building. (It’s tough to find a pita in this
town.)
The taco truck only serves two buildings.
The salad truck serves fewer buildings than the ice cream truck, so the
salad truck can’t serve all three buildings.
The hot dog truck could serve all three office buildings (see question
#20).
The ice cream truck could also serve all three buildings.
The complete list is just the hot dog truck and the ice cream truck go with (A).
Question #22 (absolute, must be false)
The last journey here is through an absolute question. Identify a lunch truck
that can’t serve both X and Z.
(A) The hot dog truck could serve all three buildings, as noted earlier.
(B) The ice cream truck does serve both X and Z. Duh.
(C) In the deduction phase, we formed
linked options with the falafel truck and
the pita truck in X and Z. This is due to
the fact that the pita truck must serve
either X or Z, but not both. Boom.
(D) The salad truck could serve X and Z as
long as the ice cream truck serves all three buildings.
(E) The taco truck does serve both X and Z. Double-duh.
That wraps up lunch break. Get back to work. We aren’t quite done with our
journey through Logic Games, but this is a big moment in the course of this
book.
First, this is another great example of a profiling
game. It’s important to note that the setup for this game
was much more ambiguous than the last, but success
was attainable with either setup. The form and depth of
deductions were similar to the first profiling game.
Once again, a couple of the groups were completely
determined from the rules, and that always feels good.
Second, this concludes our look at profiling games. Hopefully, by this
point, you will agree with our opening comments. They look scary, but
they can be mastered if you have the right strategy. So here’s one last,
quick overview of how to approach them:
1. Both variable sets are unstable? Profiling it is.
2. Check the rules to figure out the setup.
3. Sloooooow through the rules.
4. Look for tons of deductions by limiting the groups.
Third, and finally, this is the end of our voyage through grouping games.
You might remember that there are two basic tasks underlying every logic
game on the LSAT - ordering and grouping. Ordering is long gone, and we
are closing the door on grouping. We’re not done yet, but that’s pretty sweet.
21/COMBOgames
PUT IT TOGETHER
You might have sensed this coming. Somewhere, the people that write this
darn test are sitting in a dark room, thinking of ways to screw with
prospective law students like you while cackling maniacally. They’ve run out
of ordering games. Students have finally started to grasp grouping games. But
they can’t let you off that easily. Logic Games are supposed to terrorize
students, and they need a new weapon. What do they do? They put them
together.
Combo games involve both ordering and grouping
elements.
This is, of course, analogous to real life. You learn one skill, you learn
another skill, but doing them together can add a degree of difficulty. Walking
and chewing gum. Texting and driving. Going to college and maintaining
your dignity. Examples abound.
Combo games are less
common than either ordering games or
grouping games. But they tend to be
difficult, so ready yourself.
BP Minotaur:
Before we enter the land of combo games, there are a few action items we
need to discuss.
New games, old rules
Before you quiver at the mention of combo games, there’s some good news:
The rules that you are going to face won’t be new to you at all!
Combo games include ordering elements. You will see dashes, blocks,
arches, and all of that good stuff. They also include grouping elements, so
you can expect a variety of grouping relationships. But you already know
how to deal with all of that stuff.
Some rules will combine
ordering and grouping. For example,
“If Jubin and Rubin are on the same
team, then Jubin performs before
Rubin.” But that’s as close to “new” as
you will see.
Ninja Note:
This should be comforting because you have tons of practice with both
ordering and grouping rules. It will be a new challenge to deal with them at
the same time, but the vast majority of the rules will sound familiar.
Order of Operations
One mental obstacle in a combo game is that you will feel like you are
juggling a lot of rules and restrictions. And you are. This necessitates a good
plan of attack. There’s one basic principle to keep in mind when working
through a combo game: Always work with the grouping elements of a
game ahead of the ordering rules.
In a combo game, work through the grouping rules
before the ordering rules.
The reason for this is rather simple, if you think about it. In a combo game,
you have to make teams and order the players on those teams. But before you
worry about the relative order of the players on a team, you need to know
who’s on the team. In other words, you have to organize the groups before
you try to figure out the order.
Here’s a quick example. Say you are doing this game involving a sack race.
This is a classic combo game because you have two teams, and each one
is ordered from first to fourth.
At a family reunion, two teams (blue and
gold) are organized for a sack race from
among eight family members—Grandpa
Joe, Grandma Jane, Uncle Bob, Auntie
Marie, Cousin Sal, Cousin Theo, Sister
Nina, and random guy Hank. Each team
will have four members and there will be
four consecutive legs to the race.
Now, pretend you have the following partial set of rules for the game:
Uncle Bob runs an earlier leg of the race
than Cousin Theo.
Uncle Bob and Grandma Jane are on the
same team. Grandma Jane and Sister Nina
are on different teams.
If you are working through a question on this game, it will be much easier
to work with the grouping elements before the ordering elements. For
instance, if you are told that Sister Nina runs the second leg on the gold
team, what else do you know?
If you look at the first ordering restriction, you are left with nothing.
Bummer.
If you start with the grouping rules, you could deduce that Grandma
Jane is on the blue team with Uncle Bob. Then you are in good shape to
start thinking about the ordering rules.
COMMON COMBO GAMES
Combo games come in a few different varieties. Essentially, the makers of
the test can combine any type of ordering with any type of grouping.
However, there are three types of combo games that pop up more frequently
than the rest.
1. In and Out with Rankings
In and Out games involve selecting some players from a larger group. They
are brutal in nature because some players are simply left out in the cold. On
occasion, they get even more fierce. Not only are some players left behind,
but you have to rank the players that are selected.
Five of the following eight children—
Klein, Lena, Magdalena, Nantucket,
Ophelia, Peter, Quasar, and Randy— will
be selected for the debate tournament. The
children that are selected will be ranked
according to ability from first (best) to fifth
(worst) according to the following:
In this game, you have to select five of eight children, a typical process in an
In and Out grouping game. However, once the five winners are selected, they
must be ranked according to ability. So it’s not enough to know that
Nantucket will be competing in the tournament; you also have to know that
he is ranked fourth.
In and Out combo games will involve complex ordering rules based on
players being selected (if Quasar and Randy are selected, then Quasar is
ranked higher than Randy).
2. Making a Schedule
Arranging a schedule is pretty common in Logic Games - Monday through
Friday, for example. However, life gets busy. There are times when you have
to do more than one thing in a single day, and this is when you might enter
the land of combo games.
Doctor Diva must schedule
appointments with ten
different patients—A, B, C,
D, E, F, G, H, I, and J—
during one week, from
Monday through Friday.
Each day, she sees one
patient in the morning and
one patient in the
afternoon. Her schedule
conforms to the following
conditions:
In this game, you have to determine both (1) what day a patient has an
appointment, and (2) whether that appointment is in the morning or the
afternoon.
If patient B is scheduled for the morning,
then patient D is scheduled for the
afternoon.
Patient C’s appointment is before patient
F’s.
To the right, you can see some possible rules. You could be confronted
with grouping principles (the first rule) or ordering principles (the second
rule).
That just looks like one
of ‘dem tiered ordering deals from a
few chapters back.
Cleetus Comment:
That’s an astute observation. The setup to some combo games is actually
identical to tiered ordering games. However, there’s an important distinction.
In a tiered ordering game, one variable set (cars, for example) goes on one
tier and another variable set (drivers) goes on the other tier. In a combo game
like the one above, you only have one variable set, so each player could end
up in any slot. This introduces a grouping element into the game.
3. Relay Race
The third common type of combo game
is similar to the sack race introduced a
couple pages ago. Essentially, two (or
more) teams are formed, and the
members of the team have some order
assigned.
And now, we’re off. It’s time to flex those ordering and grouping muscles
that have been developed throughout this book.
The first game is about workshops. Try to contain yourself. We will use our
Blueprint Building BlocksTM approach so that we can point out key features
along the way.
JUNE 2010: GAME 1 (1-6)
It’s arts and crafts time! Grab a paintbrush and some scrapbooking materials.
While the topic of this game is very creative, the process is anything but. Put
on your logic shoes and let’s show this thing who’s the boss.
1. Setup
A community center will host six arts-and-crafts
workshops—Jewelry, Kite-making, Needlepoint,
Quilting, Rug-making, and Scrapbooking. The
workshops will be given on three consecutive
days: Wednesday, Thursday, and Friday. Each
workshop will be given once, and exactly two
workshops will be given per day, one in the
morning and one in the afternoon. The schedule
for the workshops is subject to the following
constraints:
We have
three busy days
in front of us.
Make a kite and
a rug in one
day? Crazy.
Our job in this game is to create a schedule for six arts-and-crafts workshops.
(Hint: This should quickly clue you in to the game type since scheduling was
one of our common combo games.)
There are three consecutive calendar days. Since this variable set has an
inherent order, there is definitely an ordering component to this game.
There are two workshops on each day - one in the morning and one in the
afternoon. This introduces a grouping element into the game. Three of the
workshops are in the morning group, and three are in the afternoon group.
Your setup should look like this:
I turned my setup 90 degrees
so it goes up and down.
Ditz McGee:
While there is nothing technically incorrect with constructing a vertical
setup for this game, we still would advise against it. It’s important to stay
consistent, and we almost always arrange ordering elements horizontally
(except in rare cases like floors in a building) and groups vertically. Stick
with that.
2. Rules
As with most combo games, you can expect a mix of ordering and grouping
rules. Here we go.
This first rule is loaded with helpful information. However, it’s somewhat
complicated to represent. Many students will simply transcribe the rule or
separate it into multiple parts. But that’s not necessary.
Challenge: Take a stab at
diagramming this rule. (Hint: Try
to keep it simple.)
It would be a mistake to attempt to view this as
a conditional rule. (If Jewelry, then morning and then Kite-making or
Quilting.) So please don’t try to do that. Rather, always try to visualize rules
in the same way that they are going to look when plugged into the setup.
This rule is best represented by a block. Put Jewelry on top since it is given in
the morning, and make an option for either Kite-making or Quilting in the
afternoon.
Challenge: The second rule is pretty similar, so try to
represent it correctly.
Note that the first two rules
include all six workshops. There are
two blocks introduced, and the two
workshops not involved in the blocks
will be together on the third day.
Ninja Note:
Here are the correct representations of the
first two rules. Since there are no variables
in common, there’s no easy way to
combine the rules.
And we’re already at the last rule!
It’s been a while since we’ve seen an ordering rule - hopefully you remember
how to diagram it.
This one tells us that Quilting must be given on an earlier day than Kitemaking and Needlepoint. This is a clue that Quilting will take place pretty
early in the schedule and both Kite-making and Needlepoint are pushed
back a bit.
Always beware the
distinction between “earlier” and “an
earlier day.” If one variable must be
earlier than another, it’s possible that
they could land on the same day (the
morning is, after all, earlier than the
afternoon). But when it states that
quilting is given on an earlier day,
there’s no way Quilting could be on the same day as either
of the other two.
BP Minotaur:
That’s it for the
rules. Here’s a
look at our
current setup:
3. Deductions
In this game, you get the feeling that the rules play together and there should
be some powerful deductions. But they can be tough to spot initially. The
blocks would be much more fun if they didn’t have those options in there.
However, when you have the feeling that there are deductions, you are
probably right. So let’s start slow and see what we can find.
The first thing to work with is the basic ordering principle. As we learned
in ordering games, you can build restrictions based on such a rule.
Since Quilting must be given on an earlier
day than both Kite-making and Needlepoint,
Quilting can’t be given on Friday, and neither
Kite-making nor Needlepoint can be given on
Wednesday.
Those restrictions will be helpful, but there’s
more to note at this point. Check out Quilting.
It can’t be given on Friday and it must be given
on an earlier day than both Kite-making and Needlepoint. This deserves
some attention.
If Quilting is given on Wednesday, that doesn’t look like it will be too
helpful. But if it’s given on Thursday, that might give us a useful
scenario. This will invoke the ordering restrictions, and it might help
with the blocks. It’s up to you to investigate.
Challenge: Make a hypothetical with Quilting on
Thursday.
That should have been very satisfying. This hypothetical finally helps us see
how the rules work together. Here’s an outline:
If Quilting is given on Thursday, it’s not clear right up front whether it
would be in the morning or afternoon. But we can wait on that. Both Kitemaking and Needlepoint must be given on Friday. The order on Friday
isn’t clear, but we can make linked options to represent this.
At this point, you know exactly what your blocks look
like. Since Kite-making isn’t on the same day as
Jewelry, Quilting must be on the same day as Jewelry.
Also, since Needlepoint isn’t on the same day as Rugmaking, Scrapbooking must be. Sweet.
Now Thursday is looking much better. Quilting must be given on
Thursday afternoon, and Jewelry must be given on Thursday morning.
The only place left for the other block is Wednesday, so Scrapbooking is
given on Wednesday morning and Rug-making is given on Wednesday
afternoon.
This hypothetical works out wonderfully.
Granted, it’s only one situation, but it
helps outline the deductions that follow
from putting Quilting on Thursday.
Note: This also gives us the deduction
that Quilting can’t be given on Thursday morning.
The options for Quilting turn
out to be very limited. It could be
given on Thursday afternoon, but only
if we have the arrangement above.
Otherwise, it must be given on
Wednesday. We love quilting (on the
LSAT and in real life).
Ninja Note:
Now that we have a few deductions and a better handle on how the rules play
together, it’s time to hit the questions. This is the first time we will be
juggling both ordering and grouping rules, so it will be great practice.
4. Questions
There are still a variety of challenges in front of us. We have to deal with
some grouping and some ordering. Lacking any huge deductions about the
schedule, there will be a large number of conditional questions. The goal is to
construct each hypothetical quickly and efficiently.
Question #1
1. Which one of the following is an acceptable
schedule for the workshops, with each day’s
workshops listed in the order in which they are to
be given?
Elimination
time!
First up, we have an elimination question. This is a great chance to see
how the rules can be applied to a range of answer choices. You know how
these work by now, so go for it.
Challenge: Use the elimination approach to find the
correct answer.
Jewelry must be given in the morning, on
the same day as either Kite-making or
Quilting.
Rug-making must be given in the
afternoon, on the same day as either
Needlepoint or Scrapbooking.
Quilting must be given on an earlier day
than both Kite- making and Needlepoint.
(A) Wednesday: Jewelry, Kite-making
Thursday: Quilting, Scrapbooking
Friday: Needlepoint, Rug-making
(B) Wednesday: Jewelry, Quilting
Thursday: Kite-making, Needlepoint
Friday: Scrapbooking, Rug-making
(C) Wednesday: Quilting, Needlepoint
Thursday: Scrapbooking, Rug-making
Friday: Jewelry, Kite-making
(D) Wednesday: Quilting, Scrapbooking
Thursday: Jewelry, Kite-making
Friday: Rug-making, Needlepoint
(E) Wednesday: Scrapbooking, Rugmaking Thursday: Quilting, Jewelry
Friday: Kite-making, Needlepoint
Only one answer shall stand above the rest...
If you are still getting elimination questions wrong, you need to slap
yourself. Twice. And do it pretty hard - there should be a mark tomorrow.
First up - Jewelry must be given in the morning, and either Kite-making
or Quilting is on the same day in the afternoon. In (E), Jewelry is clearly
slotted for the afternoon. Nope.
The second rule states that Rug-making is in the afternoon, with either
Needlepoint or Scrapbooking in the morning. In (D), Rug-making is in
the morning. They really aren’t picking up on this whole morningversus-afternoon distinction.
The final rule tells us that Quilting is given before Kite-making and
Needlepoint. In (A), Quilting is later than Kite-making. In (C), Quilting
and Needlepoint are on the same day. Both answers must die.
(B) is the winner.
Question #2
2. Which one of the following workshops
CANNOT be given on Thursday morning?
Time for the
deductions to
start paying off.
This is our first absolute question. It requires you to identify a workshop
that can’t be given on Thursday morning. Believe it or not, we already know
the answer. By reviewing the rules during our deduction phase, we noted that
Quilting was the most restricted workshop.
Quilting can be given on
Thursday, but only if it’s given
in the afternoon with Jewelry
that morning. This was the
hypothetical that we attempted
a few pages back.
Since Quilting can’t be given
on Thursday morning, (D) is
the glorious answer to this one.
(A) Jewelry
(B) Kite-making
(C) Needlepoint
(D) Quilting
(E) Scrapbooking
That was nice and quick.
Another lesson in utilizing a calm and measured
pace. Running in to check answers here would have been
stressful and time-consuming.
BP Minotaur:
Question #3
3. Which one of the
following pairs of workshops
CANNOT be the ones given on
Wednesday morning and
Wednesday afternoon,
respectively?
Here we go again. This one is looking
for an answer that must be false. But
now we are focusing on Wednesday.
You may find this hard to believe, but
we already have the answer to this one
as well.
Challenge: Focus on the setup and see if you can
identify a pair of workshops that can’t be given on
Wednesday. This is very, very hard (sarcasm).
(A) Jewelry, Kite-making
(B) Jewelry, Quilting
(C) Quilting, Scrapbooking
(D) Scrapbooking, Quilting
(E) Scrapbooking, Rugmaking w th f KQ N
Note that we are trying to anticipate the answer instead of having to test
each one. This will hopefully give us a very easy way through this
question.
Since both Kite-making and Needlepoint must be given on a later day
than Quilting, neither of them can be given on Wednesday.
There are still a decent number of options for the workshops on
Wednesday. However, it’s worth checking to see if they foolishly try to
schedule either Kite-making or Needlepoint for this day since we know it
won’t work.
Kite-making can’t be given at any time on Wednesday, so the
combo of Jewelry and Kite-making just isn’t going to cut it. (A) is
the winner.
Question #4
4. If Kite-making is given on Friday morning,
then which one of the following could be true?
The word “if”
shows up finally.
It’s been a longer-than-anticipated wait, but now we get a conditional
question. This one might take a little more time and effort. The new
condition is that Kite-making is given on Friday morning. First, it’s time to
see how this interfaces with our Quilting deductions.
Since Kite-making is given in the morning, Quilting must be the
workshop given in the afternoon on the day that Jewelry is given in the
morning.
The new block has two possible placements. Jewelry (with Quilting) must
be given on either Wednesday or Thursday. Since this is a could be true
question, it’s a good idea to work through both hypotheticals.
Challenge: Using the new condition and fullydetermined block, build two hypotheticals. In the first,
put Jewelry on Wednesday. In the second, move
Jewelry over to Thursday. Make sure to search for
additional deductions.
There are many deductions in both situations. Here is an overview:
The block could go on Wednesday - Jewelry
could be given on Wednesday morning, and
Quilting could be given on Wednesday
afternoon.
Rug-making must be given on Thursday afternoon.
Needlepoint and Scrapbooking are still up in the air, so you should form
linked options for the final two workshops.
Quilting squeezes into the afternoon slot.
Since Needlepoint must be given after Quilting,
Needlepoint is on Friday afternoon.
Scrapbooking is given on Wednesday morning,
and Rug-making fills in the last slot on Wednesday afternoon.
That was a lot of work, but it will
be well worth it when you glide
straight to the correct answer. We
need something that could be true
in at least one situation above.
(A) Jewelry is given on
Thursday morning.
(B) Needlepoint is given on
Thursday afternoon.
In the second hypothetical,
Jewelry is given on Thursday
morning, so (A) is correct.
(C) Quilting is given on
Wednesday morning.
(D) Rug-making is given on
Friday afternoon.
(E) Scrapbooking is given on
Wednesday afternoon.
Question #5
5. If Quilting is given in the morning, then
which one of the following workshops CANNOT
be given on Thursday?
Another
question about
Quilting?
Shocking...
Another conditional question. Here we go. As soon as you see that Quilting is
involved, you should feel good about our deductions.
Quilting could be given on Wednesday or Thursday. However, it must
be in the afternoon if it’s given on Thursday (see helpful deduction from
above). So if Quilting is given in the morning, it must be given on
Wednesday morning.
Our deductions helped us get a quick start. But there’s a long way to go.
Now, it’s your turn to take over. Remember: You need to find a workshop
that can’t be given on Thursday.
Challenge: On the next page, try to place some of the
other workshops. (Hint: Focus on Wednesday.) Then,
try to locate the correct answer.
(A) Jewelry
(B) Kite-making
(C) Needlepoint
(D) Rug-making
(E) Scrapbooking
This is a classic example of
LSAT misdirection. They ask you
about Thursday, so you are naturally
thinking about Thursday. But it’s really
all about that one spot on Wednesday
afternoon.
Ninja Note:
There aren’t a lot of additional deductions here, but there’s one huge one.
Kite-making and Needlepoint can’t be given on Wednesday, so that
knocks out two options for Wednesday afternoon.
Jewelry must be given in the morning, so Jewelry can’t be given on
Wednesday afternoon.
Rug-making is given in the afternoon, but only with either Needlepoint or
Scrapbooking in the morning. Since Quilting is given on Wednesday
morning, Rug-making can’t be given on Wednesday afternoon.
All of a sudden, through sheer will (and the powerful process of
elimination), there’s only one workshop left for Wednesday afternoon:
Scrapbooking.
Scrapbooking must be given on Wednesday.
Wednesday is not Thursday. These two facts
together definitively prove, beyond a shadow of
doubt, that Scrapbooking can’t be given on
Thursday. (E) is our answer.
Question #6
6. How many of the workshops are there that
could be the one given on Wednesday morning?
Last
question!
We have arrived at the last question. As you can see, this is an absolute
question that will test our deep knowledge of the game.
This question asks how many workshops could be given on Wednesday
morning. The first thing that you want to do is check your work on the
previous questions. You’ve worked hard, and it’s always nice when that
work pays off twice. We will give you a quick overview of what you can
learn all of from the hypotheticals since we wouldn’t want you to hurt
yourself turning back through those pages.
In question #1, Jewelry is scheduled for Wednesday morning in the
correct answer, so that’s the first one on our list.
Question #2 is no help. Boo.
In question #3, we were asked for a pair of workshops that couldn’t be
given on Wednesday. But the other four answer choices give us combos
that do work. Both Quilting and Scrapbooking are found in other answer
choices, so that’s two more. Our list grows to three.
In question #4, both Jewelry and Scrapbooking make appearances on
Wednesday morning, but they are already on our list.
In question #5, Quilting is on Wednesday morning, but that’s also been
covered. Our list is still at three.
That was very helpful, but you still have to make sure no other workshop
can slide into that time slot.
Rug-making must be given in the afternoon, so no worries there.
Both Kite-making and Needlepoint are outlawed on Wednesday.
There are three different
(A) one
workshops (Jewelry,
(B) two
Quilting, and Scrapbooking)
(C) three
that could be given on
(D) four
Wednesday morning, so (C)
(E) five
is the final answer to this
game.
So that’s a combo game. As you just experienced, our prior work on ordering
and grouping games is incredibly helpful. There are some new features in a
game like this. Ordering elements (the days of the week) are combined with
grouping elements (morning or afternoon) to throw a twist into this game.
Here are some lessons to be learned from our first combo game:
1. They aren’t that different. There’s some ordering, and there’s some
grouping. But if you stick with the setups and diagrams that we
learned earlier, all will go smoothly.
2. Deductions are still king, no matter the type of game. In this one,
there were just a few small things to notice, but they saved valuable
minutes on the questions.
3. Don’t ever schedule Quilting for Friday. It’s dangerous.
Next up, you get a shot at a combo game. This one
has some different features, but it involves both
ordering and grouping. These games are still new,
so don’t worry about timing. Focus on constructing
a proper setup and searching for deductions.
Good luck! See you on the other side.
JUNE 2004: GAME 3 (13-17)
Questions 13-17
Exactly six of an artist’s paintings, entitled Quarterion, Redemption, Sipapu,
Tesseract, Vale, and Zelkova, are sold at auction. Three of the paintings are
sold to a museum, and three are sold to a private collector. Two of the
paintings are from the artist’s first (earliest) period, two are from her second
period, and two are from her third (most recent) period. The private collector
and the museum each buy one painting from each period. The following
conditions hold:
Sipapu, which is sold to the private collector, is from an earlier period than
Zelkova, which is sold to the museum.
Quarterion is not from an earlier period than Tesseract. Vale is from the
artist’s second period.
13. Which one of the following could be an accurate list of the
paintings bought by the museum and the private collector, listed in
order of the paintings’ periods, from first to third?
(A) museum: Quarterion, Vale, Zelkova
private collector: Redemption, Sipapu, Tesseract
(B) museum: Redemption, Zelkova, Quarterion
private collector: Sipapu, Vale, Tesseract
(C) museum: Sipapu, Zelkova, Quarterion
private collector: Tesseract, Vale, Redemption
(D) museum: Tesseract, Quarterion, Zelkova
private collector: Sipapu, Redemption, Vale
(E) museum: Zelkova, Tesseract, Redemption
private collector: Sipapu, Vale, Quarterion
14. If Sipapu is from the artist’s second period, which one of the
following could be two of the three paintings bought by the private
collector?
(A)
(B)
(C)
(D)
(E)
Quarterion and Zelkova
Redemption and Tesseract
Redemption and Vale
Redemption and Zelkova
Tesseract and Zelkova
15. Which one of the following is a complete and accurate list of the
paintings, any one of which could be the painting from the artist’s
first period that is sold to the private collector?
(A)
(B)
(C)
(D)
(E)
Quarterion, Redemption
Redemption, Sipapu
Quarterion, Sipapu, Tesseract
Quarterion, Redemption, Sipapu, Tesseract
Redemption, Sipapu, Tesseract, Zelkova
16. If Sipapu is from the artist’s second period, then which one of the
following paintings could be from the period immediately
preceding Quarterion’s period and be sold to the same buyer as
Quarterion?
(A)
(B)
(C)
(D)
(E)
Redemption
Sipapu
Tesseract
Vale
Zelkova
17. If Zelkova is sold to the same buyer as Tesseract and is from the
period immediately preceding Tesseract’ s period, then which one
of the following must be true?
(A)
(B)
(C)
(D)
(E)
Quarterion is sold to the museum.
Quarterion is from the artist’s third period.
Redemption is sold to the private collector.
Redemption is from the artist’s third period.
Redemption is sold to the same buyer as Vale.
PAINTINGS AND PERIODS
You can’t ask for much more from a game on the LSAT. You have six
paintings with interesting names. You have some ordering. You’ve got some
grouping. Deductions abound. This is like a perfect storm, so let’s check it
out.
1. Setup
It takes a while for the introduction to lay out all of the features in this game.
But let’s break it down to isolate the important words and phrases.
The first sentence simply tells you that six paintings are sold. You might have
stumbled when trying to pronounce Sipapu or Zelkova, but don’t stress over
that. The second sentence claims that three of the paintings are sold to a
museum and the other three are sold to some rich dude (private collector).
Hello grouping. At this point, we have a stable grouping element to this
game. But it’s not over.
The third sentence claims that
two of the paintings are from
the artist’s first period, two are
from her second period, and
two are from her third period.
Apparently, our artist went
through a lot of phases (good
for her). This introduces an
ordering element to the game,
but it’s hard to see how this
interacts with the original
grouping challenge.
The final sentence is the key. It
tells you that the museum and the private collector each buy one painting
from each period. This simplifies the game and helps you visualize a great
setup. Once we determine the three paintings that are sold to the museum,
then we will have to order them by period. This can all be included into an
efficient combo setup.
2. Rules
There are only three rules presented, but they give you a ton of information.
It’s important to visualize how each one works in the setup because there is a
strong relationship between a number of the rules.
This rule is huge - it’s probably the most important rule in the whole game.
First, it gives you some grouping information. Sipapu is sold to the private
collector, and Zelkova is sold to the museum. Second, it also gives you an
ordering restriction - Sipapu is from an earlier period than Zelkova. You
can combine all of this information by using the representation above.
When you drill down, there’s even more you can do with this rule.
Sipapu must be the painting from the first or second period that is sold
to the private collector.
Zelkova must be the painting from the second or third period that is sold
to the private collector.
If Sipapu is from the second period, then Zelkova must be from the third
period.
If Zelkova is from the second period, then Sipapu must be from the first
period.
Since this rule provides so much information, it’s
important to revisit it with each new rule to see if
additional deductions can be made.
This is another rule that can be tricky. This is a straight ordering rule.
You gotta simplify - that
just means that Quar-ter-thingy must
be later than Tesser-whatever.
Cleetus Comment:
Cleetus is right about the first part - it is important to simplify. But you can
only simplify when you don’t change the meaning of a rule, and Cleetus falls
for that trap. In a basic 1:1 ordering game, if one variable isn’t in front of
another, then it must be behind. But in this game, two variables could be
from the same period. Thus, even if Quarterion is not from an earlier period
than Tesseract, we can’t say that it must be from a later period. They could
both be from the same period.
Here’s another way to state this rule:
Quarterion must be from the same period or a later
period than Tesseract.
If you are going through a question and you discover that Tesseract is from
the artist’s second period, then Quarterion can’t be from the artist’s first
period. But there are two more helpful deductions to note.
If Quarterion is from the artist’s first period, then Tesseract must also be
from the artist’s first period.
If Tesseract is from the artist’s third period, then Quarterion must also be
from the artist’s third period.
Just one more rule:
This rule tells us where in the order Vale falls (2), but it doesn’t help us figure
out to which group it is assigned.
On the next page, you can see our current status after the setup and the rules.
3. Deductions
Most of the work has been done by visualizing all of the rules correctly, but
there are a few additional deductions that can be helpful.
Redemption is not mentioned in any of the rules,
so it’s a random.
There’s a cool relationship between the first rule
and the last rule. If Zelkova is from the artist’s
second period, then Vale is sold to the private
collector, and Sipapu must be from the artist’s first period.
Since Quarterion can’t be from an earlier period than Tesseract,
Quarterion must be from the artist’s third period.
Alternatively, if Sipapu is from the artist’s
second period, then Vale is sold to the museum
and Zelkova is from the artist’s third period.
Since Tesseract can’t be from a later period than
Quarterion, Tesseract must be from the artist’s
first period.
At this point, we built an effective combo setup that allowed us to visualize
the ordering and grouping elements in this game. Then, we worked slowly
through the rules and examined the implications of each one. Finally, we
searched out some additional deductions. We’re in good shape to attack the
questions now.
4. Questions
The order of operations is very important for questions in combo games. You
have to make sure to prioritize the grouping rules before the ordering. Let’s
take a look.
Question #13 (elimination, could be true)
Well, look at that, an elimination question. That is a fine way to start. Our
challenge is to knock out the bad guys.
According to the first rule, Sipapu is sold to the private collector,
Zelkova is sold to the museum, and Sipapu is from an earlier period than
Zelkova. In (C), Sipapu is sold to the museum. No bueno. In (E), Sipapu
and Zelkova are both from the artist’s first period. No bueno again.
The second rule states that Quarterion is not from an earlier period than
Tesseract. In (A), Quarterion is from the first period and Tesseract is
from the third period. Get it out of here.
The final rule asserts that Vale is from the artist’s second period. In (D),
Vale is in the third slot.
After killing the four losers, (B) stands tall.
Question #14 (conditional, could be true)
This one relates nicely back to our deductions about the artist’s second
period. The new condition is that Sipapu must be from the second period.
With our new condition, we know exactly where Sipapu fits in our
setup.
Vale is also from the artist’s second period, so it must be sold to the
museum.
Zelkova must be from a later period than
Sipapu, so Zelkova must be from the
third period (and, according to the rules,
sold to the museum).
Quarterion can’t be from an earlier
period than Tesseract. They could both
be from the first period, but there’s no way that Tesseract could be from
the artist’s third period.
This question asks about the groups, so we don’t have to worry about the
order. The challenge is to identify two paintings that could both be sold to the
private collector. However, we can use the paintings that must be sold to the
museum to eliminate answer choices.
Vale and Zelkova are both sold to the museum. Thus, neither of them
could be sold to the private collector. (A), (C), (D), and (E) are gone.
Wait, that’s four answers. (B) is the only one left standing, so it must
be correct.
Question #15 (absolute, could be true)
This absolute question asks you to make a list of paintings that could be sold
to the private collector and be from the artist’s first period. It seems
complicated, but there is a smart (quick) way through this one.
Sipapu is sold to the private collector and could be from the artist’s first
period, so Sipapu is first on the list.
To test the other options, we just have to take Sipapu out of that slot.
If Sipapu isn’t from the artist’s first period, it must be from the second
period.
Luckily, you just worked with this exact same situation in question #14.
Vale must be the painting from the second
period sold to the museum, and Zelkova is
the painting from the third period sold to the
museum.
Quarterion and Tesseract could both be
from the artist’s first period, so both of these paintings could be from the
artist’s first period and sold to the private collector.
Redemption is random and could also fill that slot.
The complete list is Quarterion, Redemption, Sipapu, and Tesseract,
so (D) is the winner.
Question #16 (conditional, could be true)
The wording on this question can be tricky, but the new condition places us
squarely with the same deductions we’ve already utilized on the last two
questions. Once again, Sipapu is from the artist’s second period.
Sipapu is sold to the private collector and is
from the artist’s second period (see questions
#14 and #15).
Vale is from the artist’s second period, so it
must be sold to the museum.
Zelkova is from the artist’s third period.
Now the challenge is to untangle the wording. We’re looking for a painting
that must be from the period immediately preceding Quarterion’s period.
And it must be sold to the same buyer. But you don’t know who bought
Quarterion or what period it’s from. Hmmm...what to do?
Since there are no paintings before the artist’s first period, Quarterion
can’t be from the first period. The only option left is the artist’s third
period (sold to the private collector). So that takes care of that hurdle.
With Quarterion settled, it’s clear that
Sipapu is the painting in question (same
buyer as Quarterion and immediately
preceding period). (B) is the answer.
Question #17 (conditional, must be true)
On to the last question. This one finally tests some different deductions.
Zelkova is sold to the museum, so Tesseract
must also be sold to the museum.
Zelkova can’t be from the artist’s first
period. So if it must be from an earlier
period than Tesseract, Zelkova must be
from the second period. This places
Tesseract in the third period.
Vale must be sold to the private collector
since it must be from the artist’s second period.
Sipapu must be from an earlier period than Zelkova, so Sipapu must be
from the artist’s first period.
Normally, this is more than enough deductions. But you can actually fill in
the last two spots.
Since Quarterion can’t be from an earlier
period than Tesseract, Quarterion must be
from the artist’s third period (and sold to the
private collector).
Redemption fills the last slot - it is the painting
from the first period sold to the museum.
Quarterion is most definitely from the artist’s third period, so (B) is
correct.
And that’s a wrap. This is a very typical combo game - there are clear and
stable groups and each group has an order.
It’s been a while since you’ve played with ordering rules, so make sure to
look back over them if they are giving you any trouble.
Our journey through combo games is not over. In the next one, you get to
step into the shoes of a women’s track coach. Yep, it’s always been a dream
of ours as well. Give it a shot, and we will review once you have successfully
completed your mission.
OCTOBER 2010: GAME 3 (12-17)
Questions 12-17
The coach of a women’s track team must determine which four of five
runners—Quinn, Ramirez, Smith, Terrell, and Uzoma—will run in the four
races of an upcoming track meet. Each of the four runners chosen will run
in exactly one of the four races—the first, second, third, or fourth. The
coach’s selection is bound by the following constraints:
If Quinn runs in the track meet, then Terrell runs in the race immediately
after the race in which Quinn runs.
Smith does not run in either the second race or the fourth race.
If Uzoma does not run in the track meet, then Ramirez runs in the second
race.
If Ramirez runs in the second race, then Uzoma does not run in the track
meet.
12. Which one of the following could be the order in which the
runners run, from first to fourth?
(A)
(B)
(C)
(D)
(E)
Uzoma, Ramirez, Quinn, Terrell
Terrell, Smith, Ramirez, Uzoma
Smith, Ramirez, Terrell, Quinn
Ramirez, Uzoma, Smith, Terrell
Quinn, Terrell, Smith, Ramirez
13. Which one of the following runners must the coach select to run
in the track meet?
(A)
(B)
(C)
(D)
(E)
Quinn
Ramirez
Smith
Terrell
Uzoma
14. The question of which runners will be chosen to run in the track
meet and in what races they will run can be completely resolved if
which one of the following is true?
(A)
(B)
(C)
(D)
(E)
Ramirez runs in the first race.
Ramirez runs in the second race.
Ramirez runs in the third race.
Ramirez runs in the fourth race.
Ramirez does not run in the track meet.
15. Which one of the following CANNOT be true?
(A) Ramirez runs in the race immediately before the race in
which Smith runs.
(B) Smith runs in the race immediately before the race in which
Quinn runs.
(C) Smith runs in the race immediately before the race in which
Terrell runs.
(D) Terrell runs in the race immediately before the race in which
Ramirez runs.
(E) Uzoma runs in the race immediately before the race in which
Terrell runs.
16. If Uzoma runs in the first race, then which one of the following
must be true?
(A)
(B)
(C)
(D)
(E)
Quinn does not run in the track meet.
Smith does not run in the track meet.
Quinn runs in the second race.
Terrell runs in the second race.
Ramirez runs in the fourth race.
17. If both Quinn and Smith run in the track meet, then how many of
the runners are there any one of whom could be the one who runs
in the first race?
(A) one
(B) two
(C) three
(D) four
(E) five
TRACK MEET
This is definitely a combo game, but it has some very different features than
the first two we attempted. You get to step into the shoes of a women’s track
coach and face some complicated decisions: Should Terrell or Uzoma run the
third race? Who rides the bench? There’s obviously a lot of emotion
wrapped up in the job, but don’t let that throw you off your game.
1. Setup
The first task is to figure out the game type and setup. There are five runners:
Quinn, Ramirez, Smith, Terrell, and Uzoma. The first task is to select four of
the five runners for four upcoming races. Someone didn’t make the cut - this
is a stable In and Out grouping game. This must be reflected in your setup.
Additionally, the four races are ordered from first to fourth. Hello combo
game. This means that there will be two operations in this game:
1. Choose four runners
to compete in the four
races.
2. Order the four
runners that are
selected.
This is another common type of combo game. Here’s our setup at this point:
Since there’s only one runner left out, it’s going
to be very important to keep track of who that could be.
Ninja Note:
2. Rules
There are a few challenging rules here. As always, it’s vital to correctly
interpret each rule and simplify when possible.
This is a rule that could only exist in a combo game since it includes both
ordering and grouping. It’s commonly misdiagrammed by students.
Ditz McGee:
I skipped the arrow part and just made the block.
No, Ditz, that’s actually what you have to avoid. This rule is based on the
condition that Quinn must run in the track meet. Quinn might not run in the
track meet, in which case this rule doesn’t apply. If you simply diagram a
block, you might be led to believe that Quinn and Terrell must always form a
block. Not true - Quinn could easily be out.
Here’s the type of
deduction that does
follow from this rule:
If Quinn is found to
run in the second
race, then Terrell must
run the third race.
However, it doesn’t
work the other way: If
Terrell is found to run
in the third race, Quinn doesn’t have to run in the second race. She could,
but she might not run a race at all. You could, however, conclude that
Quinn can’t run the first or fourth race.
So that covers the basics of this rule. However, there are also some huge
deductions to be drawn solely from this rule.
If Quinn runs in the track meet, then
Terrell must run in the race immediately
after Quinn. But if Quinn tried to run the
fourth race, there would be no race after
that for Terrell to run. Thus, Quinn can’t run in the fourth race.
The second deduction is based on the grouping
parts of the game. Only one runner sits out of the
track meet. If Quinn runs in the track meet, then
Terrell must run as well. Obviously, if Quinn
doesn’t run in the track meet, then everyone else,
including Terrell, must run. Thus, Terrell must
always run in the track meet. If you tried to put Terrell in the out group,
Quinn would have to run a race, but Terrell couldn’t run right after her.
That’s a no-go.
So that would qualify as a humongous rule. Let’s keep moving.
Make restrictions to show that Smith can’t run the second race or the fourth
race. Of course, Smith doesn’t have to run the first race or the third race;
Smith could be left out of the track meet altogether.
This is an interesting conditional rule. If Uzoma is left out of the track meet,
then Ramirez must run the second race. Make sure to diagram the
contrapositive. If Ramirez does not run the second race, then Uzoma must
run in the track meet.
This rule can be simplified as an “or” statement either Uzoma runs in the track meet or Ramirez runs the
second race (or both).
Ninja Note:
I think there’s a typo that rule is the same as the last one.
Cleetus Comment:
It’s close, but it’s not exactly the same. Normally, this is a conditional fallacy
that you have to avoid, but here they give you a reciprocal relationship.
If A implies B, it doesn’t follow that B
implies A. But if you are provided with an
additional rule that B implies A, the
combination of rules leads to a love relationship.
The situation here is identical. First, you are told if Uzoma does not run in
the track meet, then Ramirez runs in the second race. Next, you are told if
Ramirez runs in the second race, then Uzoma doesn’t run in the track
meet. To simplify, Uzoma doesn’t run in the track meet if and only if
Ramirez runs in the second race.
Of course, you always have the contrapositives to
help you out. If Ramirez doesn’t run in the second
race, then Uzoma does run in the track meet. And if
Uzoma does run in the track meet, then Ramirez doesn’t run the second
race.
Since this rule tells us so
much, we will have to keep our eyes
on it for every question.
BP Minotaur:
It’s a good deal of work just to get through the rules on this game, and even
more work to understand the implications of each one. On the next page, you
can see an overview of the setup and rules.
3. Deductions
At this point, you should be armed with some deductions from applying the
rules (most notably the first rule). But there’s more gold in them hills. Your
eye should be on the last two rules. Let’s dig a little deeper.
The last two rules tell you two different things. First, either Uzoma has
to run in the track meet or Ramirez runs in the second race. Second,
both of these conditions can’t be met - it can’t be true that Uzoma runs
in the track meet and Ramirez runs in the second race.
Either Uzoma runs in the track meet or Ramirez runs in
the second race, but not both.
Once you grasp the meaning of the last two rules, it’s time to play with them
a little. Since they’re very strong rules, you will be able to see how the other
rules interact with them. First, let’s see what happens if Uzoma runs in the
track meet.
If Uzoma runs in the track meet, then Ramirez can’t run the second race.
But that’s really all you get. It’s still not clear who is out, so it’s hard to
make deductions about the other races.
So... that wasn’t amazing. But
there’s one more. Now let’s plug
Ramirez into the second race.
If Ramirez runs in the second race, then
Uzoma can’t run in the track meet at all.
Sorry, Uzoma.
If Uzoma doesn’t run in the track meet,
then the other three spots must be filled by
Quinn, Smith, and Terrell.
Since Quinn must run in the meet, we
know that Terrell must run in the race
immediately after her. There’s only one
place for this block to fit - Quinn must run the third race, and Terrell
runs the fourth race.
Smith takes the last remaining race - number one.
That went nicely. If Ramirez runs the second race, the entire setup is
determined. That will be huge to have on your side. With all of these
deductions in place, it’s time to hit the questions.
4. Questions
Let’s see how you did. If you misinterpreted any of the rules, these questions
aren’t much fun.
Question #12 (elimination, could be true)
Things get started with a nice and friendly elimination question.
If Quinn runs in the track meet, Terrell must run the race immediately
following Quinn. In (C), Quinn runs the fourth race, so Terrell can’t follow
her. (C) is out.
The second rule tells you that Smith can’t run the second or fourth race.
(B) goes away because Smith is in the second spot.
The third rule is conditional - if Uzoma doesn’t run in the track meet, then
Ramirez runs the second race. In (E), Uzoma doesn’t run in the track meet
but Ramirez is slotted into the fourth race. Nope.
The last rule is the other conditional - if Ramirez runs the second race,
Uzoma doesn’t run in the track meet. In (A), Ramirez runs the second race
but Uzoma still runs in one of the races. No way.
(D) gets the gold medal.
Question #13 (absolute, must be true)
One of the earlier deductions makes this one a breeze.
Due to the first rule (if Quinn runs in the track meet, then Terrell must run
in the race immediately after Quinn) and the fact that only one runner
doesn’t run in the track meet, you can deduce that Terrell always runs in
one of the four races.
(D) is the winner since Terrell must always run in the track meet.
Question #14 (absolute, must be true)
This question asks for an answer that would determine all four races in the
track meet. Such a question can be very time-consuming because students
commonly have to draw a hypothetical to test each answer choice. But that is
not necessary here. We already identified a situation in which all the races are
determined. During the deductions phase, the final two rules prompted us to
try a scenario with Ramirez in the second race.
If Ramirez runs the second race, then
Uzoma is out of the track meet.
Quinn must run in the third race, and Terrell
must immediately follow in the fourth race.
Smith takes the last spot in the first race.
If Ramirez runs the second race, then all
four races are determined. (B) is the answer.
Question #15 (absolute, must be false)
Here’s a big one. This is an absolute, must be false question. It’s hard to
picture some of the answer choices. But you don’t have to wait too long - the
correct answer derives from the same deductions as before.
Smith cannot run in either the second race or the fourth race, so Smith can
only run in the first or third race.
The only way for Ramirez to run immediately before Smith is if Ramirez
runs the second race and Smith runs the third race.
But wait, if Ramirez runs in the second race, we already know that Quinn
runs the third race, and Smith runs the first race.
There’s no way for Ramirez to run in the race immediately before
Smith, so (A) must be false.
When there’s a big
deduction, they just keep going back to
it. In this game, playing with the
situation where Ramirez runs the
second race really saves the day.
BP Minotaur:
Question #16 (conditional, must be true)
This is the first conditional question of the game. It requires a couple tough
deductions.
If Uzoma runs in the first race, then
Ramirez can’t run in the second race.
Ramirez and Smith can’t run in the second
race, so either Quinn or Terrell must run in
the second race.
If Quinn runs in the second race, then
Terrell must run in the third race.
If Quinn doesn’t run in the second race,
then Terrell must run in the second race, and Quinn can’t run in the
track meet.
In either situation, Smith can’t run in the fourth race, so Ramirez must
run in the fourth race.
(E) is looking good since Ramirez must be in the last spot.
Question #17 (conditional, could be true)
We end on another conditional question. For this one, both Quinn and Smith
must run in the track meet. The task is to then ascertain the number of options
for the first race. The best approach is to spot a few deductions and then test
the different possibilities.
If Quinn runs in the track meet, then Terrell
must run in the race immediately after Quinn.
Quinn could run in the first race as long as
Terrell runs in the second race. So that’s one
option.
Smith could run in the first race with plenty of
room left for the block with Quinn and Terrell.
That’s two.
Smith can’t run in the second race or the fourth
race. So if either Uzoma or Ramirez run in the
first race, Smith would have to run in the third
race. Uh-oh, that’s a problem. Then there wouldn’t be room for the
block with Quinn and Terrell.
So there is a grand total of two runners who could compete in the
first race - Quinn and Smith. That leads us to (B).
That’s the end of our track meet. It’s a tough game,
so don’t stress if you ran into some problems. Many
students considered it to be the toughest game in
that section.
Here are some lessons that became clear in that
game:
1. Another common type of combo game includes selecting a set of
players (In and Out grouping) and placing those selected in order.
2. Combo games include the same rules seen in ordering and grouping
games. Now you just get to use them together.
3. Always look for common grouping relationships. In this game, the
last two rules combined to form a love relationship.
4. When you attack the questions, keep referring back to your big
deductions. The LSAT will form different questions that all test the
same deductions.
This is a huge moment for us. Ordering games wrapped up a few chapters
back. Grouping games concluded more recently. And the track meet marks
the end of combo games.
That’s it. Well, kinda...
UP NEXT
When you take the LSAT, we would bet a lot of money that each game you
see will be an ordering, grouping, or combo game. However, once in a while,
every couple years, when they have had a bad week, something different,
something strange, something scary, pops up, and it’s time for us to take a
look...
22/NEITHER?!?
WHAT THE...
Yes, we did tell you that Logic Games are all about ordering and grouping.
And we weren’t lying. Well, okay, we were. But just a little bit. Here’s the
bad news: Every once in a while, a game pops up that involves a task that is
neither ordering nor grouping.
These games are very rare. They occur, on average, only once every two or
three years. But, just like the Boy Scouts, you always want to be prepared.
You wouldn’t want to be surprised by one of these little devils.
“Neither” games involve neither ordering nor
grouping.
We aren’t going to spend much time covering these games. In fact, we aren’t
actually going to do any of them. Rather, we will provide you with an
overview of the types of neither games you might confront. But this should
be interpreted by you as a reflection of the small amount of study time that
you should devote to these oddball games. Don’t freak out about this stuff.
Neither games introduce some form of novel gimmick. But once you get the
gimmick, you can easily do any Neither game.
Some of the Neither game
types haven’t appeared on the LSAT
since 1995. That is quite some time
ago. Coolio and TLC were sitting at
the top of the musical charts.
BP Minotaur:
COMMON NEITHER GAMES
There are four types of Neither games that have popped up over the years:
characteristic grid, mapping, cyclical, and operation.
Of the four, characteristic grid
games are by far the most common.
Ninja Note:
1. Characteristic Grid
Characteristic grid games do not venture very far from ordering and grouping
games. However, the challenge is slightly different. In an ordering or
grouping game, you are arranging a set of variables. You might be arranging
them from tallest to shortest, or separating them on to different basketball
teams. In a characteristic grid game, on the other hand, you aren’t
arranging a set of variables at all. Rather, you are assigning
characteristics to a set of variables. Rather than arranging six people, you
have to determine who is tall and who is short.
A clown will select a costume consisting of two pieces and no
others: a jacket and overalls. One piece of the costume will be
entirely one color, and the other piece will be plaid. Selection is
subject to the following restrictions:
If the jacket is plaid, then there must be exactly three colors in it.
If the overalls are plaid, then there must be exactly two colors in
them.
The jacket and overalls must have exactly one color in common.
Green, red, and violet are the only colors that can be in the
jacket.
Red, violet, and yellow are the only colors that can be in the
overalls.
Here’s an example from the December 2006 LSAT. This one is a classic.
You get to choose a costume for a clown. (We told you that the LSAT was
great training for your future legal career.) As you can see, there’s no
ordering and no grouping here.
Your job is to determine the pattern (solid or plaid) and color combination
for the clown’s jacket and overalls. (This is not a joke. This is a real game.
Stop laughing.)
In characteristic grid games, your setup will often look eerily similar to
ordering or tiered ordering games. In addition, the rules will include a lot of
restrictions on color combinations (two reds can’t be next to each other) and
conditional rules. By this point, those types of principles should be no
problem for you at all.
2. Mapping
Mapping games involve ... drum roll, please... mapping. These games will
make you reminisce about kindergarten. You’ll literally be playing ‘connect
the dots’ all over again.
When you do a mapping game, they will sometimes provide you with a
map. If not, it’s advisable to draw your own map. No need to apprentice
with a cartographer - a simple arrangement of dots will work.
Each nonstop flight offered by Zephyr Airlines departs from one
and arrives at another of five cities: Honolulu, Montreal,
Philadelphia, Toronto, and Vancouver. Any two cities are said to
be connected with each other if Zephyr offers nonstop flights
between them. Each city is connected with at least one other city.
The following conditions govern Zephyr’s nonstop flights:
In the game on the previous page (from October
2003), you have to determine the flight plan for
Zephyr Airlines (almost as bad a name as
Flyhigh). In other words, you have to connect a
bunch of dots to figure out between which cities
they offer flights.
Mapping games have appeared infrequently over the years. Many of the
deductions are similar to ordering and grouping games. For instance, you
might use a conditional rule to ascertain that a city must be connected to one
of two other cities, but not both.
3. Cyclical
Things go in cycles. Bell bottoms are really cool for a time. Then wearing
them subjects you to substantial social humiliation. Then they are all the rage
again. This doesn’t happen often in Logic Games, but it has popped up in the
past. A cyclical game tests your ability to figure out a pattern. This pattern
determines how the variables appear in each cycle. The nice thing about a
pattern is that it doesn’t change.
The population of a small country is organized into five clans—N,
O, P, S, and T. Each year exactly three of the five clans participate
in the annual harvest ceremonies. The rules specifying the order of
participation of the clans in the ceremonies are as follows:
Each clan must participate at least once in any two consecutive
years.
No clan participates for three consecutive years. Participation
takes place in cycles, with each cycle ending when each of the
five clans has participated three times. Only then does a new
cycle begin.
No clan participates more than three times within any cycle.
To the right, you can gaze at a cyclical game from December 1994. In this
one, clans participate in a harvest ceremony. The participation takes place
in cycles. Your task is to (1) figure out the structure of a cycle (seven
days? five years?), and then (2) uncover the pattern of participation in that
cycle.
For instance, in this game, it’s important to note that every clan not
participating in the first year must participate in the second and fifth years.
Easy hint for spotting a
cyclical game: Look for the word
“cycle.” It’s a dead give-away.
Ninja Note:
4. Operation
Nope - this does not refer to the magical board game in which one had to
skillfully remove a patient’s funny bone. Sorry to say it, but LSAT operation
games aren’t quite that exciting.
Most ordering and grouping games challenge you to figure out the
arrangement of players. Who is in spot 6? Who’s the captain of the squash
team? In an operation game, that’s all done. They tell you exactly how the
players are arranged. For example, they will tell you that Hilda is ranked first,
Ivan is ranked second, and Jamal is ranked third. That might sound easy, but
don’t get too excited. An operation game then challenges you to apply
different operations to the original arrangement. So it’s great that Hilda
starts out ranked first, but what could she be ranked after a couple rounds?
That’s the challenge.
Within a tennis league each of five teams occupies one of five
positions, numbered 1 through 5 in order of rank, with number 1 as
the highest position. The teams are initially in the order R, J, S, M,
L, with R in position 1. Teams change positions only when a
lower-positioned team defeats a higher-positioned team. The rules
are as follows:
In this beauty from October 1995, we sign up for a tennis league. As you
can see, the initial ranking is provided. However, the ranking will be
shaken up as the games proceed. (We are pulling for M - she’s got an
amazing backhand.)
Operation games introduce very random situations. It’s helpful to prefigure
the results that could occur after a few rounds have passed. Where could L be
ranked after two rounds of games? Could R make it all the way to number 1?
That wraps up our quick overview of Neither games. Other LSAT companies
and their inferior books will freak you out by stressing the importance of
these games. We choose not to do that. We want you to spend your valuable
time focusing on the material that is most likely to appear on your LSAT.
The chances of your LSAT score being ruined by a
terribly difficult Neither game are about the same as
you getting hit by a tidal wave in the next year. Not
gonna happen.
You can use the appendix in the back of this book to locate some Neither
games if you would like to do a little practice. But make sure to focus on
ordering and grouping.
Up next...
We are taking your game to the next level. In the following chapters, we will
cover two advanced strategies that can be used to really master Logic Games.
The ninja will be happy.
23/PLAYtheNUMBERS
MATH?!?!
We know, we know. You aren’t here to talk about numbers, and you surely
aren’t here to do math. In fact, law school is probably on the horizon because
math (and related physical sciences) never came naturally to you.
Time for some bad news. To really ace the games on the LSAT, you have to
do a little math. But don’t go searching for a new career. We aren’t talking
about calculus, or even geometry. The LSAT sticks to basic arithmetic.
Addition and subtraction. Maybe a little multiplication, but you can always
simplify that to addition if you please.1
WHAT IS IT?
As you have seen over the last several hundred pages, some games fit very
nicely. If you have seven children sitting in seven consecutive chairs, that’s
great. If you have eight politicians separated into two four-person teams,
that’s excellent.
However, games don’t always have such a natural structure. Gotta schedule
seven dates over a span of five days? That’s complicated. Have to put eight
animals into three cages of different sizes? Not so nice. Need to choose five
films to show in class, and there are three different types of films from which
to choose? Sounds more complex.
There is, however, an excellent technique to handle these situations. It’s
called Playing the Numbers. This process will help you handle the numerical
aspect of Logic Games.
Playing the Numbers is a process that can be used to
simplify the numerical part of a game.
Remember the game about
airline stops? Or alphabet soup? How
about the monkeys and pandas? Or
even the roommates and their mailbox?
In each of those games, we informally
touched on this process. However, it’s
time to define exactly when to play the
numbers and how to do it successfully.
BP Minotaur:
1 You know that trick. If you have to multiply four times three, you can just add four
plus four plus four. Genius.
THE THREE STEPS
There are three basic steps involved in this process. Way back when we were
introducing games, we stressed the importance of making games a linear
process. To conquer a game, you want to practice the same steps every time.
The same rule applies to playing the numbers.
Let’s walk through this, one step at a time.
Step 1: Recognize the challenge
You don’t have to think about playing the numbers in every game. Actually,
there are only a few types of games that require this advanced strategy. The
first step is to recognize when you are faced with a game that might require
you to play the numbers.
How do you know? Good news - there are only three types of games in
which you have to think about playing the numbers. Here are the big
three:
1. Underbooked or Overbooked Games
1:1 games are nice. Everything fits nicely - you have one and only one slot
for each variable. But underbooked games involve variables that can occur
more than once. And overbooked ordering games give you slots that can be
assigned more than one player. All of a sudden you are dealing with
questions about how many.
{
How many times can Randy appear? }
How many workshops can be offered on
Wednesday?
Here’s an example of an underbooked ordering game:
A traveling circus makes seven
consecutive stops during a tour. Each stop
is in one of the following cities: Wichita,
Sioux Falls, Boise, and Cheyenne. The
circus visits each city at least once during
the tour. The following conditions must
follow:
In this ordering game, the task is to determine which city the circus heads to
for each of the seven consecutive stops. However, there are only four cities.
So the circus is definitely headed to some cities more than once.
How many times does the circus go to Boise? Does the circus visit Cheyenne
more times than Sioux Falls? What is the maximum number of trips the circus
could make to Wichita? These questions will be answered by playing the
numbers.
2. Players from Different Categories
Some games present you with a bland cast of characters from which to
choose. But other games give you a list of players from different categories.
Rather than just having eight students with odd, alphabetically-ordered
names, you might have eight students: three freshmen, two sophomores, and
three juniors. When this happens, you might face questions about how many
players can be chosen from each category.
{
Can you select three juniors?
}
Could you have more monkeys than
raccoons?
Here’s an example of a grouping game with different categories:
Five food products must be chosen for a
lab test. The products must be chosen from
three dairy products (milk, cheese, and
yogurt), four meats (chicken, lamb, turkey,
and veal), and three desserts (cake, ice
cream, and apple pie). The following
conditions govern the selection:
Since you have to select five of the ten food products, this is an In and Out
grouping game. The food products come from three different categories dairy products, meats, and desserts. It’s key to track how many products
are chosen from each of the three categories.
How many dairy products must be selected? Could more than two desserts be
selected? Which category could have the most products selected? After you
play the numbers, you will uncover the answers to these important questions.
Different categories are
presented most often in grouping
games (In and Out grouping games are
the most common). However, this can
occur in ordering games as well.
BP Minotaur:
3. Unstable Grouping Games
When you know how many players are assigned to a group, that’s
comforting. You get to make a set number of slots and take a deep breath.
But when they don’t define the number of players in each group, that’s scary.
You have to think about how many there could be.
How many diplomats could visit
}
Bangladesh?
What is the minimum number of players on
the gold team?
{
Here’s an example of an unstable grouping game:
There are three different science courses
offered during fourth period at the local
high school: biology, chemistry, and
physics. Eight students—Zander, Yolena,
Xade, Wendy, Vanda, Umeko, Taylor, and
Shiba—must each choose to take one of
the courses, according to the following
conditions:
In this grouping game, eight players are separated into three groups. It’s far
from clear, however, how many students choose each one of the courses. Not
only will it be important to track who takes what course, but questions will
also relate to how many students take each course.
What is the minimum number of students taking chemistry? If three
students take physics, how many students take biology? If you play the
numbers correctly, the answers to these questions will be at your fingertips.
That’s the first step. You have to keep a watch out for games in which
playing the numbers will be helpful. It’s not a guarantee that you will play the
numbers in any of these game types, but it’s a strong possibility.
Step 2: Look for Principles of Distribution
After identifying a game in which it might help to play the numbers, you
need to watch for clues that it’s going to be helpful. Luckily, they will give
you lots of them. You just have to notice these helpful hints.
A hint to play the numbers comes in the form of a principle of distribution.
Most rules in games revolve around who goes where. You have seen a ton
of rules that look like those to the right.
The circus stops in Boise at some point
after any stops that it makes in Cheyenne.
If the circus stops in Sioux Falls second,
then it stops in Wichita sixth.
Chicken and ice cream cannot both be
selected.
If apple pie is not selected, then cheese
must be selected. Xade does not take
physics.
Zander takes chemistry unless Umeko
takes biology.
Most rules in games give restrictions on who goes
where (which slot of which group).
A principle of distribution, however, will sound different. It won’t relate to
the placement of the variables at all. Rather, it will restrict the distribution of
variables. It won’t tell you that Johnny appears on day 4; it will tell you that
no contestant can appear on more than two days. It won’t tell you that Ralph
is assigned to the green squadron; it will tell you that the green squadron
must have at least as many members as the fuschia squadron.
A principle of distribution restricts the numbers in a
game:
(1) how many times a variable can appear,
(2) how many variables can be selected from a
category, or
(3) how many players can be assigned to a group.
Principles of distribution will sound different depending on the type of game
you are attempting. Let’s use our previous games to look at some examples.
In the underbooked ordering game, a circus had to make seven stops in
four different cities during a tour. Clearly, this requires more than one stop
in some of the cities. Here are some principles of distribution that would
govern how many stops could be made in each of the cities:
The circus makes a maximum of three stops in any city.
The circus makes more stops in Boise than Wichita.
In the second game about various food products, five winners had to be
chosen from ten contestants. The challenge is to discern how many
variables could be chosen from each of three different categories. To the
left are some principles of distribution you could see.
Exactly one dairy product is tested. At most two desserts can be
tested. If more than one meat is tested, then only one dessert is
tested.
The third game sends you back to school. Eight students must be assigned
to one of three science classes. But the groups are entirely unstable. You
will likely encounter principles of distribution restricting how many
students take each class.
At least one student takes each course. Exactly twice as many
students take chemistry as physics.
That’s step two. You might have noticed that you haven’t actually done much
up to this point. The first step is just to recognize the type of game, and the
second step is to watch out for principles of distribution. But don’t get lazy step three is when it’s time for action.
Step 3: Do the math
It’s time to work out the numbers. At this point, you have a grasp on the basic
challenge in the game, and you know the restrictions governing the different
distributions. It’s time to crunch the numbers.
When you play the numbers, always distribute the
larger variable set into the smaller variable set (for
example, spread out eight players among the five days).
When you play the numbers, follow these three easy steps to make sure that
you identify all possible distributions.
1. Start with the most extreme distribution (use the
maximum or minimum).
2. Take from the biggest set and give variables to the
smaller sets.
3. When you reach the most equal distribution, you are
done.
Let’s see how this works in real time. Pretend you are attempting this game.
Outlined are the introduction and a couple rules. Here’s a step-by-step
analysis of the correct approach:
Step 1: Recognize the challenge
This is an overbooked ordering game. There are nine players (chimpanzees)
that must be distributed over six slots (acts). This means we might have to
play the numbers.
Over the course of six consecutive acts (numbered first through
sixth), exactly nine chimpanzees—A, B, C, D, E, F, G, H, and J—
must perform. The performances are governed by the following
conditions:
At least one chimpanzee must perform during each act.
No more than three chimpanzees can perform during any act.
Step 2: Look for principles of distribution
Both rules should catch your eye - they are principles of distribution. Rather
than telling you when a monkey does or doesn’t perform, they put restrictions
on how many monkeys can perform during each act.
Step 3: Do the math
First, start with the most extreme distribution. Since there is a maximum
of three chimpanzees, that’s a good way to start. One act could have
three chimpanzees, another act could have two chimpanzees, and the
other four acts would then have just one chimpanzee. Lame.
Next, you want to take one chimp from the biggest group and give it to
one of the smaller ones. You could also have three acts with two chimps
each, and three with just one chimp.
Since the second distribution is the most equal way to distribute the
chimpanzees, you are done.
Sweet.
That’s the process. You now have
isolated the two possible distributions
in this game.
Make sure you understand
the function of playing the numbers: to
tell us the possible distributions. Here,
it tells us the two ways to distribute the
chimpanzees over the six acts.
However, I caution against going too
far. We still don’t know which acts
feature multiple chimpanzees.
BP Minotaur:
Let’s do another example - a grouping game this time.
Step 1: Recognize the challenge
A chef is preparing a dinner that will have three courses—
appetizer, entree, and dessert. The chef will use six different spices:
basil, cayenne pepper, coriander, paprika, sage, and thyme. Each
spice is used in exactly one of the three courses. The following
must obtain:
Each course must include at least one of the spices.
Here, we have a grouping game. There are six players (spices) and three
groups (courses). But the game is totally unstable - it’s not clear how many
spices are in each course. It might be time to play the numbers.
Step 2: Look for principles of distribution
The first rule is a principle of distribution. A simple one, but a principle of
distribution nonetheless. It specifies that each course must include at least
one of the spices.
Step 3: Do the math
First, start with the most extreme distribution.
Since each course must have at least one
spice, the most spices that one dish could
feature is four. The other two dishes would be
left with one lonely spice.
Next, you want to borrow from the biggest
group. You could also have a dish with three
spices, one with two, and the third dish with only one spice.
The third possibility is that all three dishes include two spices. This is
the most equal distribution, so we’re done.
These distributions will help
you in two big ways. First, they will
eliminate wrong answer choices where
the math doesn’t match up. Second,
they will give you a head start on
making hypotheticals for conditional
questions.
Ninja Note:
There’s one more example to cover. This one is slightly different. In the first
two examples, you were given general principles of distribution that applied
to each slot or group. In other games, you will find principles that are specific
to one slot, or to one group. When this happens, you should play the numbers
for the specific slots or groups.
Here’s an example. Note how the process changes slightly.
Step 1: Recognize the challenge
You should recognize this one as an In and Out grouping game - six of ten
sports are selected. But the sports are pulled from different categories. Might
be time to play the numbers for how many can be selected from each.
Six sports will receive funding from the Olympic committee. The
six must be selected from four summer events (basketball,
swimming, track and field, and volleyball), three winter events
(bobsled, curling, and skeleton), and three experimental events
(bungee jumping, kite surfing, and sandboarding). The selection
must accord with the following:
Exactly two winter events are selected.
At least as many summer events as experimental events must be
selected.
Step 2: Look for principles of distribution
Boom. Both rules are principles of distribution. Note: Since the principles of
distribution give restrictions on the specific categories, our approach should
change. Rather than testing the general limits of the game, we can test the
limits for each category.
Step 3: Do the math
The winter events are always set to two, which is nice. The most
extreme distribution would include all four summer events and no
experimental events.
Working towards the most equal distribution,
three summer events and one experimental
event could be selected.
The most equal distribution would be two
events from each category, which is also
possible.
Don’t always assume each
group must have at least one. Unless
otherwise stated, zero is an option.
Ninja Note:
That’s an overview of how to play the numbers, but now you get to practice
on your own.
PLAY THE NUMBERS DRILL
Here we go. This drill is designed to develop your math skills. Well, at least
to the point where it will help you on the LSAT. For each of the following
games, use the setup and rules to determine the possible distributions.
The first few examples will provide you with a template. After that, it’s up to
you.
1. Chuck has to wear pants on each day of a
five-day work week: Monday through
Friday. He has three different types of pants
from which to choose—corduroys, jeans, and
khakis. His wardrobe is constrained by the
following:
Chuck must wear each type of pants at
least once.
2. On a game show, a contestant must select
seven balls from three different boxes. Box 1
contains three balls (A, B, and C), box 2
contains three balls (D, E, and F), and box 3
contains three balls (G, H, and I). The
contestant must abide by the following rules:
3. The seven children in the Cathey family—
J, K, L, M, N, O, and P—are all in school.
Each child is either in high school or college,
but not both. The following is known about
the family:
M is in high school.
There is an odd number of children in
college.
4. At a local sea park, a dolphin show is given every afternoon
from Sunday through Saturday. During one week, there are four
dolphins available to perform— Red, Sippy, Tigger, and Ugo.
Exactly one dolphin performs each day, according to the following:
No dolphin performs more than three times.
Sippy performs exactly twice.
5. A psychologist has five consecutive appointments scheduled for
Tuesday. The psychologist will see seven patients—M, N, O, P, Q,
R, and S—over the course of the appointments. Her schedule is
governed by the following conditions:
Each appointment is scheduled for either one patient or two
patients together.
6. Tim is heading out for a backpacking trip across Europe and
space in his pack is tight. He can only pack five items. His options
include three kitchen items (A, B, and C), three clothing items (D,
E, and F), and three personal items (J, K, and L). In choosing the
items, Tim must follow these guidelines:
Exactly one kitchen item must be packed.
K and L cannot both be packed.
7. Eight new students—M, N, O, P, Q, R, S, and T—will be added
to four different classrooms at the local elementary school: 1, 2, 3,
and 4. Each student will be assigned to one classroom, according to
the following:
At least one student is added to each of the four classrooms.
8. Eight politicians—Bush, Cheney, Gore, Jefferson, Lincoln,
Madison, Obama, and Pelosi—are assigned to three different
committees: agriculture, education, and traffic. Each politician is
assigned to one committee. The following must obtain:
Exactly twice as many politicians are assigned to the education
committee as the traffic committee.
Madison and Obama cannot be assigned to the same committee.
9. A magazine is selling ad space to ten different advertisers: A, B,
C, D, E, F, G, H, I, and J. Each of the advertisers buy some ad
space spread over four consecutive pages of an upcoming issue.
The pages are numbered 31, 32, 33, and 34. The following rules
govern the sale of the ad space:
Each page features either one, two, or four advertisements.
Each advertiser buys only one advertisement.
10. There are five roles available in the upcoming school play.
Nine students are trying out from three different classes. Of the
nine students, three are freshmen (B, C, and D), three are
sophomores (J, K, and L), and three are juniors (W, X, and Y). The
selection of roles must accord with the following conditions:
At least one student is chosen from each class.
If more than one freshman is chosen, then no more than one
sophomore can be chosen.
If X is selected, then Y cannot be selected.
ANSWER KEY
Here’s an overview of how to play the numbers on the preceding games.
Make sure to look over any examples that you missed.
Chuck must spend hours trying to pick out his pants
every morning. Since he must wear each type of pants
at least once, he could go heavy on one type of pants
(likely khakis), wearing them three times and the other two types only once.
Or Chuck could be more egalitarian and wear two types of pants twice and
the third type of pants only once.
This one can be confusing because there are no rules.
But this is a grouping game with different categories,
and it is helpful to play the numbers. Since there are
only three balls in each box, there are only two ways to come up with seven
balls: either pull three balls from two boxes and one from the third, or pull
three balls from one of the boxes and two balls from each of the other two.
This is a busy family. Since there must be an odd number
of children in college, it’s best to start there. The first odd
number is one, so there could be one child in college and
six in high school. The next odd number is three, so there
could be three in college and four in high school. Then
there could be five in college and two in high school. It might seem like all
seven could be in college (seven is, after all, another odd number), but that
doesn’t fly because we need M to be in high school.
The goal here is to calculate in how many shows
each dolphin performs. Sippy is set at two, but
the rest of the dolphins are up in the air. Since no
dolphin can perform more than three times, let’s
start there. One dolphin performs on three days,
another on two days, and the final dolphin has
the week off (it never said that each dolphin has to perform). Or it could be
the case that one dolphin performs on three days and the other two on just
one day each. Finally, it could be true that Sippy and two other dolphins
perform on two days, and the final dolphin only performs once.
Wait a second - there’s only one distribution
here! That will happen once in a while.
Since each patient must be seen and each appointment is scheduled for either
one or two patients, there must be two joint sessions and three solo sessions.
The kitchen items are the easy part - only one is
selected. The other rule tells you that K and L, which
are both personal items, cannot be packed together, so
at most two personal items can be selected. Thus,
there are only two distributions for the remaining four items - either three
clothing items and one personal item, or two of each.
This one has a large number of distributions,
more than you are likely to see in a real game.
But it’s great practice. To split the eight students
into four classrooms, there could be a maximum
of five students in one of the classrooms. To find
the other distributions, simply borrow from the
larger groups. Once you get to the completely
even distribution (two students in each classroom), you can feel comfortable
that you found all five possibilities. And take a deep breath.
Whoa, now it’s time for multiplication. In a situation
like this, the easiest way to proceed is to plug one
politician into the traffic committee. (It’s always
easiest to start with one when multiplication is
involved.) One times two is two, so that means two for the education
committee, and five left over for agriculture. Next, plug two politicians
(hopefully Jefferson and Lincoln) into the traffic committee. Then you would
have four on the education committee (hopefully not Bush) and two on the
agriculture committee.
Or you could just have zero
on the traffic committee.
Ditz McGee:
Actually, no. It might seem like that would work, but zero multiplied by two
would still give you zero on the education committee. That leaves all seven
politicians for the agriculture committee, and you know they aren’t all going
to get along. Since Madison and Obama can’t be on the same committee, this
one doesn’t work.
This is a severely overbooked ordering game.
There are ten players (advertisers) for the four
slots (pages). The principle of distribution is
rather complicated - each page features one, two, or four advertisers. When
you crunch the numbers, there are two possibilities. Either two pages have
four ads and the other two only have one, or one page has four ads and the
other three each have two.
One of the big obstacles in this example is where to
begin. The last rule tells us that X and Y can’t both
be selected, limiting the number of juniors to either
one or two. That’s the best place to start. If one
junior is selected, you aren’t allowed to select two
freshmen and two sophomores (due to the second
rule). You either have to go with all three freshmen
and one sophomore, or all three sophomores and one freshman. If two juniors
are selected, then there must be either two freshmen and one sophomore, or
one freshman and two sophomores.
Hope that drill went well. Playing the numbers can be challenging for
students, but it’s very helpful. When you are doing a game and the time is
ticking away, you have to fight the urge to jump straight into the questions.
They will look tempting - after all, that’s where the points are. But skipping
right to them is a mistake you have to avoid.
During the deduction phase, it’s crucial to play the numbers in certain games.
Here’s a reminder of the types of games that might require this extra step:
1. Underbooked or Overbooked Games
2. Games with Players from Different Categories
3. Unstable Grouping Games.
Make sure to review the proper steps to play the numbers. But now it’s time
to see this powerful technique in action. Next, we will look at a couple games
where success or failure is determined by the ability to properly play the
numbers.
JUNE 2011: GAME 4 (18-23)
Here we go. We kinda gave away the challenge here, but it’s important to
practice the appropriate steps. We have to recognize why this game requires
us to play the numbers and execute the process perfectly. Since this is our
first game with playing the numbers, we’ll walk through it using the
Blueprint Building BlocksTM approach.
Games that require you to
play the numbers tend to be difficult
and thus tend to be found later in the
section (this game was last).
BP Minotaur:
1. Setup
A street entertainer has six boxes stacked one on
top of the other and numbered consecutively 1
through 6, from the lowest box up to the highest.
Each box contains a single ball, and each ball is
one of three colors—green, red, or white.
Onlookers are to guess the color of each ball in
each box, given that the following conditions
hold:
Street
performers and
colored balls?
You might have
fallen for this
one before.
As always, the first task is to identify the type of
game. As soon as you see that there are six consecutively numbered boxes (1
through 6), you know it’s an ordering game.
It would be nice to have six different
balls to fill those six boxes. But life ain’t
always nice. Since there are only three
colors to go into the six boxes, this is an
underbooked ordering game. Clearly,
some colors will be found in more than
one box.
Since we are now facing an underbooked
ordering game, we know that it might be
necessary to play the numbers. You want
to wait until you check the rules, but it
will probably be helpful to do some math and figure out how many balls of
each color can be selected. Two red? Three white? Only one green? We
will soon see.
At this point, there are no
restrictions on the colors. They don’t
tell you that each color must show up,
and there is no maximum. There could,
at this point, be zero green balls, or
there could be six green balls. Keep a
keen eye out for ways to limit these
possibilities.
Ninja Note:
2. Rules
It’s been a while since we worked through an ordering game. We should
expect a variety of the usual suspects (blocks, restrictions, arches). In
addition, we want to watch closely for any hints about how many balls of
each color could be found in the boxes (principle of distribution).
Boom. Here we see the first principle of distribution. It’s a huge hint that
we will need to play the numbers. Don’t jump in and do the math yet - it
might get more restricted in later rules.
This rule is a bit weird right now since we don’t know that there are any
white balls. But there are still two deductions that follow:
Since there must be more red balls than white balls, there must be at
least one red ball.
The maximum number of white balls is two. If there were three white
balls, it wouldn’t be possible to have more red balls than white balls.
It’s important to represent this rule in a manner that is visually consistent
with the setup (vertically). Also, be careful of the wording. There is a green
ball lower in the stack than any red ball. If the lowest red ball is in box 3,
there must be a green ball below that (in box 1 or 2). There’s also one
deduction that you can draw about the colors at this point.
Based on the first rule, there must
be at least one red ball. Also, since
there must be a green ball below
the lowest red ball, there must also
be at least one green ball.
There’s one more deduction at this
stage. To make sure you aren’t
snoozing, you get to find it.
Challenge: Try to make
restrictions based on the first two rules.
When you are making deductions in a game with multiple distributions,
always attempt to limit the possibilities. We don’t know how many green
and red balls are found in the boxes. However, there is a deduction that
follows from the first two rules.
Since there must always be a green ball lower than any red ball, a red
ball can’t be in box 1. Thus, either a green or white ball must be in box
1.
And you can’t put a
green ball up in that top box.
Cleetus Comment:
Be careful. While there must be one green ball lower than any red balls, there
could be more than one green ball. You could, for example, have a green ball
in box 1 (below any red balls) and another green ball up top in box 6.
On to the last rule.
This last rule gives us a nice block. However, it doesn’t lead to any new
restrictions. The block could land in a number of different places, and there
could be additional green or white balls.
According to this rule, there must be
at least one white ball. Now we
know there must be at least one of
each color: green, red, and white.
Here’s a quick look at our setup and
rules. We are now heading into the
deduction phase, which will definitely
involve playing the numbers.
3. Deductions
This game is just screaming at you to play the numbers. “You better play the
numbers, sucka, or I am going to ruin your future!” Hear that?
First, this is an underbooked ordering game, so we had to be on the lookout
for clues to play the numbers. Second, the rules gave partial restrictions on
the number of each color that could be found in the boxes. All of this spells...
playing the numbers.
Play the Numbers
There are six boxes and three colors to choose from. We deduced while
working through the rules that there is at least one ball of each color. Finally,
the first rule restricts us to a maximum of two white balls.
Here’s what you don’t know. How many
green, red, and white balls could be in the
six boxes ? That’s up to you.
Challenge: Play the numbers to
determine the possible
distributions (refer to the above
setup and rules).
There are a number of different ways for this one to work. The white balls are
the best place to start because there must be either one white ball (because of
the block) or two white balls (that’s the maximum because you have to have
more red balls).
If there is only one white ball, there are three
distributions that meet the criteria: (1) two red balls
and three green balls, (2) three red balls and two
green balls, and (3) four red balls and one green ball.
If there are two white balls, there’s only one
distribution: There must be three red balls and only
one green ball.
Even though there are a lot of possible distributions in the game, playing
the numbers will still be very helpful.
Four distributions is on the
high side for a game. You will
frequently only find one or two
possibilities.
BP Minotaur:
If you uncover only two distributions in a game, it’s worth the time to
investigate each of them further. But with four distributions, that would take
too much valuable time. This outline will help you quickly work through
questions. Here’s a few quick examples of further deductions made possible
by playing the numbers:
In the first two distributions, the only white ball must be part of the
block with a green ball.
In the third distribution, all four red balls would have to be in the top
four boxes, and the block with the green ball and white ball would have
to be in box 1 and box 2.
In the fourth distribution, the only green ball would have to be part of
the block with a white ball, and the block would have to be below all of
the red balls.
With a solid setup, some good deductions, and a firm grasp on the
distributions in this game, we should feel confident moving forward. There
aren’t any more helpful deductions to be found.
Next up, we’re going to try something different. To enjoy the rewards of
playing the numbers, you are going to complete the questions on your own.
Make sure to use the deductions and distributions to get through the questions
efficiently.
Challenge: Complete the questions to this game on the
next page.
JUNE 2011: GAME 4 (18-23)
Questions 18-23
A street entertainer has six boxes stacked one on top of the other and
numbered consecutively 1 through 6, from the lowest box up to the highest.
Each box contains a single ball, and each ball is one of three colors—green,
red, or white. Onlookers are to guess the color of each ball in each box, given
that the following conditions hold:
There are more red balls than white balls.
There is a box containing a green ball that is lower in the stack than any
box that contains a red ball.
There is a white ball in a box that is immediately below a box that contains
a green ball.
18. If there are exactly two white balls, then which one of the
following boxes could contain a green ball?
(A)
(B)
(C)
(D)
(E)
box 1
box 3
box 4
box 5
box 6
19. If there are green balls in boxes 5 and 6, then which one of the
following could be true?
(A)
(B)
(C)
(D)
(E)
There are red balls in boxes 1 and 4.
There are red balls in boxes 2 and 4.
There is a white ball in box 1.
There is a white ball in box 2.
There is a white ball in box 3.
20. The ball in which one of the following boxes must be the same
color as at least one of the other balls?
(A)
(B)
(C)
(D)
(E)
box 2
box 3
box 4
box 5
box 6
21. Which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
There is a green ball in a box that is lower than box 4.
There is a green ball in a box that is higher than box 4.
There is a red ball in a box that is lower than box 4.
There is a red ball in a box that is higher than box 4.
There is a white ball in a box that is lower than box 4.
22. If there are red balls in boxes 2 and 3, then which one of the
following could be true?
(A)
(B)
(C)
(D)
(E)
There is a red ball in box 1.
There is a white ball in box 1
There is a green ball in box 4
There is a red ball in box 5
There is a white ball in box 6.
23. If boxes 2, 3, and 4 all contain balls that are the same color as
each other, then which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
Exactly two of the boxes contain a green ball
Exactly three of the boxes contain a green ball.
Exactly three of the boxes contain a red ball.
Exactly one of the boxes contains a white ball.
Exactly two of the boxes contain a white ball.
4. Questions
Hopefully the balls bounced nicely for you. This is a tough game - it was the
last one on the June 2011 LSAT. But the deductions and distributions should
have been very helpful. Let’s take a look.
Question #18 (conditional, could be true)
No elimination question? Lame. They throw you right into the mix here with
a conditional question. The new condition is that there are exactly two white
balls.1
If there are exactly two white balls, the fourth distribution is in play.
There must be one green ball and three red balls.
The one green ball is involved in the block with one of the white balls.
In addition, the green ball must be lower than any of the three red balls.
Once you realize that the block has to be below all
three of the red balls, it’s time to think about where
the block could be placed. Luckily, there are only
two options:
The block could go in box 1 and box 2. If that
happens, the other white ball could be in any of
the other boxes.
The block could also go in box 2 and box 3. The
three red balls have to go in boxes 4, 5, and 6 to be above the single
green ball. The other white ball takes the last box - box 1.
The sole green ball must be in either box 2 or box 3, so (B) is the
correct answer.
1 You have no idea how hard it is to resist making bad jokes about all of the colored
balls in this game (especially the notable lack of blue balls). But we are going to do it.
We swear.
Question #19 (conditional, could be true)
Next up is another conditional question. Playing the numbers is the key yet
again. For this one, there must be green balls in boxes 5 and 6.
If there are green balls in boxes 5 and 6, there
are already at least two green balls. Only the
first two distributions are possible.
There must be a third green ball below any red
balls. The distribution is now clear - three
green balls, two red balls, and one white ball.
There are two possibilities, depending on the
placement of the green and white block.
The green and white block could be placed in
boxes 4 and 5. The two red balls go in boxes 2 and 3, and the final green
ball is below them in box 1. Alternatively, the green and white block
could be placed below the red balls. The red balls are in boxes 3 and 4,
with a green ball in box 2 and a white ball in box1.
A white ball could be in box 1, so (C) is the winner for this one.
Knowing the correct
distribution gave us a quick and easy
way to attack this question.
Ninja Note:
Question #20 (absolute, must be true)
This one is brutal. Awful. Painful. The most challenging part of the question
is just figuring out the correct approach. Most students attack this question in
a very inefficient manner.
The question asks you to identify a box that must have the same color ball as
at least one other box. In other words, you have to find a box that can’t be
unique. What’s the best way to identify that box? You should try to make
each box unique. If it doesn’t work, there’s your answer.
According to our distributions, you could have
just one green ball or just one white ball. So if
a box is going to be unique, it must have
either a green or white ball.
In question #18, there was only one green ball,
and that green ball could be in either box 2 or
box 3. Either of those two boxes could be
unique, so (A) and (B) are out.
Box 4 could also be unique. If box 4 is the
only white ball, then a green ball must be in box 5. However, a mix of
red and green balls could fill the other boxes as long as another green
ball is in box 1.
Box 5 is very similar to box 4. Ifbox5isthe only white ball, then a green
ball must be in box 6. As long as box 1 is also green and there are at
least two red balls, all is fine with this hypothetical as well.
If you try to make the ball in box 6 unique, it just doesn’t work. It
can’t be red because there are at least two red balls. It can’t be
green because you have to have a green ball below any red balls.
And it can’t be white because then you wouldn’t have the block with
a green ball above a white ball. Thus, (E) is the answer.
Question #21 (absolute, must be true)
Yet another absolute question. This one can take a short lifetime to complete
unless you use all of your previous work. Since they are asking for what must
be true, we can use previous hypotheticals to eliminate answer choices.
On question #18, there is only one green ball, and it is in either box 2 or
box 3. Thus, it doesn’t have to be true that there is a green ball higher
than box 4. (B) is out.
Also on question #18, the second hypothetical had three red balls in
boxes 4, 5, and 6. Thus, it doesn’t have to be true that a red ball is lower
than box 4. (C) is gone.
On question #19, red balls are found only in box 3 and box 4. Thus, a
red ball doesn’t have to be higher than box 4, and (D) is out.
On question #20, the second hypothetical has a sole white ball in box 5.
Thus, there need not be a white ball lower than box 4 and (E) is another
loser.
Look at that. Glancing through our old hypotheticals killed the four
incorrect answer choices. (A) is looking good at this point, but let’s prove
it anyway.
There must be at least two red balls. The red balls could be placed
up top in boxes 5 and 6. But there must be a green ball below any
red balls. Thus, there must be a green ball in a box lower than 4,
since there would be three green balls. (A) must be true.
Question #22 (conditional, could be true)
Two more questions to cover. This is another conditional question. Now, red
balls are found in boxes 2 and 3.
There must be a green ball below the lowest red
ball, so a green ball must be in box 1.
The block could go in two spots - either boxes 4
and 5, or boxes 5 and 6.
Since there must be more red balls than white
balls, the remaining box can’t hold a white ball.
It must be either red or green.
In the second hypothetical, a green ball could
be in box 4, so (C) is the champion.
Question #23 (conditional, must be true)
For the last challenge, we see one more conditional question. For this one,
boxes 2, 3, and 4 all contain balls of the same color.
The distributions help you get started again. You
can’t have three white balls, so these three boxes
must be either green or red.
If boxes 2, 3, and 4 are all green, then box 1 must
be white to complete the block. There must be at
least two red balls, so they must be in boxes 5
and 6.
If red balls are in boxes 2, 3, and 4, a green ball
must be below them in box 1. And the block must go up top.
It turns out that all you need to find the correct answer is the
distributions. In both situations, there is only one white ball. (D) is
the final answer.
That wraps up our ball game. Or box game. Or colored ball game. Whatever
you want to call it, that game is behind us.
Hopefully you can agree that it went well. Most notably, it went better than it
would have if we didn’t play the numbers. By outlining the possible
distributions during our deduction phase, we were able to gain a better grasp
on the game and get a head start on a number of questions.
Now it’s time to rewind and try again. However, next up will be an entirely
different type of game. But playing the numbers will again be an important
step in the process.
SEPTEMBER 2006: GAME 2 (6-11)
It’s time for another challenge. Officially, we are going to do this game
together using the Blueprint Building BlocksTM approach. But the guidance is
going to be minimal. Really, we will just check in a few times along the road
to make sure that all is progressing well. Let’s do this.
1. Setup
First up, let’s take a gander at the introduction. What wonders might we have
in store? Parakeets? Camp counselors? Flavors of ice cream?
Challenge: Read through the introduction to this game.
Then, represent the variable sets and build the
appropriate setup.
The members of a five-person committee
will be selected from among three parents
—F, G, and H—three students—K, L, and
M—and four teachers—U, W, X, and Z.
The selection of committee members will
meet the following conditions:
Here are the key things to notice at this point:
(1) This is an In and Out grouping game. You are selecting a five-person
committee. There are ten different people from which to choose. Some of
them will make the committee, others will not. Thus, your setup should
have an In group with five slots and an Out group with five slots.
(2) The players in this game come from different categories. Rather than ten
random people, there are parents, students, and teachers.
How many teachers could be
selected? Could you select two parents
and only one student? Once we get
some principles of distribution, we will
be able to answer these burning
questions.
Ninja Note:
2. Rules
Next, it’s time to check out the rules. You should expect some of our good
old grouping relationships (stalker, love, hate, baby) as well as restrictions
about how many players can be selected from each category (principles of
distribution).
Note: Watch for principles of distribution or any other
rules that restrict how many players can be selected
from each category.
Challenge: Represent the rules in this game. Combine
rules if possible.
The committee must include exactly one
student. F and H cannot both be selected.
M and Z cannot both be selected.
U and W cannot both be selected.
F cannot be selected unless Z is also
selected.
W cannot be selected unless H is also
selected.
Most of the rules in this game should be easy to visualize. These are grouping
rules you have seen a number of times before.
The first rule is a straight principle of distribution. It tells you that
exactly one student is selected. Huge.
The second rule gives you a hate relationship between F and H. They
cannot both be selected. It’s also important to note that F and H are both
parents. Since there are only three parents, this restricts the committee to
a maximum of two parents. This is another principle to be used when
you play the numbers.
The third rule gives you more hate - M and Z just don’t get along.
They’re from different categories, however, so this doesn’t limit our
distributions.
Even more hate in the fourth rule - U and W cannot both be selected.
Both U and W are teachers, so the selection can now include a
maximum of three teachers.
The fifth rule can be tricky to diagram. Remember to replace “unless”
with “if not” to make it easier to diagram. If Z is not selected, then F is
not selected. And if F is selected, Z must be selected.
The sixth rule follows the same formula as the last one. If H is not
selected, then W is not selected. If W is selected, then H is selected as
well.
3. Deductions
Now it’s time for the big show. It’s time to slap this game around a bit. There
are some deductions based on combining the rules, but let’s jump straight to
the big stuff. Here are the clues telling us to play the numbers:
(1) We have players in a game that come from different categories.
(2) The rules gave us principles of distribution that restrict how many
players can be selected from the different categories.
Play the Numbers
The next step is to determine what combinations could make up this fiveperson committee. It’s your turn again. On the left side are the restrictions
governing this game. On the right, you should do the math and find every
possible distribution.
Challenge: Play the numbers to determine how many
parents, students, and teachers could be selected to
form the committee.
The committee has five members.
Exactly one student is selected.
At most two parents can be selected
(because F and H cannot both be
selected).
At most three teachers can be
selected (because U and W cannot
both be selected).
Since you are selecting five committee members from ten different options, it
might seem like there are a ton of options. But it’s actually pretty restricted.
Since there is a strict limit on the parents
(maximum of two), that’s the best place to start.
Zero parents won’t work because then all four
teachers would have to be selected. But U and
W can’t both be selected. No dice. So either one
or two parents must be selected.
If one parent is selected, then three teachers must be selected.
If two parents are selected, then two teachers must be selected.
Having those two distributions at our disposal will be very helpful. Now that
we’re done playing the numbers, here are a couple additional deductions that
you should have spotted while diagramming the rules:
If F is selected, then Z must be selected. If Z is
selected, then M cannot be selected. Thus, if F is
selected, then M cannot be selected. And if M is
selected, then F can’t be selected.
If W is selected, then H is selected. But if H is
selected, F can’t be selected. Thus, W and F
cannot both be selected.
We’ve got a lot of good work under our belts. Let’s take a look at our current
progress. Below you can see the setup, rules, and first round of deductions.
But we aren’t done yet. We are really going to push you to make deductions
in this game. You have the rules down. You played the numbers and found
two great distributions. But do you think all of this stuff can play
together? Might the rules give you deductions based on which distribution is
in play? If three teachers are selected, do you think we might be able to
figure out who those teachers are? It’s your turn to find out.
Challenge: Use the rules to make deductions in each of
the two distributions. Take your time and work through
each one slowly.
There are a couple quick deductions in each scenario:
In the first distribution, you should start with the teachers because three of
the four are selected. Also, remember that U and W can’t both be selected,
though one of them must be (since only one teacher is left out). X and Z
must be selected to fill the other slots.
Since Z is selected, M can’t be selected. The one student must be either K
or L.
In the second distribution, the parents should catch your eye since we are
selecting two out of three. F and H can’t both be selected, though one of
them must be (since only one parent is left out). The two parents must be
G and either F or H.
Two teachers must be selected. U and W can’t both be selected. Thus,
either X or Z (or both) must be selected.
Those are some impressive deductions. The game started off as a monster,
but the questions will be easy with these deductions in hand. Here’s a quick
recap of the steps we took:
1. We classified this game as an In and Out grouping
game. Importantly, the players came from different
categories. Rather than a set of ten generic people, you
were given three parents, three students, and four
teachers. This is a hint that you might have to play the
numbers.
2. There were a number of rules restricting the number
of players that could be selected from each category.
The first rule limited us to selecting exactly one
student. In addition, other rules limited the committee
to two parents and three teachers.
3. After the rules were diagrammed, we didn’t run
straight into the questions. In a much better maneuver,
you played the numbers and found that only two
distributions are possible. That’s nice to know.
4. We took an extra moment and visualized the two
possibilities. Applying the rules to each one led to even
more deductions.
It’s a beautiful process when it all comes together. Each one of these steps is
necessary to conquer a difficult game on the LSAT.
This game is a great lesson
in time management. All of this work
will take you up to four or five
minutes. And you haven’t even
glanced at a question yet. But this is
time well spent. You will be fast and
accurate when you attack the
questions.
BP Minotaur:
You’ve done all of the hard work up to this point. It’s only fair that you also
get to experience the payoff. On the next page, you can see the setup, rules,
and deductions. Take a few moments and work through the questions. Try to
get through each one efficiently, and don’t waste any time or energy.
Questions 6-11
The members of a five-person committee will be selected from among three
parents—F, G, and H—three students—K, L, and M—and four teachers—U,
W, X, and Z. The selection of committee members will meet the following
conditions:
The committee must include exactly one student. F and H cannot both be
selected.
M and Z cannot both be selected.
U and W cannot both be selected.
F cannot be selected unless Z is also selected. W cannot be selected unless
H is also selected.
6. Which one of the following is an acceptable selection of committee
members?
(A)
(B)
(C)
(D)
F, G, K, L, Z
F, G, K, U, X
G, K, W, X, Z
H, K, U, W, X
(E) H, L, W, X, Z
7. If W and Z are selected, which one of the following is a pair of
people who could also be selected?
(A)
(B)
(C)
(D)
(E)
U and X
K and L
G and M
G and K
F and G
8. Which one of the following is a pair of people who CANNOT both
be selected?
(A) F and G
(B) F and M
(C) G and K
(D) H and L
(E) M and U
9. If W is selected, then any one of the following could also be
selected EXCEPT:
(A)
(B)
(C)
(D)
(E)
F
G
L
M
Z
10. If the committee is to include exactly one parent, which one of the
following is a person who must also be selected?
(A)
(B)
(C)
(D)
(E)
K
L
M
U
X
11. If M is selected, then the committee must also include both
(A)
(B)
(C)
(D)
(E)
F and G
G and H
H and K
K and U
U and X
4. Questions
There’s a good mix of questions in this game. Some of them are very quick
when armed with the deductions, but others require a bit more work.
Question #6 (elimination, could be true)
The first one is a straightforward elimination question.
The first rule states that exactly one student is selected. In (A), both K
and L are selected, so (A) is gone.
The second rule (F and H cannot both be selected) doesn’t eliminate
anything.
The third rule (M and Z cannot both be selected) also doesn’t help.
U and W cannot both be selected. This fourth rule kills (D).
The fifth rule (if F is selected, Z must be selected) is violated by answer
choice (B).
The final rule (if W is selected, H must be selected) eliminates (C).
After a short scuffle, (E) is the champ.
Question #7 (conditional, could be true)
W and Z are both teachers. If they are both selected, it’s not clear which of
the two distributions is in play. However, the rules and our distributions
quickly kick out a couple answer choices on this one. Once that happens,
process of elimination is the most effective approach.
If Z is selected, M cannot be selected. (C) is out.
If W is selected, neither U nor F can be selected. (A) and (E) are out.
Both K and L are students. Only one student is selected, so (B) also
doesn’t work.
The only answer left standing is (D).
Question #8 (absolute, must be false)
This is a quick one. It comes straight from an earlier
deduction.
If F is selected, then Z must be selected. If Z is selected, then M cannot
be selected.
F and M cannot both be selected, so (B) is the winner.
Question #9 (absolute, must be false)
This is another quick one. Again, an earlier deduction supplies us with the
correct answer.
If W is selected, then H must be selected. If H is selected, then F
cannot be selected.
W and F cannot both be selected, so (A) is the correct answer.
Question #10 (conditional, must be true)
For this one, all of the deductions are necessary. The new condition is that
one parent is on the committee. Luckily, we already worked through this
hypothetical.
If one parent is selected, then one student
and three teachers are also selected (first
distribution).
Since three teachers are selected, but U and W cannot both be selected,
both X and Z must be selected with either U or W.
X must be selected, so (E) is correct.
Question #11 (conditional, must be true)
The last question requires a bit more work. For this one, M is selected.
M can only be selected in the second
distribution (two parents, one student, and
two teachers).
Since M is selected, both Z and F are out.
G and H must be the two parents selected.
Since Z is out, X must be selected with either U or W.
G and H are both selected, so (B) is the final answer.
And that’s it. That’s all the math you’ll have to deal with on the LSAT. At
the beginning of this chapter, we told you that playing the numbers wouldn’t
be that painful, but it would be very helpful. Hopefully, you are now
convinced of both points.
Make sure to closely review the beginning of this chapter. You should
memorize (1) the types of games that commonly require you to play the
numbers and (2) the correct way to complete the process. Up next, we have
one more advanced technique.
24/SCENARIOS
THE BIG TIME
You know that big movie scene when the master finally entrusts the student
with the key to victory and then sends him off into battle?1 It’s a big moment.
Well, we have arrived at that crossroads. It took 23 chapters, lots of games,
and a smattering of bad jokes to get here, but now you are ready.
We saved the best for last. Don’t fade on us now - this is the most
important chapter in the whole book.
The basic routine for attacking games should be second nature by now.
That’s a good thing - repetition and a good game plan are crucial to your
success. But now we are going to introduce a powerful new technique. Right
in the middle of some games, we are going to ask you to ditch the normal
strategy, make a sharp left turn, and head in a totally different direction. We
call this process making “scenarios” in a game.
Scenarios will enable you to work through a game faster and more accurately
than you ever thought possible. Why wait until now? It’s vital that you get
lots of practice with the normal steps before we introduce this advanced
technique. You have to learn to walk before you can run. But now, as Mufasa
once famously told Simba, “It is time.”
The true difference between a
novice and a ninja is found in this
chapter - it’s all about scenarios.
Ninja Note:
WHAT ARE THEY?
Logic Games are all about constraint. When a game is first introduced, there
are thousands of possibilities. Each player can be placed in any slot or on any
team. Shauna could be selected along with Peter, or Robert, or Taylor, or all
of them (that hussy). The freedom is exhilarating, but it’s frightening at the
same time. And then the constraint starts to creep in.
The rules limit these possibilities. All of a sudden, Mr. Bantam can’t be
scheduled for Tuesday. And he can’t be scheduled for Thursday. Then the
deductions phase begins and the constraint really starts to pile up. Since
Shauna can’t be with Taylor and she never hangs out with Robert, she must
choose between Umlad and Veronica. This dialogue should sound familiar it is the basis for our approach.
You have come to appreciate and look for this type of constraint. Important
deductions are found when certain players are constrained to only a few slots.
When you discover that the alligator must be in cage 4 or cage 6, you’ve just
deduced some very valuable information.
Since the beginning of this
journey, you’ve been trained to spot
the areas of constraint in a game.
Scenarios allow you to use those skills
in a new and powerful way.
BP Minotaur:
Some games are more constrained than others. Some of the games we have
attempted were heavy on constraints and deductions. Other games were more
flexible and required a lot of work to answer the questions (generally in the
form of hypotheticals).
Certain constraints limit a game so much that there are only a few ways
for it to work. That’s when scenarios come into play.
When a game is limited to only a small number of
possibilities, scenarios should be created for each
possibility before attacking the questions.
You should have a lot of questions swirling around your head right now.
When do you make scenarios? How do you make scenarios? What am I going
to have for dinner? Try to forget about dinner, because you need to focus on
the task at hand. We could spend the next ten pages outlining the exact steps
involved in making scenarios and how to determine when to use this
technique. But let’s save that for later - we have a better idea.
This is a big moment, and too much talk can ruin a big moment (think of the
lowlights of your dating life). Scenarios are best appreciated when you see
them in action. So we are going to jump straight into a game. You are mostly
going to be a spectator for this one, so sit back and relax. The following game
will illustrate the process of making scenarios and the incredible benefits that
follow. After you are sold on the technique, we will give you the complete
guide on when and how to do it on your own. Turn the page when you’re
ready.
It’s time for a chess tournament...
1 We always picture Karate Kid at this point. But that might just be our obsession
with Mr. Miyagi. Feel free to substitute your own bad action flick.
DECEMBER 2004: GAME 2 (7-12)
Make sure to follow along closely. This game will be used to illustrate
scenarios, our powerful new technique. The most important points are why
we make scenarios in this game and how to execute them properly. Scenarios
will be found in a variety of game types, but the process of creating scenarios
is similar.
This is a classic game about a hotly-contested chess tournament. Hold on to
your seats.
1. Setup
Exactly six people—Lulu, Nam, Ofelia, Pachai,
Santiago, and Tyrone—are the only contestants
in a chess tournament. The tournament consists
of four games, played one after the other. Exactly
two people play in each game, and each person
plays in at least one game. The following
conditions must apply:
We hear that
Ofelia is a beast
with her bishop.
At this early stage, our priorities are the same: identify the variable sets and
determine the best setup for the game. There are six contestants with a
diverse set of names.1
There are four separate chess games. The fact that the games occur “one
after the other” is our clue that we have an ordering game.
It’s important to do the math. There are eight spots in the four games.
Since there are only six people, this is an underbooked ordering game.
Each of the six contestants must play in at least one game. What to do?
Play the numbers, of course.
In this game, it’s helpful to
play the numbers before you even hit
the rules.
Ninja Note:
Playing the numbers was outlined in the last
chapter. This is a pretty simple, yet still helpful,
version. To fill the eight spots, either two
contestants will have to play in two games each, or one contestant will
have to go crazy and play in three of the games.
To build the proper setup,
visualize the four games
with two slots each.
Some students have an urge
to build a vertical setup, but
try to stay consistent with
earlier games. It’s much
easier to visualize the
ordering elements of the
game (earlier versus later) with a horizontal setup.
In this game, there is no difference between the
top slot and the bottom slot. The two slots simply imply
BP Minotaur:
that the two players battle each other in
that game.
2. Rules
There’s been nothing new so far. We have an effective setup and two
distributions. It’s time to approach the rules. Here’s what you can expect:
1. Ordering restrictions on the players.
2. Rules restricting who can play whom.
3. Hints about which players can play in more than one game.
Let’s take a look.
The first rule should be used to form restrictions for Tyrone on the first and
third games. This implies that Tyrone must play in the second or fourth game
(or both).
Let’s move on to rule #2.
Save the best for last - Lulu plays in the last game. Since some players will
compete in more than one game, Lulu isn’t necessarily finished - she could
play in other games.
Things are progressing nicely. The third rule is up next.
There goes one of our dream matchups. This rule gives you two bits of
information, so it’s important to diagram both. First, Nam plays in only one
game. That’s the first person we know can’t play multiple games. Second,
Nam does not play against Pachai. Keep those two away from each other.
We have already arrived at the final rule.
Whoa, this is a big one. First, Santiago plays in two games. Second, since it
says the “only” game that Ofelia plays in, Ofelia plays in just one game.
Third, we get a sweet SOS block.2
Nice little deduction about our distributions:
With the last rule, you should note that the first
distribution is eliminated. Since Santiago plays in
exactly two games, you know that one player other
than Santiago will also play in two games. The
other four people only play in one game.
I don’t get it. Where’s this
new scenario stuff? This all seems the
same to me.
Ditz McGee:
Patience, Ms. McGee - it’s a virtue. To be fair, she is right. Everything we’ve
done so far should be very familiar to you. We built a nice setup, worked
through the rules, and even narrowed down the distribution. And now the
twist is coming. In general, you shouldn’t start thinking about scenarios
until the deductions phase of a game. Guess what’s next?
3. Deductions
At this point in the process, we would normally start to gather deductions.
Some of the rules probably can be combined. Maybe we can make some
restrictions. Shoot, we might even be able to fill in a spot or two. But we’ve
got bigger plans for this game.
In this game, we are going to forget about the normal deductions. Instead, we
are going to apply a new strategy. It’s time for scenarios. It’s a different
approach, and it is going to destroy this game. Quickly.
The first topic to discuss is why we
are making scenarios in this game.
Remember, we are looking for
constraint. It all boils down to the
SOS block. That is a big block. It
covers three of the four games. Take
a look and think through all of the
possible landing spots for the block.
As you can see, this aggressively-sized block places a
huge constraint on the game. That’s why we make
scenarios.
There are only two places the block can be placed:
(1) The SOS block can go in slots 1, 2, and 3; or
(2) The block could slide over into slots 2, 3, and 4.
That’s all! There are only two possible placements for the biggest rule in this
game. Once you have that realization, here is a rough approximation of the
internal dialogue that should follow (we know you sound smarter than this, of
course):
Wow! The block can only go in two places. Those
questions look mighty tempting - maybe I should just
jump into the questions and forget about it. Oh crap, that
Blueprint book I read told me not to do that. I liked that
book. Okay, fine - I’ll wait. You know, if the block goes
up front, there might be some more deductions. And if it
moves over there, the rules are going to fill in a bunch of
slots. I bet it would be really nice to figure all of that stuff
out before the questions. Scenarios, here I come. Thanks,
Blueprint!
At this point, we are going to jump into the process of building scenarios.
Note the characteristics of this game that brought us to this point. First, we
noticed a big form of constraint (the large block) that severely limited the
possible scenarios in the game. Second, we have a clear idea of how many
scenarios will need to be created. Third, it looks like there will be additional
deductions to be made in each scenario.
Scenarios can result from one
rule, a combination of rules, or one big
deduction. From this point forward,
you should always watch for
opportunities to build scenarios.
Ninja Note:
Rather than keeping track of the big deduction that we made about the SOS
block, we are going to run with it. Once you have spotted the two big
scenarios, it’s time to work through each one. This can be frightening for
students because all of this work will be done before you ever glance at a
question. But, if done correctly, all of the work will pay off big time.
Check it out.
Scenario #1
In the first scenario, the SOS block is placed in slots 1, 2, and 3. Now, it’s
time to search for additional deductions.
Make sure to proceed in an organized fashion.
Let’s work through a checklist of the remaining
rules.
Since Tyrone can’t play in the first or third
games, he must play in the second or fourth
games (or both). Visualize this with arches.
Nam must play exactly one game, and Pachai
must play in at least one of the games. They also
can’t play each other, but that’s not an issue with
the remaining open slots.
There are still some open slots, but this gives us a much better idea of what
follows if the SOS block is placed in slots, 1, 2, and 3. In addition, juggling
the remaining rules should be relatively simple.
When you do scenarios,
don’t expect to fill out every slot. The
goal is to identify all of the deductions
that are possible in each scenario. If
you do that, victory is yours.
BP Minotaur:
Now it’s time for the second scenario.
Scenario #2
The only other possibility for our large SOS block is
slots 2, 3, and 4. This is the basis for our second
scenario.
When you apply the rules to this one, the deductions
are amazing. You can determine who plays in almost every game.
Lulu and Santiago square off in the final game.
Since Tyrone can’t play in the fourth game, he
must take on Santiago in the second game.
Nam and Pachai each have to play in a game.
Since they can’t play each other, one must play in
the first game and the other must play in the third
game. Linked options take care of this deduction.
The other spot in the first game can’t be Nam or
Pachai (they don’t play each other and it would be odd to play yourself).
Tyrone, Santiago, and Ofelia are also out, so Lulu must fill the last spot.
And that’s a wrap. We’ve covered the only two possible
scenarios in this game. As you can see, the first scenario
has a little wiggle room, but the second one is very close
to complete.
Before we move on, make sure you understand the
importance of these two scenarios:
These two scenarios are not two of the ways the game
can work. These are the only two ways the game can
work. When we hit the questions, this is it. They can’t
ask about any other possibilities because there are no
other possibilities.
When you run into an absolute question, you check the two scenarios. When
they throw a conditional question at you, quickly use the two scenarios.
You can probably imagine the payoff that results from making great
scenarios. But you have to experience it to really believe it. Scenarios take a
lot of work, but the questions will feel like a walk in the park.
Once you have made effective scenarios in a game, you
can generally run through all of the questions in no
more than a minute or two.
Don’t believe it? Let’s put our scenarios to the test.
1 Can you imagine some of the classic battles here? Lulu versus Tyrone. Nam versus
Pachai. This would easily be the most entertaining chess tournament ever.
2 This is a joke, courtesy of the LSAT. Since they think you will be crying for help in
a game like this, they create an SOS block. Not very funny, LSAT.
4. Questions
When you do scenarios, the time investment in the deductions phase is great
(as much as five or six minutes). But the ultimate goal is to cut down the time
needed to answer the questions. Not only will you be able to answer the
questions more quickly, scenarios should also help with your accuracy. Let’s
run through the questions and see this in action.
It’s important to use the
scenarios properly when you attack the
questions. For absolute questions, you
have to skim all scenarios. For
conditional questions, you have to
determine which scenarios are in play
and work from there.
Ninja Note:
Question #7
7. Which one of the following could be an accurate list of the
contestants who play in each of the four games?
(A) first game: Pachai, Santiago; second game: Ofelia, Tyrone;
third game: Pachai, Santiago; fourth game: Lulu, Nam
(B) first game: Lulu, Nam; second game: Pachai, Santiago; third
game: Ofelia, Tyrone; fourth game: Lulu, Santiago
(C) first game: Pachai, Santiago; second game: Lulu, Tyrone;
third game: Nam, Ofelia; fourth game: Lulu, Nam
(D) first game: Nam, Santiago; second game: Nam, Ofelia; third
game: Pachai, Santiago; fourth game: Lulu, Tyrone
(E) first game: Lulu, Nam; second game: Santiago, Tyrone; third
game: Lulu, Ofelia; fourth game: Pachai, Santiago
This is an elimination question, which isn’t a very exciting way to start. No
matter how great your scenarios are, it’s still best to use our good old
elimination technique. Here’s a quick overview:
Tyrone doesn’t play in the first or third game. (B) is gone.
Lulu must play in the last game. (E) breaks that rule.
Nam plays in only one game, which kills both (C) and (D).
We didn’t get to use our scenarios, but we arrive quickly at the correct
answer. It’s (A).
Question #8
8. Which one of the following contestants
could play in two consecutive games?
(A)
(B)
(C)
(D)
(E)
Lulu
Nam
Ofelia
Santiago
Tyrone
Now we get to use the scenarios. This absolute question is looking for a
contestant who could play in two consecutive games.
In the second scenario, none of the contestants play in two consecutive
games.
In the first scenario, Tyrone, Santiago, Ofelia, and Nam (who only plays
one game) are out. But either Lulu or Pachai could play two consecutive
games.
Lulu shows up in the answers. (A) looks good.
Question #9
9. If Tyrone plays in the fourth game, then
which one of the following could be true?
(A)
(B)
(C)
(D)
(E)
Nam plays in the second game.
Ofelia plays in the third game.
Santiago plays in the second game.
Nam plays a game against Lulu.
Pachai plays a game against Lulu.
Next up is the first conditional question. The scenarios quickly take care of
this one.
Tyrone can only play in the fourth game in the first scenario.
(B), (C), (D), and (E) all must be false in the first scenario. But Nam
could play in the second game. So (A) is the answer.
Question #10
10. Which one of the following could be true?
(A) Pachai plays against Lulu in the first
game.
(B) Pachai plays against Nam in the
second game.
(C) Santiago plays against Ofelia in the
second game.
(D) Pachai plays against Lulu in the third
game.
(E) Nam plays against Santiago in the
fourth game.
Here is a big absolute question. You are searching for anything that could be
true. This question would be torture without scenarios. With scenarios, it’s
simple.
(B), (C), (D), and (E) are impossible in both scenarios.
(A) says that Pachai plays against Lulu in the first game. In the
second scenario, that could be true. Lulu could play against either
Nam or Pachai.
Question #11
11. Which one of the following is a complete
and accurate list of the contestants who
CANNOT play against Tyrone in any
game?
(A)
(B)
(C)
(D)
(E)
Lulu, Pachai
Nam, Ofelia
Nam, Pachai
Nam, Santiago
Ofelia, Pachai
Here’s another absolute question. Gotta search both scenarios again.
In the first scenario, Tyrone could play against Ofelia or Lulu.
In the second scenario, Tyrone plays Santiago.
Tyrone can only play Lulu, Ofelia, or Santiago. So he can’t ever
play Nam or Pachai. That leads us to (C).
Question #12
12. If Ofelia plays in the third game, which
one of the following must be true?
(A)
(B)
(C)
(D)
(E)
Lulu plays in the third game.
Nam plays in the third game.
Pachai plays in the first game.
Pachai plays in the third game.
Tyrone plays in the second game.
The scenarios make quick work of the last question.
Ofelia doesn’t play in the third game in the first scenario, but she does
in the second scenario.
In the second scenario, Tyrone plays in the second game. So (E)
must be true.
What did we say? Wasn’t that fun?
That concludes your first look at scenarios. Life would be grand if every
game could go so smoothly. That game has a complex setup and some
complicated rules. But we destroyed it by working through two big scenarios
before attempting the questions.
If you weren’t keeping track
at home, the questions took us about 53
seconds to complete.
BP Minotaur:
You should now be convinced that making scenarios is an amazing process
that can drastically improve your performance in Logic Games, so it’s time to
talk details. The next step is to learn why, when, and how to make scenarios
on your own. Here’s an outline of what’s next:
1. Why are scenarios so important?
2. How often is it possible to create scenarios?
3. What types of games have scenarios?
4. When should you make scenarios?
4. How do you make scenarios?
WHY?
Hopefully, some of the benefits of making scenarios are already apparent
from the chess game. We easily took down Lulu and Tyrone by forming two
scenarios. The basic benefits are twofold.
First, the time you spend on a game will be drastically reduced. A tough
game can easily take you 12 minutes or more. But with scenarios, that
same game might only take six or seven minutes. Second, your accuracy
will be greatly improved. When you have to work through lots of
hypotheticals, it’s easy to make mistakes. But when you do all of the work
beforehand with scenarios, you will get those same questions correct.
What you might not have realized is that the benefits go much farther.
Making effective scenarios for a game doesn’t just change your performance
on that game - it can change your performance on the entire section. By
saving time and energy on one game, you will have more of both to devote to
the other games in the section.
Let’s use a quick example. Meet Joan. Joan is a very nice young lady who
has been studying her butt off for the LSAT. At first, she hated games, but
she has improved dramatically. Joan can now finish most games with decent
accuracy. She is always rushed a the end of the section, and she normally has
to skip a few questions or a whole game. Here are two possible results:
Games Section #1: Joan completes the first game in a respectable eight
minutes. She then tackles the second game. It’s a tough tiered ordering
game, and she gets stuck for a few minutes. Eventually, Joan works her
way through it in 12 minutes. Joan starts to freak out because she is
running a bit behind. The third game takes her another 11 minutes. She
drops a few questions because she is rushing. Joan only has four minutes
left for the final game, so she randomly guesses on most of the questions.3
Game 1
Game 2
Game 3
Game 4
8 minutes
12 minutes
11 minutes
4 min
Games Section #2: Joan still completes the first game in eight minutes.
Good work, Joan. When she attacks the second game, she notices a
deduction that leads to three scenarios. She works through the scenarios
and they uncover more helpful deductions. She flies through the questions
and finishes in six minutes. Joan is thrilled. The third game is tough and
still takes her 11 minutes. But she is cool, calm, and collected - she still has
10 minutes to leisurely finish the final game.4
Game 1
Game 2
Game 3
Game 4
8 minutes
6 minutes
11 minutes
10 minutes
Obviously, situation #2 is going to be preferable for Joan. You get the point scenarios are incredibly important. Creating scenarios on just one game can
dramatically improve your performance in the games section, which will
improve your LSAT score (and your life).
HOW OFTEN?
Scenarios occur a lot - more often than you would think.
There’s no way they let you
do that very much. It’s like cheating.
Ditz McGee:
That’s what many students assume when they first see the power of
scenarios. But they are mistaken. Let’s cover a few points about the
frequency of scenarios.
1. It’s almost guaranteed
Here are some stats on the prevalence of scenarios. You might be surprised at
how often you can break one of these challenging little logical puzzles into
just a few scenarios.
Note: Statistics are based on all games from released LSATs in the modern
era: PT 1 (June 1991) through PT 68 (December 2012).
42% of all games can be broken into helpful scenarios.
Wow! More than 2 out of 5 games in the history of the LSAT can be broken
into scenarios. Every time you approach a new game, there’s nearly a
50% chance that you should break the game into scenarios. That’s
wonderful - you have many chances to use this powerful technique. Of
course, this also puts a lot of pressure on you to identify when and how to
create scenarios.
Here’s the stats on how often scenarios have appeared on each Logic Games
section since 1991:
If you were in Vegas,
you would never see odds
this amazing.
Games
with
Scenarios
Tests
Percentage
0
10
14.7%
1
20
29.4%
2
24
35.3%
3
9
13.2%
4
5
7.4%
}
If you were in Vegas,
you would never see
odds this amazing.
Those are impressive stats. 85% of games sections include at least one game
that should be broken into scenarios. 56% of sections have given students two
or more opportunities to make scenarios. That’s a lot of games and even more
scenarios.
When you take the LSAT,
ask not whether you will get a game
with scenarios. Ask how many.
BP Minotaur:
2. It’s getting more common
Many other LSAT prep courses and materials don’t make a big deal out of
scenarios. That’s probably because they’re old. Scenarios were much less
prevalent back when Clinton was doing dirty deeds in the Oval Office. These
days, however, scenarios are all over the place.
Here are the stats:
Years
Total Games
Scenarios
Percentage
1991 - 1997
96
28
29%
1998 - 2002
60
21
35%
2003 - 2012
104
66
57%
That’s crazy. The prevalence of scenarios has nearly doubled over the years.
Since 2003, over half of the games on the LSAT have broken down nicely
into scenarios. Here are some additional stats about recent exams:
The September 2006 LSAT, the June 2009, and the June 2012 LSAT
each featured four games that could all be broken into scenarios.
Since 2007, the average number of games with scenarios per section is
2.41.
The last LSAT that didn’t have any games with helpful scenarios was
October 2001.5
It’s not clear why scenarios are becoming more common, but the numbers
don’t lie.
2. It’s not really new
You might be wondering, “If scenarios are so helpful and so common, how
have we made it through 400 pages of games without a mention?” The
answer is that we haven’t.
There have been a number of games covered in this book that involve
scenarios. However, we dealt with them informally or didn’t complete the
process. This book is designed to turn you into a games ninja. That doesn’t
happen overnight. It’s important to learn and practice the basics before
introducing the advanced strategies. So we tiptoed our way around the
issue. Our sincerest apologies, but it’s good for you.
After you complete this chapter, a great exercise is going back to some of the
earlier games in this book and completing them again using scenarios.
Here’s a list of the earlier games in this book that can be broken down using
scenarios:
You have probably gotten the point of this section by now. To put it bluntly,
you are very likely to have at least one game with scenarios, and your
performance on the whole section depends on your execution. But no stress.
We are here to help.
WHAT TYPES?
All of them. Scenarios are not limited to just one or two different types of
games. They are found across the board, from ordering to grouping to combo
and even neither games. So you always have to be on the lookout for
opportunities to make scenarios.
As we move forward, you will see that the theme of scenarios is the same.
Some form of constraint is placed on a game that limits the possibilities and
allows you to make scenarios. This constraint can be present in any type of
game.
The more detailed answer to this question is a little different. While you
should always watch for scenarios, certain types of games lend themselves to
scenarios more frequently than others. Take a look at the numbers on the next
page.
Game Type
Scenarios
Basic Ordering
0%
1:1 Ordering
53%
Underbooked Ordering
47%
Overbooked Ordering
48%
Tiered Ordering
57%
In and Out Grouping
39%
Stable Grouping
52%
Unstable Grouping
54%
Profiling
0%
Combo
40%
Characteristic Grid
24%
Weird Neither Games
21%
As you can see, the big
games types (1:1 ordering,
underbooked and overbooked ordering,
tiered ordering, In and Out grouping,
stable grouping, and unstable
grouping) all float around 50%. The
less common types of games (basic
ordering, profiling, combo, and neither
games) are much less likely to have scenarios. The latter
games try to challenge you with the rules or a gimmick.
BP Minotaur:
Scenarios pop up in almost all game types, but they are more prevalent in
some categories. For example, if you are doing a tiered ordering game, you
should really be on the watch for scenarios. But if you are attacking a basic
ordering game or a characteristic grid game, the odds are much lower.
WHEN?
That is a very important question. Scenarios are very common and super
helpful when formed correctly. The crucial step is determining when to invest
the time and effort into making scenarios.
Constraint is the name of the game. Certain rules constrain a game to
such an extent that a limited number of scenarios exist.
First, we are going to discuss the general requirements for scenarios. After
that, we will dig a little deeper and look at the types of rules and deductions
that commonly lead to scenarios.
Before you start to make scenarios in a game, you want to make sure that the
game satisfies both of the following requirements. If it does, scenarios will be
very helpful. If not, you are wasting your time chasing a pot of gold that
doesn’t exist.
1. There are four or fewer scenarios.
Scenarios are not designed to test some of the ways a game can work. The
goal of the process is to outline the only ways a game can work.
Here is the correct mindset when building scenarios:
“Boy, that block is pretty restricted in this game. It can
only be placed in spots 1 and 2, 4 and 5, or 5 and 6.
That’s all. No other way this game can work.”
When you have a realization like that, it’s time to jump in and build
scenarios. Compare that with the following internal dialogue:
“Hmm... looks like that big block has to take up three
spots. It can go in the front, or it could slide back in the
middle. And it could probably go in some other spots
as well.”
That’s no good. In this situation, the student is trying out some of the ways a
game can work. This won’t be very helpful because there could be other
possibilities. The questions will be a struggle. Here’s one more glimpse into
the mind of a student:
“Oooh... if Charlie is assigned to the blue team, then
both Daisy and Eddie have to go on the red team. And
then Farrah would have to be on the yellow team.”
While these might be helpful deductions, these are not scenarios. If Charlie is
on the blue team, other team members can be determined. But what if Charlie
isn’t on the blue team? If you aren’t covering all of the possibilities, you
aren’t making effective scenarios.
When you start to test the different constraints on a game, a good general rule
is that you only want to create four or fewer scenarios. Anything more than
that is going to take too much time and effort.
Only make scenarios when you know you are covering
the only ways that a game can work and there are four
or less scenarios.
Ninja Note:
It’s very common to only make two or three
scenarios.
2. It’s going to lead to more.
Scenarios aren’t designed to give you an opportunity
to master just one rule or deduction. The purpose is to
uncover additional deductions that you wouldn’t spot
otherwise. When you make scenarios, you want to
have a reasonable belief that there will be helpful
deductions in most, if not all, of the scenarios.
Once again, let’s play the mindset game. Here is one:
“Sharon and Mark have to go to either Dallas or
Houston. If they go to Dallas, there’s only one spot left
and it would have to be Nina. If they go to Houston, it
looks like I could apply the third and fourth rule to
figure out who goes to Austin.”
That’s the dream. Here, creating two scenarios leads to other deductions.
Such deductions would be difficult to spot without scenarios, giving you a
huge advantage. Contrast that with the following situation:
“Well, it looks like Bella has to sit in seat 3, 4, or 5.
But I don’t have any other rules about Bella or those
seats.”
This is not an effective way to make scenarios. If you can’t activate other
rules to find more deductions, it’s unlikely that the scenarios will be helpful.
Only make scenarios when you have a strong suspicion
that they will activate other rules and lead to more
deductions.
Think back to our chess
tournament game and how it satisfied
these two requirements. First, the SOS
block had only two possible
placements. Second, there were other
rules that were also solved by the
placement of the block, such as the
restrictions on Tyrone, Nam, and
Pachai.
BP Minotaur:
Now that you know the general requirements, it’s time to look at the rules
and deductions that normally generate scenarios. There are three situations
that will generally be the trigger.
1. Constrained Player(s)
The first, and most common, way you will find scenarios is with a
constrained player or players. This can result directly from a rule or more
indirectly from a combination of rules.
In some games, one player will be confined to only a few possible slots. In
other games, combinations of players will be similarly constrained. This was
our big rule in the chess tournament - Santiago and Ofelia were very
constrained. Here’s an overview:
Ordering Games
There are a few rules to watch in ordering games. Anytime you see one of the
following rules or deductions, scenarios should be on your mind.
1. Blocks: One of the most common ways to form scenarios
is with blocks. They are big. They are ugly. And they tend
to not have many places they can land. Blocks occur in all
ordering games (1:1, underbooked, overbooked, and
tiered) as well as some combo games. Every single time you are presented
with a block, you need to examine where it can be placed.
2. Arches: Another common culprit is arches. This
type of rule puts a strong constraint on a player he, she, or it must be assigned to one of only two
spots. Arches are also a common form of
deduction. Through a combination of ordering
rules, many games allow you to reduce the
possibilities for a player down to just two.
Grouping Games
In grouping games, the rules aren’t about spatial relationships. But there are
still some common grouping relationships that limit the possibilities for a
player or players.
1. Love (Must Be Together): A very common grouping
relationship asserts that two players must be together. This can
be very helpful. If this is an In and Out game, you either select both players
or neither one. If there are multiple groups, there might not be many
groups that can accommodate these lovebirds.
2. Baby (At Least One): The baby relationship is very
important because it establishes that one of two conditions
must always be met. This rule has a tendency to set up two scenarios (one
for each option) or three scenarios (the third is formed by satisfying both
conditions) depending on the game.
3. Two of Three: In grouping games, you will frequently
have to select two of three variables. This is more
commonly a deduction than a rule, but it occurs both
ways. Whenever this happens, it sets up three great
scenarios. You could select the first and second variable, the first and third
variable, or the second and third variable.
This is not an exhaustive list of the ways that a player or players can be
constrained and lead to scenarios. Remember, one of these rules in isolation
is not sufficient to justify scenarios. But this is a good guide for common
ways that constrained players lead to successful scenarios.
Spotting a constrained player
is by far the most common way to
create scenarios.
BP Minotaur:
2. Constrained Slot(s)
The next road to scenarios focuses on the slots rather than the players. Some
slots, through a rule or a combination of rules, become very restricted. Only
two or three players can occupy the slot. Earlier in the book, we used options
to represent this important deduction. Since only a few possibilities exist, this
gives you a great chance to make scenarios.
Ordering Games
1. Options: Some rules will simply provide you with an
option - one of two players must occupy a certain slot.
But options can also be deduced by eliminating all but
two players from one lucky slot.
Grouping Games
1. Designated Spots: Some grouping games will
designate one member of the group as special in
some way. A committee might have a leader, a team
could have a captain, or the mob might have a head
honcho. When this happens, watch for rules that restrict the possibilities
for this slot. If there are only a few, this could lead to scenarios.
2. One Spot Left: Many times you will fill some, but
not all, of the spaces available in one of your groups.
When this happens, make sure to run through the
possibilities for the remaining slots. If you only have one spot left and
there are only two or three options, it’s time for scenarios.
Since slots are a constant throughout games, you always have to be on the
watch for constrained slots. This type of constraint is easy to miss because it
arises through a combination of rules.
3. Constrained Distributions
In the last chapter, you learned how to play the numbers. It’s an effective
technique used to identify all of the possible distributions in a game.
Sometimes there will be a large number of distributions. That’s a bummer.
But many other games present you with only a limited number of
distributions. This is a great way to create scenarios.
To review, here are the types of games that commonly require you to play the
numbers:
1. Underbooked or Overbooked Games: In an underbooked game, you are
presented with fewer players than slots, so some of the players must
occupy more than one slot. In an overbooked game, there are more players
than slots. So some slots must make room for more than one player. Either
of these situations could give rise to scenarios if there are only a few
possible distributions.
2. Players from Different Categories: When the players in a game come
from different categories (boy and girls, heroes and villains), you will
commonly play the numbers to decipher how many players can be selected
from each category. This happens most frequently with In and Out
grouping games, and it commonly leads to scenarios.
3. Unstable Grouping Games: In unstable grouping games, you don’t know
the sizes of the groups. That sucks. But playing the numbers helps
determine the possibilities. If there are only a few, it’s time for scenarios.
When you play the numbers to determine
the distributions that are present, always
stop yourself when there are four or fewer.
Applying the rules to each scenario can
yield a big payoff.
If you are able to play the numbers and use that to
build scenarios, you are quickly approaching ninja status.
Ninja Note:
That’s an overview of when you should make
scenarios. It will take some practice to train yourself
to spot scenarios, but make sure to follow the general
guidelines. Constraint is the name of the game.
HOW?
Earlier, you saw one example of how to execute scenarios when we visited
the chess tournament. The general idea is to work through each scenario and
plug in additional deductions as you go. Here are some additional tips to help
you sharpen your skills:
1. Don’t expect to fill every slot. The goal is to identify all of the
deductions that you can make in each scenario. Sometimes you will
determine nearly everything, but other times you will only fill a few slots.
Think back to the chess tournament. One scenario was nearly completed,
but the second one had lots of open spaces. Even if your scenarios are
incomplete, they will give you a jump start on the questions and aid your
understanding of the game.
2. Scenarios can lead to more scenarios. When you start building
scenarios, you might note that you arrive at another fork in the road. If
placing a block in one place means that another slot now has only two
options, don’t hesitate to break that one scenario into two scenarios. As
long as you are uncovering deductions along the way, the time investment
will be worth it.
3. Utilize the scenarios for as many questions as possible. When you hit
the questions, it’s time for the big payoff. You want to reap the rewards of
all your hard work. For each question, examine the scenarios before doing
any additional work. For absolute questions, glance through all of the
scenarios. For conditional questions, quickly look through the scenarios
and determine which ones are relevant for that question.
Scenarios will help you get through all
questions very quickly, except for elimination questions.
Stick with the original strategy for those little buggers.
BP Minotaur:
That concludes our overview of scenarios. You are
probably excited to get started, but cool your jets.
Before we attempt real games, there’s a drill on the
next page to help you develop your new skills.
3 Joan goes home after the test, cries, drinks heavily for a few days, and then signs up
to retake the LSAT.
4 Joan still drinks heavily after the LSAT, but in the good way.
5 To give some context, 2001 was the year that American Idol and The Bachelor first
premiered in the US. Both shows are now in season 91 or so.
SCENARIOS DRILL
This drill is designed to develop your scenario-building skills. For each of the
following:
1. Build the appropriate setup.
2. Diagram the rules.
3. Identify the biggest constraint on the game and build scenarios.
1. Five children’s toys—an abacus, a ball, a chess set, a doll, and
an elephant—are arranged side-by-side on a shelf. The toys are
arranged from left (first) to right (fifth), according to the following
conditions:
The doll is the second toy from the left.
The chess set is immediately to the right of the ball.
The abacus is somewhere to the left of the ball.
2. Six employees of the Hergot Corporation are assigned to
complete team-building exercises. The six employees— Heather,
Ivan, Jackie, Karl, Lin, and Matt—will be split into three twoperson teams: team 1, team 2, and team 3. Each employee is
assigned to one team. The following must obtain:
Ivan and Lin must be assigned to the same team.
Karl must be assigned to team 2.
Heather and Karl cannot be assigned to the same team.
3. At a fancy dinner event, guests are served seven consecutive
courses. The seven courses are the following: jicama, kale, lentils,
macaroni, nutella, omelettes, and pizza. The courses are served one
at a time, according to the following:
Jicama is served before pizza. Macaroni is served before nutella.
Kale is served first or last.
Lentils are served fifth.
Nutella and omelettes are served consecutively.
4. Six contestants—U, V, W, X, Y, and Z—perform six
consecutive songs in a singing competition. Two of the contestants
sing hip-hop, two sing reggae, and two sing show tunes. The
following conditions apply:
The two reggae songs must be separated by at least two other
songs.
X sings fifth.
The first song performed is a hip-hop song.
A show tune is the only song performed between the
performances by V and W.
V performs after U but before Z.
U sings either a reggae song or a show tune.
5. A table at the holiday party offers four different snacks: nachos,
poppers, quesadillas, and wings. Ted eats six snacks, consecutively
and one at a time. He eats each type of snack at least once. The
following conditions govern his snacking:
Ted enjoys poppers at least twice in a row, but he doesn’t eat
poppers fourth.
Ted cannot eat wings later than third.
If Ted eats nachos fourth, then he also eats nachos second.
6. Exactly four of seven products—beer, chicken, dentistry, eggs,
fire alarms, gelatin, and ham—are featured in commercials during
halftime of the big game. The selection of commercials is
consistent with the following:
If beer is featured, then ham is not.
If chicken is featured, then dentistry is not.
If fire alarms are not featured, then eggs are featured. Of the
three commercials, beer, chicken, and gelatin, exactly two are
selected.
7. Eight inmates—F, G, H, I, J, K, L, and M—are assigned to three
different work crews on Tuesday. Two inmates are assigned to
Rose Street, three are assigned to Sawyer Avenue, and three are
assigned to Tunnel Road. The following conditions govern the
assignment:
J is assigned to Rose Street.
Neither L nor M is assigned to Sawyer Avenue.
If G is assigned to Tunnel Road, then H is assigned to Rose
Street.
F and I are assigned to the same crew.
8. Six puppets—Fergie, Gregg, Howie, Ivanna, Jules, and Kid—are
scheduled to perform at a local show that is running from Monday
through Wednesday. Each day, one puppet will perform in the
morning and one will perform in the afternoon. Each puppet will
perform exactly once. The following conditions must hold:
Fergie performs in the afternoon on the same day that either
Gregg or Howie performs in the morning.
Jules and Kid perform on an earlier day than Gregg. If Ivanna
performs in the afternoon, then Howie also performs in the
afternoon.
9. Five travel destinations are being chosen for a “best of the best”
list in a travel magazine. The five will be chosen from among three
tropical destinations (Anguilla, Bermuda, and Curacao), three
adventure destinations (Peru, Roatan, and South Africa), and three
romantic destinations (Maldives, Napa, and Ojai). The selection is
in accord with the following:
At least one destination from each category is selected. At least
as many tropical destinations are selected as adventure
destinations.
If Anguilla is selected, then Ojai is not.
If Curacao is selected, then Bermuda is not. If Napa is selected,
then Peru is also selected.
10. Seven clowns—Ricky, Suzy, Tuffy, Verby, Willy, Yoogy, and
Zimmy—are assigned to three different birthday parties on
Saturday. The three parties are for Bobby, Cindy, and Eddie. Each
clown is assigned to one party, and each party is assigned at least
one clown, according to the following:
Eddie’s party is assigned exactly one more clown than Cindy’s.
Zimmy is assigned to Bobby’s party.
Verby cannot be assigned to Eddie’s party.
Tuffy and Verby must be assigned to the same party. Willy and
Yoogy cannot be assigned to the same party. If Ricky is assigned
to Bobby’s party, then Yoogy must be assigned to Cindy’s party.
ANSWER KEY
Here are the solutions to the previous drill. Make sure you formed scenarios
correctly and found the resulting deductions.
This is a 1:1 ordering game.
The block leads to two constrained
players (B and C). The block has only
two possible placements.•
In scenario 1, the block goes in slots 3
and 4. A must be in slot 1, and E goes in
slot 5.
In scenario 2, the block slides over to slots 4 and 5. There should be
linked options for A and E in slots 1 and 3.
This is a stable grouping game
with three groups.
The I and L combo are highly
constrained - they must be on team
1 or team 3.
In both scenarios, H must stay away from K, and linked options can be
formed for M and J.
This is a 1:1 ordering game.
The block with N and O leads to
three scenarios.
In scenario 1, N and O are in
slots 2 and 3. M must go in slot
1, and K is pushed to slot 7. J is
served before P, so J is in 4 and
P is in 6.
In scenario 2, N and O are in
slots 3 and 4. M must go in slot
1 or 2, and P must be in slot 6 or 7 (served later than J).
In scenario 3, N and O are in slots 6 and 7. K must go in slot 1. J is
earlier than P, so either M or J is in slot 2, and either M or P is in slot 4.
Now it’s time for tiered ordering.
The V and W block has only two
possibilities.
In scenario 1, V and W are in slots 1
and 3. But U must precede V, so V
must beinslot3. U is a show tune in
slot 2. The two reggae songs are
separated in slots 3 and 6. Two pairs
of linked options finish off the
deductions.
In scenario 2, V and W are in slots 2 and 4. U can’t sing hip-hop, so U
can’t go in 1. This pushes U into slot 3 and V into 4. Z is in slot 6 (after
V). To keep the reggaes separated, one of them must go in slot 2.
This is a delicious underbooked
ordering game.
Since P and W can’t go in slot 4, it
must be either N or Q. This
constrained slot leads to two
scenarios.
In scenario 1, N is in slot 4, so N must also go in slot 2. The PP block is
in slots 5 and 6 and there are linked options for slots 1 and 3.
In scenario 2, Q goes in slot 4. There’s no further deductions since the
PP block still has three possible placements.
Here is an In and Out grouping game.
The final rule sets up three scenarios - the selection could include B and
C, B and G, or C and G.
In scenario 1, B and C are selected, but G is not. D and H feel the hate
and are left out. The out group is full, so E and F must be selected.
In scenario 2, B and G are selected, but C is not. H can’t be selected
with B. Either E or F (or both) must be
selected.
In scenario 3, C and G are selected, but B
is not. D can’t be selected with C. Again,
either E or F (or both) must be selected.
This is a stable grouping game with three
groups.
The final rule places a strong constraint on
two players. F and I must be assigned to
Sawyer Avenue or Tunnel Road. That sets up
two scenarios.
In scenario 1, F and I go to Sawyer Avenue.
Neither L nor M can be assigned to Sawyer
Avenue, so at least one of them must go to
Tunnel Road.
In scenario 2, F and I go to Tunnel Road. L and M can’t be assigned to
Sawyer Avenue, so they form linked options on Rose Street and Tunnel
Road. The only remaining spots are on Sawyer Avenue, so the
remaining inmates (G, H, and K) are also determined.
This is a combo scheduling game.
The first rule provides a block with F in the afternoon and either G or H
in the morning. This block could go on any of the three days, and that
leads to three scenarios.
In scenario 1, F is on Monday afternoon. G can’t be scheduled for
Monday, so H takes the Monday morning slot. J and K must be on an
earlier day than G, so they form linked options on Tuesday. Since H is
in the morning, I must be on Wednesday morning, and G fills in
Wednesday afternoon.
If F is on Tuesday afternoon, either G or H
could be on Tuesday morning. Split this
scenario into two scenarios; now there will be
a total of four.
In scenario 2, G is on Tuesday with F. J and K
form linked options on Monday to stay in
front of G. The conditional rule dictates that I
must go on Wednesday morning and H on
Wednesday afternoon.
In scenario 3, H is on Tuesday with F. J and K
are still linked options on Monday. Since H is
in the morning, I must be in the morning on
Wednesday. G takes the last spot on
Wednesday afternoon.
In scenario 4, F is on Wednesday afternoon
with either G or H in the morning. Since there
appears to be a number of options for the
remaining variables, this is a good time to
stop.
Note: It is possible to split the final scenario into two more scenarios (G or H
on Wednesday morning), but try to limit yourself to four or less scenarios.
Next up is an In and Out grouping game
with players from different categories.
It’s important to play the numbers. There
are three distributions that lead to three
scenarios.
In scenario 1, the selection includes one
tropical, one adventure, and all three
romantic destinations: M, N, and O. If N
is selected, P is selected. A can’t be
selected with O, so B or C takes the last slot.
In scenario 2, two tropical, one adventure, and two romantic
destinations are chosen. B and C can’t both be selected, so A is selected
withoneofthem. O is out, so M and Nare selected. P fills the final slot.
In scenario 3, there are two tropical, two adventure, and one romantic
destination. A is selected with either B or C. A forces O out, so M or N
is the romantic spot chosen.
The final example is an unstable grouping
game.
After playing the numbers, two
distributions emerge. That spells two
scenarios.
In scenario 1, Bobby gets four clowns
(lucky guy), Cindy gets one, and Eddie
gets two. V can’t entertain Eddie, and
there’s not enough room for both T and V
at Cindy’s party, so T and V head to Bobby’s party. The other slots are
still up in the air.
In scenario 2, Bobby gets two clowns, Cindy gets two clowns, and
Eddie gets three. T and V can only fit at Cindy’s party. W and Y form
linked options for Bobby and Eddie. Finally, R and S take the last two
slots with Eddie.
That’s it for the big scenarios drill. It’s a tough one, but those are the sort of
deductions that can make the difference in your Logic Games performance. If
your scenarios were slightly different, that’s normal. Just looking for
scenarios is half the battle. But make sure to go back through the examples
and practice the correct way to attack each game.
At this point, we have completed the introduction of scenarios. Now, it’s time
to practice them. We are going to cover a variety of games that can be
mastered using scenarios.
DECEMBER 2007: GAME 1 (1-5)
This is our first game using scenarios. Very exciting. Even though you know
it’s coming, pretend like you don’t. Your focus here should be how to spot
the opening for scenarios and how to execute properly.
For the first couple games, we will give you a little guidance using the
Blueprint Building BlocksTM method. But the assistance will decrease as we
move forward, and you will soon be doing them on your own. First up, we
take a trip to a talent agency.
1. Setup
Five performers—
Traugott, West, Xavier,
Young, and Zinser—
are recruited by three
talent agencies—Fame
Agency, Premier
Agency, and Star
Agency. Each
performer signs with
exactly one of the
agencies and each
agency signs at least
one of the performers.
The performers’
signing with the agents
is in accord with the
following:
Traugott? Zinser?
These are clearly fake stage
names.
No big surprises here. It’s a grouping
game. Five players are assigned to three
different talent agencies. You don’t know
how many performers sign with each
talent agency, so this is an unstable
grouping game. Each agency signs at
least one performer, but the exact number
is unclear. You can quickly play the
numbers to find the two possible distributions: (1) one agency signs three
performers and the other two agencies each sign one, or (2) two agencies
sign two performers and the third agency only signs one.
Once that is all set, it’s on to the rules.
2. Rules
Our focus here is on scenarios, so we are going to fly through the rules. You
know the drill. This is a grouping game, so you should expect various
grouping relationships. Read through the rules and check the diagramming on
the next page.
Xavier signs with Fame Agency.
Xavier and Young do not sign with the
same agency as each other.
Zinser signs with the same agency as
Young.
If Traugott signs with Star Agency, West
also signs with Star Agency.
The first rule tells you that Xavier signs with Fame Agency. That can be
plugged right into the setup.
According to rule #2, Xavier and Young hate each other. Place a
restriction for Young next to Fame Agency.
The third rule tells you Zinser and Young sign with the same agency
(love).
The final rule is conditional. If Traugott signs with Star Agency, then West
also signs with Star Agency. Don’t forget the contrapositive: If West
doesn’t sign with Star Agency, then Traugott doesn’t sign with Star
Agency.
That gets us quickly through the
setup and the rules. But now it’s
time for the big show. There’s no
way we are jumping into the
questions. In fact, we aren’t even
going to go down the normal
road of deductions. Rather, it’s
time for scenarios.
3. Deductions
First, they tell you that Xavier signs with Fame Agency. Then, they tell you
that Xavier and Young do not sign with the same agency. So Young can’t
sign with Fame Agency. Finally, they tell you that Zinser and Young sign
with the same agency. But that only leaves two options for Zinser and
Young: Premier Agency and Star Agency. Boom. We have our two
scenarios.
Challenge: Work through the two scenarios using the
setup and rules above. Scenario #1 Scenario #2
The scenarios in this
grouping game stemmed from a love
(must be together) relationship. This is
an example of constrained players.
BP Minotaur:
Here’s an overview of the two scenarios:
Scenario #1
Zinser and Young sign with Premier Agency.
Either Traugott or West must sign with Star
Agency. But if Traugott signs with Star Agency,
West must sign with Star Agency. And if
Traugott doesn’t sign with Star Agency, then
West signs with Star Agency. So West must sign
with Star Agency.
Traugott could sign with any of the three agencies.
Scenario #2
Zinser and Young sign with Star Agency.
Either Traugott or West must sign with
Premier Agency.
If Traugott signed with Star Agency, then
West would have to sign with Star
Agency. But that would be four
performers, which violates our distributions. So Traugott can’t sign with
Star Agency.
Those are two beautiful scenarios. By recognizing the constraint formed by
the first three rules, you can identify the only two ways it can work.
I missed a few of those
little things with T and W. Are those
important?
Cleetus Comment:
Cleetus, as you should know by now, there are no little deductions. When
you make scenarios, each “little” deduction will save you time on the
questions.
4. Questions
You should be filled with excitement. You’ve done a great job, and the
payoff is coming. You don’t know exactly what the questions are going to
ask, but you don’t really care. In the world of the LSAT, this is as good as it
gets. It’s time for the victory lap. The questions are all yours.
Challenge: Use the scenarios to quickly answer all of
the questions.
Question #1
Xavier signs with Fame Agency.
Xavier and Young do not sign with the
same agency as each other.
Zinser signs with the same agency as
Young.
If Traugott signs with Star Agency, West
also signs with Star Agency.
1. Which one of the following could be a
complete and accurate list of the
performers who sign with each agency?
(A) Fame Agency: Xavier PremierAgency:
West Star Agency: Traugott, Young,
Zinser
(B) Fame Agency: Xavier PremierAgency:
Traugott,West Star Agency: Young, Zinser
(C) Fame Agency: Xavier PremierAgency:
Traugott,Young Star Agency: West, Zinser
(D) Fame Agency: Young, Zinser
PremierAgency: Xavier StarAgency:
Traugott,West
(E) Fame Agency: Xavier, Young, Zinser
PremierAgency: Traugott Star Agency:
West
Questions #2 and #3
2. Which one of the following could be
true?
(A) West is the only performer
who signs with Star Agency.
(B) West, Young, and Zinser all
sign with Premier Agency.
(C) Xavier signs with the same
agency as Zinser
(D) Zinser is the only performer
who signs with Star Agency.
(E) Three of the performers sign
with Fame Agency.
3. Which one of the following must be true?
(A) West and Zinser do not sign with the same agency as each
other.
(B) Fame Agency signs at most two of the performers.
(C) Fame Agency signs the same number of the performers as
Star Agency.
(D) Traugott signs with the same agency as West.
(E) West does not sign with Fame Agency.
Question #4
4. The agency with which each of the
performers signs is completely
determined if which one of the
following is true?
(A) Traugott signs with Fame
Agency.
(B) Traugott signs with Star
Agency.
(C) West signs with Premier
Agency.
(D) Xavier signs with Fame
Agency.
(E) Zinser signs with Premier
Agency.
Question #5
5. If Zinser signs with Star Agency,
which one of the following must be
false?
(A) Premier Agency signs exactly
one performer.
(B) Star Agency signs exactly
three of the performers.
(C) Traugott signs with Star
Agency.
(D) West signs with Star Agency.
(E) None of the other performers
signs with the same agency as
Xavier.
Xavier signs with Fame Agency.
Xavier and Young do not sign
with the same agency as each
other.
Zinser signs with the same
agency as Young.
If Traugott signs with Star
Agency, West also signs with
Star Agency.
Hopefully that was enjoyable. The scenarios allow you to fly through the
questions with little additional work. Here’s a quick review of the questions.
Check your answers, but also make sure that you used the scenarios in the
best way possible.
Question #1 (elimination, could be true)
Remember, elimination questions are the one time when it’s not helpful to
use your scenarios.
Xavier signs with Fame Agency, so (D) is gone.
Xavier and Young cannot sign with the same agency, which kicks out
(E).
(C) is eliminated because Zinser and Young must be together.
In (A), Traugott signs with Star Agency but West does not. No way.
(B) is the correct answer.
Question #2 (absolute, could be true)
(A) In scenario 1, West could be the only performer to sign with
Star Agency as long as Traugott signs with either Fame Agency
or Premier Agency.
(B) If Young and Zinser sign with Premier Agency (scenario 1), West
must sign with Star Agency.
(C) Xavier signs with Fame Agency, and Zinser can’t sign with Fame
Agency.
(D) Zinser always signs with the same agency as Young.
(E) In both scenarios, the maximum number of performers who could
sign with Fame Agency is two.
Question #3 (absolute, must be true)
(A) In scenario 2, West and Zinser could both sign with Star Agency.
(B) In both scenarios, the maximum number of performers that
could sign with Fame Agency is two. Since Zinser and Young
both have to sign with Premier Agency or Star Agency, there’s
no way for three performers to sign with Fame Agency.
(C) In scenario 1, Fame Agency could sign one or two performers, and
Star Agency could sign one or two performers. In scenario 2, Fame
Agency could sign one or two performers, and Star Agency could
sign two or three performers. Both agencies could easily sign
different numbers of performers.
(D) Traugott and West could sign with different agencies in both
scenarios.
(E) In scenario 2, West could sign with any of the three agencies,
including Fame Agency.
Question #4 (absolute, must be true)
For this question, you need to find an answer that determines the placement
of all the players. With scenarios, this means that the answer must force you
into only one scenario and completely fill it out. That’s a lot to ask from an
answer.
(A) Traugott could sign with Fame Agency in either scenario.
(B) Traugott can only sign with Star Agency in scenario 1. If that
occurs, the other four performers are already determined.
(C) West can only sign with Premier Agency in scenario 2. But
Traugott is still not determined.
(D) Xavier always signs with Fame Agency, so that doesn’t help at all.
(E) Zinser only signs with Premier Agency in scenario 1. But Traugott
is still a question mark.
Question #5 (conditional, must be false)
Zinser only signs with Star Agency in scenario
2.
Traugott cannot sign with Star Agency in
scenario 2, so (C) must be false.
That was fun. Building the proper scenarios completely
demolishes this game. Hopefully, that was one of your most
enjoyable games experiences. And that is exactly the idea.
A true ninja aces this game in
about five minutes, leaving plenty of
time for the three remaining games in
the section.
Ninja Note:
We should continue to develop your newly-acquired skills. Up next is an
entirely different type of game. But scenarios will still be the key to success.
As we progress, less assistance will be provided, and you will take on more
of the work.
JUNE 2008: GAME 4 (18-23)
The next challenge is the final game from the June 2008 LSAT. Even when
you are working on the last game in a section and time is scarce, it’s
important to keep scenarios in mind.
You will complete most of this game on your own. However, we will check
in once or twice to make sure you are on the right path.
1 and 2. Setup and Rules
Below you will see the introduction and the rules. It’s time to review some
contract bids. As you create the setup and diagram the rules, watch for rules
that place big constraints on the game.
Challenge: Build the proper setup for this game and
diagram the rules.
A panel reviews six contract bids—H, J, K, R, S, and T. No two bids
have the same cost. Exactly one of the bids is accepted. The following
conditions must hold:
The accepted bid is either K or R and is either the second or the
third lowest in cost.
H is lower in cost than each of J and K.
If J is the fourth lowest in cost, then J is lower in cost than each of
S and T.
If J is not the fourth lowest in cost, then J is higher in cost than
each of S and T.
Either R or S is the fifth lowest in cost.
The introduction to this game is surprisingly short. There are six contract bids
- those are the players in the game. No two bids have the same cost. That’s a
hint that you are going to be ordering the bids according to cost. If you glance
at the rules, you see that they refer to ranking the bids by cost. So our original
suspicion is confirmed. Since the bids are being ordered and no bids have
the same cost, this is a 1:1 ordering game.
There is a twist: One of the six bids is accepted. This introduces another
variable set (accepted or not accepted). Normally, we would create a
second tier. But the first rule tells you that the accepted bid is either second
or third lowest in cost. You don’t have to worry about the other slots, so
just make a second tier for the second and third slots.
According to the first rule, either K or R is the accepted bid. Since the
accepted bid must be second or third lowest in cost, this can be
visualized with arches and an option.
The second rule outlines some basic ordering principles: H is lower in
cost than J and K.
The third and fourth rules are interesting conditional rules. If J is the
fourth lowest in cost, then J is lower in cost than S and T. But if J is not
the fourth lowest in cost, then J is higher in cost than S and T.
Interesting.
The final rule gives you an option of either R or S in the fifth spot.
3. Deductions
It’s go time. There are a few rules that place big constraints on this game.
First, there’s the mystery of the accepted bid. It must be either K or R, and it
must be either second or third. But that sounds like four scenarios, and it
doesn’t look like that would activate many other rules. Second, there’s the
option for either R or S in the fifth spot. That looks tempting, but you should
be even more intrigued by J. J is a very constrained player. Check it out:
There are only two options for J: Either J is fourth lowest in cost, or it’s
not. If J is the fourth lowest in cost, then it is lower in cost than S and T.
If J is not the fourth lowest in cost, then it is higher in cost than S and T.
Both scenarios should uncover even more.
Challenge: Work through the two scenarios using the
other rules in this game. Scenario #1: J is the fourth
lowest. Scenario #2: J is not the fourth lowest.
There are many big deductions based on the placement of J. Note that it was
a combination of two rules that led to the scenarios in this game, but both
severely constrained the same player.
Scenario #1
Since J is the fourth lowest in cost, J must
be lower in cost than both S and T.
Either R or S must fill the fifth spot, so S
must be the fifth lowest in cost, and T is
the highest in cost.
The three remaining bids are H, K, and R. H must be lower in cost than
K, so either H or R must be the lowest in cost, and either K or R must be
the third lowest in cost.
Either K or R could still be accepted.
Scenario #2
If J is not the fourth lowest in cost, J must
be higher in cost than both S and T.
H is also lower in cost than J.
H, S, and T are all lower in cost than J.
So J cannot be the lowest, second lowest, or third lowest in cost.
Either R or S must be the fifth lowest in cost, so J also cannot go in the
fifth slot. Only one option left: J must be the highest in cost.
J must go in either slot 4 or
slot 6 - that’s huge. There’s no way to
spot that deduction without scenarios.
Ninja Note:
The first scenario works great. The second scenario still has a lot of variables
up in the air. But that’s fine. By working through the scenarios, you get a
huge deduction about J, and this will give you a great head start on the
questions. We are leaving those for you.
Challenge: Now it’s time for the fun part. Use the
scenarios to run through the questions.
JUNE 2008: GAME 4 (18-23)
Questions 18-23
A panel reviews six contract bids—H, J, K, R, S, and T. No two bids have the
same cost. Exactly one of the bids is accepted. The following conditions must
hold:
The accepted bid is either K or R and is either the second or the third
lowest in cost.
H is lower in cost than each of J and K.
If J is the fourth lowest in cost, then J is lower in cost than each of S and T.
If J is not the fourth lowest in cost, then J is higher in cost than each of S
and T.
Either R or S is the fifth lowest in cost.
18. Which one of the following could be an accurate list of the bids in
order from lowest to highest in cost?
(A) T, K, H, S, J, R
(B) H, T, K, S, R, J
(C) H, S, T, K, R, J
(D) H, K, S, J, R, T
(E) H, J, K, R, S, T
19. Which one of the following bids CANNOT be the fourth lowest
in cost?
(A)
(B)
(C)
(D)
(E)
H
J
K
R
T
20. Which one of the following bids CANNOT be the second lowest
in cost?
(A)
(B)
(C)
(D)
(E)
H
J
K
R
T
21. If R is the accepted bid, then which one of the following must be
true?
(A)
(B)
(C)
(D)
(E)
T is the lowest in cost.
K is the second lowest in cost.
R is the third lowest in cost.
S is the fifth lowest in cost.
J is the highest in cost.
22. Which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
H is lower in cost than S.
H is lower in cost than T.
K is lower in cost than J.
S is lower in cost than J.
S is lower in cost than K.
23. If R is the lowest in cost, then which one of the following could
be false?
(A)
(B)
(C)
(D)
(E)
J is the highest in cost.
S is the fifth lowest in cost.
K is the third lowest in cost.
H is the second lowest in cost.
K is the accepted bid.
That should have a been a nice joyride through the questions. Let’s take a
look.
Question #18 (elimination, could be true)
K or R must be in the second or third slot. Neither one shows up in
those slots in (C).
The second rule says that H is lower in cost than J and K. (A) is gone.
If J is the fourth lowest in cost, then J should be lower than S and T. (D)
breaks this rule.
If J is not the fourth lowest in cost, then J must be higher in cost than S
and T. In (E), J is lower than both.
(B) is the correct answer.
Question #19 (absolute, must be false)
In scenario 1, J is the fourth lowest in cost.
In scenario 2, many bids could be the fourth lowest in cost. But R or S
must be the fifth lowest in cost, and J is highest in cost. H still must be
lower in cost than K. K could be the fourth lowest in cost, so H must fill
one of the first three slots.
H can’t be the fourth lowest in cost in either scenario, so (A) is the
winner.
Question #20 (absolute, must be false)
The biggest deduction that came from our scenarios is that J is either the
fourth lowest in cost or the highest in cost.
J can’t be the second lowest in cost, so (B) is correct.
Question #21 (conditional, must be true)
Either R or S must be the fifth lowest in cost. If R is the accepted bid,
then R must be the second or third lowest in cost. Thus, S must be the
fifth lowest in cost.
(D) is the champ.
Question #22 (absolute, must be true)
(A) In scenario 1, H must be lower in cost than S. But in scenario 2, H
could be higher in cost than S. S could even be the lowest in cost.
(B) This one is similar to (A). In scenario 1, H is lower in cost than T.
But T could be lower in cost than H in scenario 2.
(C) In scenario 1, the only two bids that are higher in cost than J
are S and T. In scenario 2, J is the highest in cost. This means
that S and T are the only two bids that can ever be higher in
cost than J. The other bids (including K) must be lower in cost
than J.
(D) In scenario 2, S is lower in cost than J. But in scenario 1, S is
higher in cost than J.
(E) In scenario 1, S is higher in cost than K. And in scenario 2, S could
be higher in cost than K.
Question #23 (conditional, could be false)
R could be the lowest in cost in both scenarios.
If R is the lowest in cost, then S must be the fifth lowest in cost.
If R is the lowest in cost, then K must be the accepted bid.
H must be lower in cost than K, and the accepted bid must be second or
third lowest in cost. So H must be second lowest in cost, and K must be
third lowest in cost.
In scenario 1, everything is determined.
In scenario 2, T fills the fourth slot.
J could be the fourth lowest in cost, so
it could be false that J is the highest in
cost. (A) could be false.
Excellent work. Let’s review for a second - it’s important to take lessons
from each game. Here are some interesting points about this game:
1. This is an entirely different type of game. The last game was an
unstable grouping game with three distinct groups. This is a 1:1
ordering game with a twist that adds a small second tier. What was
the key to both games? Spotting the constraint in the game and
using that to build scenarios.
2. The scenarios weren’t completely determined. The first scenario
allows you to fill in a couple slots, but the second scenario really
only uncovers one big deduction. That’s the point. Scenarios give
you a big head start on the questions, even if they don’t fill up every
slot.
First, we took you to a chess tournament for a demonstration. Then we visited
talent agencies and contract bids together. You will complete the next game
on your own. Take your time. The goal is to practice making helpful
scenarios.
JUNE 2005: GAME 3 (12-16)
Questions 12-16
One afternoon, a single thunderstorm passes over exactly five towns—
Jackson, Lofton, Nordique, Oceana, and Plattesville—dropping some form of
precipitation on each. The storm is the only source of precipitation in the
towns that afternoon. On some towns, it drops both hail and rain; on the
remaining towns, it drops only rain. It passes over each town exactly once
and does not pass over any two towns at the same time. The following must
obtain:
The third town the storm passes over is Plattesville.
The storm drops hail and rain on the second town it passes over.
The storm drops only rain on both Lofton and Oceana.
The storm passes over Jackson at some time after it passes over Lofton and
at some time after it passes over Nordique.
12. Which one of the following could be the order, from first to fifth,
in which the storm passes over the towns?
(A)
(B)
(C)
(D)
(E)
Lofton, Nordique, Plattesville, Oceana, Jackson
Lofton, Oceana, Plattesville, Nordique, Jackson
Nordique, Jackson, Plattesville, Oceana, Lofton
Nordique, Lofton, Plattesville, Jackson, Oceana
Nordique, Plattesville, Lofton, Oceana, Jackson
13. If the storm passes over Oceana at some time before it passes over
Jackson, then each of the following could be true EXCEPT:
(A)
(B)
(C)
(D)
(E)
The first town the storm passes over is Oceana.
The fourth town the storm passes over is Lofton.
The fourth town the storm passes over receives hail and rain.
The fifth town the storm passes over is Jackson.
The fifth town the storm passes over receives only rain.
14. If the storm drops only rain on each town it passes over after
passing over Lofton, then which one of the following could be
false?
(A)
(B)
(C)
(D)
(E)
The first town the storm passes over is Oceana.
The fourth town the storm passes over receives only rain.
The fifth town the storm passes over is Jackson.
Jackson receives only rain.
Plattesville receives only rain.
15. If the storm passes over Jackson at some time before it passes
over Oceana, then which one of the following could be false?
(A) The storm passes over Lofton at some time before it passes
over Jackson.
(B) The storm passes over Lofton at some time before it passes
over Oceana.
(C) The storm passes over Nordique at some time before it passes
over Oceana.
(D) The fourth town the storm passes over receives only rain.
(E) The fifth town the storm passes over receives only rain.
16. If the storm passes over Oceana at some time before it passes over
Lofton, then which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
The third town the storm passes over receives only rain.
The fourth town the storm passes over receives only rain.
The fourth town the storm passes over receives hail and rain.
The fifth town the storm passes over receives only rain.
The fifth town the storm passes over receives hail and rain.
THE PERFECT STORM
There’s a thunderstorm and it’s headed in your direction. Don’t be scared.
It’s actually headed in the direction of five mythical towns with weird names.
Your job is to deduce the order in which it passes over the towns and what
kind of mess they have in store. This one is pretty intense.
Your goals for this game were the following. Did you accomplish them?
1. Figure out the type of game and build the appropriate setup.
2. Represent the rules correctly and make deductions as you work
through the rules.
3. Recognize the big constraint in this game that leads to scenarios.
4. Execute the scenarios properly.
5. Use the scenarios to answer the questions quickly and accurately.
Yes, that’s a lot of goals for one little game. But we are nearing the end of
this book, and we need to hold you to a higher standard.
1. Setup
In the first sentence, you are given five towns. However, you don’t get a very
clear idea of what you are doing with the towns until a few sentences later.
The key is when they state the storm passes over each town exactly once and
doesn’t pass over any two towns at the same time. We have an ordering
game.
The challenge is complicated when they task you with determining
whether the storm drops both hail and rain (really sucks) or just rain (not
quite as bad) on the town. This adds another variable set and should be
represented with a second tier in
your setup.
For the forms of
precipitation, we will use “H” to
denote hail and rain, and “R” to signify
just rain. Please refrain from drawing
pictures of hail and/or rain.
BP Minotaur:
2. Rules
There are a few important deductions that stem from the later rules. It’s
imperative to catch them since they set the stage for scenarios.
The third town the storm passes over is
Plattesville.
The storm drops hail and rain on the
second town it passes over.
The storm drops only rain on both Lofton
and Oceana.
The storm passes over Jackson at some
time after it passes over Lofton and at
some time after it passes over Nordique.
The first two rules can be plugged straight into the setup.
The third rule states that the storm drops only rain on Lofton and
Oceana. Since the storm drops hail and rain on the second town, neither
Oceana nor Lofton can be second. Make restrictions in the setup.
The final rule gives an ordering principle with three of the towns.
For the last rule, I wrote that J
is after L, and L is after N.
Ditz McGee:
Be careful - that’s a common mistake. There’s no direct relationship between
Lofton and
Nordique. The rule only states that the storm passes over Jackson after each
of them.
3. Deductions
There are a few general deductions you need to spot before you can form
scenarios.
Since the storm passes over Jackson at some point after Lofton and
Nordique, Jackson can’t be the first or second town.
Since Lofton, Oceana, and Jackson can’t be the second town and
Plattesville is busy being the third town, the one remaining town
(Nordique) must be the second town that the storm passes over.
Once you spot that huge deduction about Nordique, you know things are
starting to happen. The next place you want to look is the most-constrained
player: J. Jackson can only be the fourth or fifth town, and that means...
two scenarios!
Scenario #1: The storm passes over Jackson fourth.
The storm has to pass over Lofton before
Jackson, so Lofton is the first town.
Oceana takes the final slot.
The storm drops just rain on both Lofton
and Oceana.
Scenario #2: The storm passes over Jackson fifth.
Lofton and Oceana are the two remaining
towns, so they form linked options for the
first town and the fourth town.
Since the storm drops only rain on both
towns, the storm drops only rain on the first and fourth towns.
Boom! At this point, you should feel great. The scenarios arose from a
constrained player (Jackson). Each scenario uncovered more deductions.
Time for the questions.
4. Questions
Question #12 (elimination, could be true)
The elimination approach always works. But when you get a huge deduction,
it’s worth checking that deduction first. It will generally eliminate a number
of answer choices.
Nordique must be the second town that the storm passes over, which
eliminates (B), (C), (D), and (E).
That was a quick road to (A).
Question #13 (conditional, must be false)
The storm can only pass over Oceana before Jackson in scenario 2.
In scenario 2, the fourth town the storm passes over is either Lofton or
Oceana. The storm drops only rain on both towns, so the storm drops
only rain on the fourth town.
The fourth town cannot receive hail and rain, so (C) must be false.
Question #14 (conditional, could be false)
The storm can drop only rain on each
town it passes over after passing over
Lofton only in scenario 2, and only if
Lofton is the fourth town.
The fifth town (Jackson) must receive only rain.
Plattesville could receive only rain or hail and rain, so (E) could be
false.
Question #15 (conditional, could be false)
The storm can only pass over Jackson before Oceana in scenario 1.
In scenario 1, Jackson, the fourth town, could receive only rain or
hail and rain. Thus, (D) could be false.
Question #16 (conditional, must be true)
The storm can pass over Oceana before
Lofton only in scenario 2, and only if
Oceana is the first town and Lofton is the
fourth town.
Lofton, the fourth town, receives only
rain. (B) must be true.
Excellent! At this point, you should be more comfortable with creating
scenarios. Big deductions are very common in tiered ordering games, and
scenarios are common. Practice makes perfect, so let’s keep the train rolling.
There’s another thrilling surprise waiting for you on the next page. (Hint:
There might just be scenarios.)
DECEMBER 2011: GAME 3 (12-16)
Questions 12-16
The organizer of a luncheon will select exactly five foods to be served from
among exactly eight foods: two desserts—F and G; three main courses—N,
O, and P; three side dishes—T, V, and W. Only F, N, and T are hot foods.
The following requirements will be satisfied:
At least one dessert, at least one main course, and at least one side dish
must be selected.
At least one hot food must be selected.
If either P or W is selected, both must be selected.
If G is selected, O must be selected.
If N is selected, V cannot be selected.
12. Which one of the following is a list of foods that could be the
foods selected?
(A)
(B)
(C)
(D)
(E)
F, N, O, T, V
F, O, P, T, W
G, N, P, T, W
G, O, P, T, V
G, O, P, V, W
13. Which one of the following is a pair of foods of which the
organizer of the luncheon must select at least one?
(A)
(B)
(C)
(D)
(E)
F, T
G, O
N, T
O, P
V, W
14. If O is the only main course selected, then which one of the
following CANNOT be selected?
(A)
(B)
(C)
(D)
(E)
F
G
T
V
W
15. If F is not selected, which one of the following could be true?
(A)
(B)
(C)
(D)
(E)
P is the only main course selected.
T is the only side dish selected.
Exactly two hot foods are selected.
Exactly three main courses are selected.
Exactly three side dishes are selected.
16. If T and V are the only side dishes selected, then which one of the
following is a pair of foods each of which must be selected?
(A)
(B)
(C)
(D)
(E)
F and G
F and N
F and P
N and O
O and P
LUNCHEON MENU
It’s time for a feast. Make sure you recognized the important elements in this
game and created scenarios successfully.
1. Setup
Your job is to select five of eight foods
to serve at a luncheon, making this an
In and Out grouping game. It’s also
important to note that the players in this
game come from different categories
(desserts, main courses, and side
dishes).
The categories in this game are an indication that you might need to play
the numbers. How many desserts could be selected?
There’s also a twist. Three of the options are hot foods. It’s not clear why
that’s important (maybe a lack of hot mitts), but it should be noted. We
decided to underline the hot foods.
2. Rules
The first rule is a principle of distribution. At least one food from each
category must be selected. This makes us consider playing the numbers.
But it’s pretty weak, so we are hoping for something better.
The second rule says at least one hot food is selected. You knew
something was coming related to those crazy hot foods. You can
diagram this by saying that F or N or T (or some combination) must be
selected.
Next up is a love relationship with P and W.
Either they are both In, or they are both Out.
That’s big.
The fourth rule is a stalker relationship. Make
sure to diagram the contrapositive.
The final rule tells you N and V can’t both be
selected. They are from different categories, so
that’s not super helpful.
3. Deductions
There’s a load of information here. There are eight variables from different
categories. There’s a principle of distribution. There’s five fairly complicated
rules. This can be overwhelming. But always look for the biggest constraint
on the game. Where is it?
The third rule! If P or W is selected, they must both be selected. There are
only two possibilities: either both foods are In, or both foods are Out.
That’s going to fill two slots in one of the two groups, and further
deductions will likely be found since they each show up in other rules.
This is a great chance to break the game into two scenarios.
Scenario #1: Both P and W are selected; F is selected.
P is a main course and W is a side dish. But the selection of foods must
still include at least one dessert, so either F or G must be included.
Depending on whether F or G is selected, more deductions will likely be
found. What to do?
At this point, it’s a good idea to split this one scenario into two scenarios.
If F is selected, then the other two spots
are still up in the air.
The final two rules are still in play.
Scenario #2: Both P and W are selected; G is selected.
If G is selected, then O is selected.
A hot food must be selected, so the final
spot must be F, N or T.
V can’t be selected.
Scenario #3: P and W are out.
There’s only one spot left in the Out
group. N and V can’t both be selected, so
they form linked options.
The Out group is now full, so F, G, O, and T are all selected.
Breaking one scenario up to
find more deductions - ninja maneuver.
Ninja Note:
As the ninja points out, this is an advanced technique. If your scenarios
looked a bit different, that’s not the end of the world. As long as you are
investing the time to build scenarios, there will always be a big payoff.
Here are some important notes about the scenarios in this game:
The scenarios arise from two constrained players. The love relationship
let us know both P and W had to be together. They were both selected or
both left out. Love relationships are great to use when forming
scenarios.
When you start to build the first scenario, a constrained slot pops up
because you must have a dessert and there are only two options (F and
G). This should be a clue to break the first scenario into two separate
scenarios.
At this point, it’s time to ace the questions. Check your answers and your
approach.
4. Questions
Question #12 (elimination, could be true)
In (E), no hot foods are selected. It’s out.
(D) has P but no W. That’s not cool.
If G is selected, O must be selected. This rule eliminates (C).
N and V can’t both be selected. Say goodbye to (A).
(B) is left standing.
Question #13 (absolute, must be true)
In scenarios 1 and 2, P is selected.
In scenario 3, P is not selected, but O is selected.
Either P or O (or both) is always selected, so (D) is the answer.
Note: This can be a time-consuming question without scenarios. It asks
you to uncover a baby (at least one) relationship. Remember, the best way
to test answers is to put both variables in the Out group. If P is out, W is
out. And if O is out, G is out. That is four foods, which doesn’t leave us
with enough options to serve a proper luncheon.
Question #14 (conditional, must be false)
P is a main course. In scenarios 1 and 2, P
is selected. Thus, the only time O can be
the sole main course selected is in scenario
3.
N is also a main course, so N is out. V must be selected.
The three foods that are not selected are P, W, and N. (E) is the
answer.
Question #15 (conditional, could be true)
F can only be in the Out group in scenario
2.
In scenario 2, P and O must be
selected. The third main course (N) could also be selected. So all
three main courses could be selected, and (D) is the winner.
Question #16 (conditional, must be true)
W is selected in scenarios 1 and 2, so only
scenario 3 allows you to have T and V as
the only side dishes selected.
If V is selected, N is not.
In scenario 3, both F and G must be selected. So (A) is the answer.
You should have noticed by now that scenarios can pop up anywhere. Our
chess tournament was an underbooked ordering game. The talent agency was
an unstable grouping game. We checked out some 1:1 ordering with the
contract bids. Our stormy game was tiered ordering. And we just finished a
stable, In and Out grouping game. The game types are varied, but the name of
the game is constraint. Make sure to pay attention to the specific rules and
deductions that lead to scenarios.
On the next page, you will see one more game. One
more chance to make sweet scenarios.
OCTOBER 2011: GAME 4 (19-23)
Questions 19-23
Exactly eight books—F, G, H, I, K, L, M, O—are placed on a bookcase with
exactly three shelves—the top shelf, the middle shelf, and the bottom shelf.
At least two books are placed on each shelf. The following conditions must
apply:
More of the books are placed on the bottom shelf than the top shelf.
I is placed on the middle shelf.
K is placed on a higher shelf than F. O is placed on a higher shelf than L.
F is placed on the same shelf as M.
19. Which one of the following could be a complete and accurate list
of the books placed on the bottom shelf?
(A)
(B)
(C)
(D)
(E)
F, M
F, H, M
G, H, K
F, G, M, O
G, H, L, M
20. It is fully determined which of the shelves each of the books is
placed on if which one of the following is true?
(A) I and M are placed on the same shelf as each other.
(B)
(C)
(D)
(E)
K and G are placed on the same shelf as each other.
L and F are placed on the same shelf as each other.
M and H are placed on the same shelf as each other.
H and O are placed on the same shelf as each other.
21. Which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
O is placed on a shelf higher than the shelf M is placed on.
K is placed on a shelf higher than the shelf G is placed on.
I is placed on a shelf higher than the shelf F is placed on.
G is placed on a shelf higher than the shelf O is placed on.
F is placed on a shelf higher than the shelf L is placed on.
22. If G is placed on the top shelf, then which one of the following
could be a complete and accurate list of the books placed on the
middle shelf?
(A)
(B)
(C)
(D)
(E)
H, I
I, L
H, I, L
I, K, L
F, I, M
23. If L is placed on a shelf higher than the shelf H is placed on, then
which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
F and G are placed on the same shelf as each other.
G and H are placed on the same shelf as each other.
H and M are placed on the same shelf as each other.
I and G are placed on the same shelf as each other.
K and O are placed on the same shelf as each other.
FILLING THE SHELVES
Scenarios are crucial on this final game from the October 2011 exam. There
are a few different avenues you can take to uncover the big deductions. Let’s
check it out.
1. Setup
There are eight books being arranged on three different bookshelves: the top
shelf, the middle shelf, and the bottom shelf.
Eight books onto three
shelves - this is just a straight grouping
game.
Cleetus Comment:
Actually, no. But you can’t be faulted much for thinking that. At first glance,
this looks like a grouping game. The setup even makes you think grouping.
But this is actually an ordering game. The shelves have an order to them
(bottom to top). The books are arranged spatially. There are more books
than shelves, so this is an overbooked ordering game. This is one of the
rare ordering games that is better visualized vertically.
Each shelf holds at least two books, so
make sure to put two slots next to each
shelf. Because the game is
overbooked, playing the numbers
might be necessary.
2. Rules
The first rule is a great principle of distribution. You
don’t want the bookshelf to fall over, so more books are
placed on the bottom shelf than the top shelf. This is a
big hint to play the numbers.
For the second rule, just throw I on the middle shelf.
The third and fourth rules give you ordering principles. It’s important to
always represent the rules in a way that is consistent with the setup.
Since we have a vertical setup, you want to visualize these rules
vertically.
The final rule should be combined with the third
rule. The third rule tells you that K is placed on
a higher shelf than F. The final rule tells you that
F is placed on the same shelf as M. So F and M
form a horizontal block and K must be higher
than this block.
That’s all for the rules. The biggest obstacle in the early stages of this game is
visualizing the proper setup.
3. Deductions
There are many elements that should catch your eye in this game. It’s an
overbooked ordering game and the first rule gives you a strong principle of
distribution. There’s also a nice block to investigate. Lots of restrictions
could be formed with the ordering rules. But it’s important to work through
the deductions in an orderly fashion.
The first step is to play the numbers. Before you worry about other
deductions, it’s going to be very helpful to figure out the constraints on the
shelves.
Play the Numbers
There are two big restrictions on the shelves. First, each shelf must have at
least two books. Second, the bottom shelf must hold more books than the top
shelf. Pursuant to these rules, there are only two possibilities:
In the first distribution, two books are placed on
the top shelf and the middle shelf, and four
books are placed on the bottom shelf.
In the second distribution, two books are placed
on the top shelf, and three books are placed on
the middle shelf and the bottom shelf.
Wait a second... there are only two distributions? What
could we possibly do with just two distributions to gain a
deeper understanding of this game and make the next five
minutes of our lives much more enjoyable?
Of course, now it’s time to use these two distributions
to make scenarios.
The scenarios in this game
stem from constrained distributions.
Anytime you have only two
distributions, make two scenarios.
BP Minotaur:
The distributions set up two great scenarios. However, the F and M block
splits the second scenario into two. So there are a total of three scenarios.
Scenario #1: Four on the bottom shelf.
Since there is only one more spot on
the middle shelf, both F and M must
be placed on the bottom shelf.
K and O can’t be placed on the
bottom shelf.
L can’t be on the top shelf.
Scenario #2: Three on the bottom shelf; F and M in the
middle.
Since F and M are placed on the middle
shelf, K must be above them on the top
shelf.
Only the top and bottom shelves have slots
left. O must be higher than L, so O is on the
top shelf and L is on the bottom shelf.
G and H must take the last two spots on the bottom shelf.
Scenario #3: Three on the bottom shelf; F and M on the
bottom.
K must be placed higher than F and
M, so K can’t be on the bottom shelf.
O must be placed higher than L, so O
can’t be on the bottom shelf and L
can’t be on the top shelf.
With these huge deductions in hand, confidence should be high running into
the questions.
4. Questions
Question #19 (elimination, could be true)
The bottom shelf must have more books than the top shelf, so it must
have at least three. In (A), the list only includes two books. No good.
K is placed on a higher shelf than F, so K can’t be on the bottom shelf.
(C) is gone.
O is placed on a higher shelf than L, so O can’t be on the bottom shelf.
Sorry (D).
F and M are placed on the same shelf. Poor (E) has M without F. Loser.
(B) is the champion.
Question #20 (absolute, must be true)
Scenarios can be very helpful on questions like this one. The most
constrained scenario is scenario 2. In that one, all eight books are fully
determined. So you should search for an answer that forces you into only
scenario 2.
I and M are only placed on the same shelf in scenario 2.
(A) determines the placement of all eight books, so (A) is correct.
Answers (B), (C), (D), and (E) are all possible in scenarios 1 and 3.
Since the scenario isn’t determined, there’s no way to determine all of
the books.
Question #21 (absolute, must be true)
(A) In all three scenarios, O is placed on a higher shelf than M.
Excellent.
(B) In scenarios 1 and 3, K could be placed on a lower shelf than G.
(C) In scenario 2, I and F are placed on the same shelf.
(D) In scenario 2, G is placed on a lower shelf than O.
(E) In scenarios 1 and 3, F is on the bottom shelf and thus can’t be on
a higher shelf than L.
Question #22 (conditional, could be true)
This is the toughest question in this game. Even when you have great
scenarios, they will throw you a challenge or two.
G could be on the top shelf in scenario 1 or
scenario 3.
In scenario 1, K and O can’t be on the bottom
shelf. You should form linked options for K and
O on the top and middle shelves.
Hand L both have to be on the bottom shelf.
If there are two books on the middle shelf, those two books must be I
and either K or O. (A) gives you H and I, and (B) offers you I and L.
Neither of these combos work, so (A) and (B) are eliminated.
In scenario 3, K and O still can’t be on the
bottom shelf. K or O could be on the top shelf,
but at least one of them must be on the middle
shelf.
Answer choice (C) offers you H, I and L. But if three books are on the
middle shelf, either K or O (or both) must be on the middle shelf. So (C)
is out.
In both scenarios, F and M are on the bottom shelf. There goes (E).
It took some hard work, but it’s now clear that (D) is the only
possibility. In scenario 3, I, K, and L could be on the middle shelf
(above).
Question #23 (conditional, must be true)
L could be placed on a higher shelf than H
in scenarios 1 and 3.
L can’t be on the top shelf. Thus, L must
be on the middle shelf, and H is on the
bottom shelf. This is true in both
scenarios.
Both H and M are on the bottom shelf. Thus, (C) must be true.
And that, friends, is going to conclude our journey
through scenarios. It takes a good deal of practice to
master this advanced technique, but you’ve seen the
huge benefits that follow from successful execution.
This is a chapter that you should review frequently.
While there is no easy guide for when and how to create
scenarios, the general guidelines covered in the
beginning of this chapter provide a great outline.
As you move forward in your practice with games, it’s important to alter your
standard of review. From the start of this journey, we urged you to judge your
performance by more than just right and wrong answers. It’s important to
assess your strategy. When you do a game, it’s great to get the questions
right. But could you have made scenarios? If you ever miss the opportunity
to build scenarios, it’s important to learn from that mistake.
Before we move on, here’s a quick review of scenarios:
1. Scenarios are a technique that
allows you to identify a limited
It’s really important.
number of possibilities in a game.
2. Making scenarios in a game will increase both your speed and
accuracy.
3. Saving time by making scenarios in one game can improve your
performance in the entire games section by affording you more time for
the other games.
4. Historically, scenarios can be
utilized on 41% of all games.
5. You are almost guaranteed to
get at least one game with
scenarios.
It’s going to pop up on
your LSAT.
6. Scenarios can occur in all types of games, but they are more prevalent
in some game types.
7. To look for scenarios, keep an
eye out for strong forms of
constraint.
Watch for strong
constraint.
8. Only make scenarios when you
can visualize that a game has four or less scenarios and when the
scenarios will lead to additional deductions.
9. There are three common forms of constraint that lead to scenarios: (1)
constrained player (s), (2) constrained slot (s), and (3) constrained
distributions.
10. To make effective scenarios,
search for all additional
deductions in each scenario.
Milk it for everything
you can.
11. Refer back to the scenarios as
often as possible when working through the questions (for every nonelimination question).
This marks another important moment in our journey. We completed all
game types a few chapters back. Now, we have looked at all of Blueprint’s
advanced strategies for Logic Games. We’re almost done.
But there’s one more topic that we should discuss. It’s
a scary topic, but it’s important. It’s time to check out
the most terrifying and brutal games that have ever
appeared on the LSAT. It’s time for the tough stuff.
Brace yourself.
25/180GAMES THE SCARY ONES
If you want to win the title, you gotta beat the champ. The same can be said
for the LSAT. If you want to achieve an elite score, there’s a decent chance
you will have to prove your mettle against a very difficult game.
The LSAT is a temperamental beast. Once in a while, the LSAT starts to get
a bit insecure. The LSAT thinks she is losing her edge.1 Students seem to be
figuring out these games. There are rumors circulating that they are getting
easier. People stop studying very much. But the LSAT can’t let that stand. So
she unleashes her fury by unveiling a monstrosity of a game. Students weep.
New careers are chosen. Years of nightmares and psychological counseling
follow.
Okay, we are exaggerating. But they can be pretty tough.
A killer game pops up about
once per year. Good news: They
normally balance out the section with
some easier games.
BP Minotaur:
When you take the LSAT, you might not get a killer game. They don’t appear
on every test. But you might. And if you do, you need to be ready. This
chapter will help you understand the enemy and how you can defeat it.
OLD PLUS OLD EQUALS NEW
You might think there would be no way to prepare for tough games. Luckily,
that’s not the case. There is a general formula the makers of the LSAT use to
create a very difficult game. They don’t create a brand new challenge. No
calculus. No organic chemistry. No translating ancient languages. Rather,
they use the tools they already have at their disposal.
It’s pretty easy to pat yourself on the head. It’s rather simple to rub your
belly. But it is shockingly difficult to do both at the same time (and utterly
hilarious to watch). The LSAT uses this same principle to create difficult
games.
It’s hard to do two things at the same time.
That’s why combo games are difficult. In this chapter, you will find that the
LSAT takes it one step further. There are lots of features that you understand
and recognize at this point: ordering, additional variable sets, tiers,
underbooked, overbooked, In and Out grouping, stable groups, unstable
groups, profiling, categories, and on and on.
Normally, these elements are kept apart. But not always.
Difficult games on the LSAT combine elements not
normally found together in the same game.
Sounds simple enough, but the result can be very complicated. Here are a few
examples of this phenomenon from recent exams:
1. 1:1 ordering game with complicated conditional grouping relationships
2. Underbooked ordering game with players from different categories
3. In and Out grouping game with tiers
4. Profiling game with ordering elements for the groups
5. Tiered ordering game with an overbooked variable set
If you build a new kind of
setup or develop a new strategy for a
tough game, you are screwed. A ninja
never deviates from the plan. You can
still use the same approach.
Ninja Note:
Thank you, ninja. As long as you utilize the same strategies we have
practiced, you can make it through any difficult game thrown at you.
After all this practice, we think you are up for the challenge. We are going to
do two monster games. Even if done well, they will take you a long time.
And if they don’t go great, that doesn’t mean you are destined for failure. It
just means you need a bit more practice.
When you are ready, try the game on the
next page. Slay the beast!
1 We have always assumed the LSAT is a woman. Mostly because of her incredible
power. And a little bit because of her mood swings.
DECEMBER 2008: GAME 4 (17-23)
Questions 17-23
Five executives—Quinn, Rodriguez, Sasada, Taylor, and Vandercar—are
being scheduled to make site visits to three of their company’s manufacturing
plants—Farmington, Homestead, and Morningside. Each site will be visited
by at least one of the executives and each executive will visit just one site.
Each of the three site visits will take place on a different day. The schedule of
site visits must conform to the following requirements:
The Farmington visit must take place before the Homestead visit.
The Farmington visit will include only one of the executives.
The site visit that includes Quinn must take place before any site visit that
includes either Rodriguez or Taylor.
The site visit that includes Sasada cannot take place after any site visit that
includes Vandercar.
17. Which one of the following could be the executives included in
each of the site visits, with the sites listed in the order in which
they are visited?
(A) Farmington: Quinn Homestead: Rodriguez, Sasada
Morningside: Taylor, Vandercar
(B) Farmington: Quinn Homestead: Rodriguez, Vandercar
Morningside: Sasada, Taylor
(C) Farmington: Rodriguez Morningside: Quinn, Taylor
Homestead: Sasada, Vandercar
(D) Homestead: Sasada Farmington: Quinn Morningside:
Rodriguez, Taylor, Vandercar
(E) Morningside: Quinn Farmington: Rodriguez, Sasada
Homestead: Taylor, Vandercar
18. If the second of the three site visits includes both Rodriguez and
Taylor, which one of the following must be true?
(A)
(B)
(C)
(D)
(E)
The Farmington visit includes Quinn.
The Homestead visit includes Vandercar.
The Morningside visit includes Sasada.
The second of the three site visits includes Sasada.
The second of the three site visits includes exactly three of
the executives.
19. If one of the site visits includes both Quinn and Sasada, which
one of the following could be true?
(A)
(B)
(C)
(D)
(E)
The Farmington visit is the first of the three site visits.
The Homestead visit is the second of the three site visits.
One of the site visits includes only Vandercar.
The second of the three site visits includes Sasada.
The second of the three site visits includes exactly two of the
executives.
20. The executives who visit Homestead CANNOT be
(A)
(B)
(C)
(D)
(E)
Quinn and Vandercar only
Rodriguez and Taylor only
Sasada and Taylor only
Quinn, Sasada, and Vandercar
Rodriguez, Sasada, and Taylor
21. If the Morningside visit includes both Quinn and Vandercar,
which one of the following could be true?
(A) One of the site visits includes both Rodriguez and Sasada.
(B) The second of the three site visits includes exactly three of
the executives.
(C) The last of the three site visits includes exactly three of the
executives.
(D) The Homestead visit takes place earlier than the Morningside
visit.
(E) The Morningside visit takes place earlier than the Farmington
visit.
22. Which one of the following must be true?
(A) The Farmington visit takes place earlier than the Morningside
visit.
(B) The site visit that includes Vandercar takes place earlier than
the site visit that includes Rodriguez.
(C) One of the first two site visits includes Sasada.
(D) The second of the three site visits includes at least two of the
executives.
(E) At least one of the first two site visits includes only one of the
executives.
23. If the Farmington visit includes Sasada, which one of the
following must be true?
(A)
(B)
(C)
(D)
(E)
One of the site visits includes exactly three of the executives.
The last of the three site visits includes Rodriguez.
The Homestead visit includes Quinn.
The Morningside visit includes Taylor.
The site visit that includes Vandercar also includes Quinn.
BIG SHOTS
In this game, the big shots have decided to head down and visit the common
folk. A group of “executives” is making visits to the company’s
manufacturing plants. If this game was tough for you, join the club. But the
important thing is to note the similarities between this game and challenges
you have seen before.
1. Setup
Without a doubt, the biggest early obstacle in this game is the setup. When
you read through the introduction, it seems like there are two separate tasks
here. First, you have to figure out the order for the site visits. Second, you
have to group the executives together for the visits. This mode of thinking
can send you down a terrible path.
I just made two separate
setups. One for the ordering part and
one for the grouping part.
Ditz McGee:
That’s exactly what you have to avoid. If you ever catch yourself making two
setups for a game, please stop. Just put the pencil down and walk away. It
won’t go well from there.
Here’s another common mistake committed by students. They make a
setup using the sites as the base. While this allows you
to figure out the assignment of executives to sites, it
totally m: overlooks the ordering element of the game.
So the challenge is to create a setup that reflects all of this
information. Why is this setup complicated? Because the executives are
overbooked. Five executives are assigned to three site visits. If there were
only three executives, or if there were five site visits, this would be easier.
We would build a tiered setup with one tier for the sites and a second for the
executives.
Don’t change that setup. This is a
tiered ordering game with an
overbooked variable set
(executives). This additional feature
complicates the game and confuses
students. If you build the wrong setup,
the deductions are difficult to spot.
This setup will allow us to visualize all of the rules.
Since the executives are overbooked, it’s also important to
play the numbers. Each site visit is assigned at least one
executive. So there are two options: (1) three executives
visit one site and only one executive visits the other two
sites, or (2) two executives visit two sites and only one executive visits the
third.
2. Rules
With the proper setup, the rules are very helpful and easy to visualize.
The first rule gives an ordering principle:
Farmington before Homestead.
The second rule tells you that only one
executive visits Farmington. This interfaces
with our distributions, and Farmington is
going to be important.
According to the third rule, Quinn is before
Rodriguez and Taylor. Since there are only 3.
three site visits, this is big.
Be careful with the fourth rule. Sasada cannot
go after Vandercar. But Sasada doesn’t have to go in front of Vandercar.
Some site visits will have more than one executive, so they could be
together.
The most important part of this game, as you might expect, comes next.
3. Deductions
There are only three site visits in this game. That’s a huge advantage for you.
It’s easy to find constraint when you are only ordering three variables. It
turns out that the first rule places a huge constraint on Farmington - it must be
the first or second site visit. This should make you think about scenarios. The
next rule also involves Farmington, so the scenarios are likely to uncover
more deductions. Guess what’s next?
It’s sometimes wise to break a game into
scenarios as you work through the rules. As soon as you
read the first two rules, you know that the placement of
Farmington will be crucial. Go ahead and make scenarios
before working through the other rules.
Ninja Note:
Scenario #1: Farmington is first.
Homestead and Morningside
should form linked options
for the second and third
visits.
The first visit only gets one executive.
Rodriguez and Taylor have to follow Quinn, so they can’t go to the first
site visit.
Vandercar can’t attend an earlier site visit than Sasada. Since only one
executive attends the first site visit, that executive can’t be Vandercar.
Quinn or Sasada are the only options for the first site visit. That’s a big
deduction to uncover in this scenario.
Scenario #2: Farmington is second.
Homestead is the third visit and Morningside is the
first visit.
Only one executive attends the second site visit.
Quinn has to precede Rodriguez and Taylor, so
Quinn can’t attend the third site visit, and neither
Rodriguez nor Taylor can attend the first visit.
The final rule doesn’t furnish you with any restrictions. Vandercar could
go to the first site visit as long as Sasada also does. And Sasada could go
to the third site visit as long as Vandercar tags along.
We are in good shape. The big turning point was building the proper setup.
Without a good setup, it’s difficult to spot the scenarios and find the
deductions.
4. Questions
Question #17 (elimination, could be true)
Farmington must be visited before Homestead, which kills (D).
Only one lucky executive goes to the Farmington visit. In (E), both
Rodriguez and Sasada are scheduled for Farmington. Get rid of it.
Quinn must precede both Rodriguez and Taylor, which kicks out (C).
Vandercar can’t go to a visit before Sasada, so (B) is the last one out.
(A) is the correct answer.
Question #18 (conditional, must be true)
In scenario 2, the second visit only includes one
executive. So both Rodriguez and Taylor can
only attend the second site visit in scenario 1.
Quinn must attend the first site visit.
Someone has to go to the third site visit. Sasada
and Vandercar are the only executives left. Vandercar can’t precede
Sasada as part of the second site visit, so Vandercar must attend the
third site visit (with or without Sasada).
Quinn attends the Farmington visit, so (A) is correct.
Question #19 (conditional, could be true)
In scenario 1, either Quinn or Sasada (but not both) attends the first site
visit. Thus, they can’t attend the same site visit. So Quinn and Sasada
can only attend the same site visit in scenario 2.
The second site visit includes only one
executive, and Quinn can’t go to the third site
visit, so Quinn and Sasada attend the first site
visit.
The three remaining executives are Rodriguez,
Taylor, and Vandercar. Rodriguez and Taylor could attend the second or
third site visit, but only one executive attends the second visit.
Vandercar could attend any of the three site visits.
If both Rodriguez and Taylor attend the third site visit, the second
site visit could include only Vandercar. (C) is the winner.
Question #20 (absolute, must be false)
This is a tough question even with the aid of our scenarios.
(A) In scenario 1, Quinn and Vandercar could be the only executives
who visit Homestead. Homestead must be the second site visit, and
Morningside must be the third. Sasada goes to Farmington, and
both Rodriguez and Taylor go to Morningside.
(B) In either scenario, Rodriguez and Taylor could be the only
executives who visit Homestead (see hypothetical for question
#19).
(C) In either scenario, Sasada and Taylor could be the only executives
to visit Homestead.
(D) In scenario 1, either Quinn or Sasada visits Farmington. In
scenario 2, Quinn can’t visit Homestead. So there is no way for
both Quinn and Sasada to visit Homestead, let alone the two of
them with Vandercar.
(E) As long as Homestead is the second visit and Vandercar goes to
Morningside as the third visit, then Rodriguez, Sasada, and Taylor
could all visit Homestead in scenario 1.
Question #21 (conditional, could be true)
The Morningside visit could include both
Quinn and Vandercar in either scenario, so
you have to work through both of them.
In scenario 1, the Morningside visit would
have to be the second visit.
Since Quinn doesn’t go to the first site visit
(Farmington), Sasada must.
Rodriguez and Taylor must follow Quinn, so
they both attend the third visit.
In scenario 2, both Quinn and Vandercar
attend the first site visit.
Vandercar can’t precede Sasada, so Sasada joins them for the first visit.
At least one executive must attend the second and third site visits, so
you can form linked options for Rodriguez and Taylor.
In scenario 2, the Morningside visit does take place earlier than the
Farmington visit. (E) could be true.
Question #22 (absolute, must be true)
(A) The Farmington visit must take place earlier than the Morningside
visit in scenario 1 but not in scenario 2.
(B) Remember to use your work on previous questions. In question
#18, Rodriguez goes to the second visit and Vandercar goes to the
third, so Vandercar need not be earlier than Rodriguez.
(C) The same question comes to our rescue again. In question #18,
Sasada could go to the second site visit or the third. So this answer
could be false.
(D) Nope. In scenario 2, the second site visit includes only one
executive.
(E) They make you wait for it, but they give you an answer that
flows easily from the scenarios. Since Farmington is the first or
second site visit and only one executive visits Farmington, only
one executive visits one of the first two sites.
Geez... too many
questions on this one.
Cleetus Comment:
Yep, we have to agree. When this game appeared on the LSAT, not only was
it supremely difficult, but there were also a lot of points on the line.
Question #23 (conditional, must be true)
The Farmington visit could include Sasada
in both scenarios. So you have to work
through two hypotheticals again.
In scenario 1, Sasada goes to the first site
visit.
Quinn has to do a site visit before Rodriguez
and Taylor, so Quinn goes to the second site
visit, and Rodriguez and Taylor go to the
third.
Vandercar could attend the second or third
site visit.
In scenario 2, Sasada goes to the second site visit.
No other executives go to the second site visit.
Quinn still has to go to a site visit before Rodriguez and Taylor, so
Quinn goes to the first site visit, and both Rodriguez and Taylor go to
the third site visit.
Vandercar can’t go to a site visit before Sasada, so Vandercar must also
go to the third site visit.
In both scenarios, Rodriguez goes to the third site visit. Thus, (B) is
the final answer to this wonderful game.
If you struggled on that game, remember this chapter is devoted to the most
difficult logic games. Don’t overreact. If you lost a cooking competition to
Wolfgang Puck, would you give up on ever making a delicious chicken
piccata? Of course not. Most games are much easier.
It’s important to understand why this game is
difficult. We worked through tiered ordering games
many chapters ago. But nearly all tiered ordering
games have 1:1 correspondence. This game is
complicated because one of the variable sets is
overbooked. There are five executives for three site
visits. That sends people into a tailspin. But as long
as you utilize the tools covered earlier, you can beat a game like this one.
This game also featured seven questions, which is the most you will ever
see on a game. This game is very time-consuming even when done
perfectly.
It’s time for one more monster game. On the next page, you will find one of
the most infamous games in the history of the LSAT. If you ever meet
someone who took the test in June of 2009, don’t mention mauve dinosaurs.
Don’t even mention dinosaurs at all. In fact, you might want to avoid all
discussions of the Mesozoic era.
This one is all on you. Keep an eye out for the important elements in this
game. Use the proper strategies, work slowly through the rules, and search
for deductions. Good luck...
JUNE 2009: GAME 3 (12-17)
Questions 12–17
Each of seven toy dinosaurs—an iguanadon, a lambeosaur, a plateosaur, a
stegosaur, a tyrannosaur, an ultrasaur, and a velociraptor—is completely
colored either green, mauve, red, or yellow. A display is to consist entirely of
exactly five of these toys. The display must meet the following
specifications:
Exactly two mauve toys are included.
The stegosaur is red and is included.
The iguanadon is included only if it is green.
The plateosaur is included only if it is yellow.
The velociraptor is included only if the ultrasaur is not. If both the
lambeosaur and the ultrasaur are included, at least one of them is not
mauve.
12. Which one of the following could be the toys included in the
display?
(A) the lambeosaur, the plateosaur, the stegosaur, the ultrasaur,
the velociraptor
(B) the lambeosaur, the plateosaur, the stegosaur, the
tyrannosaur, the ultrasaur
(C) the iguanadon, the lambeosaur, the plateosaur, the stegosaur,
the ultrasaur
(D) the iguanadon, the lambeosaur, the plateosaur, the
tyrannosaur, the velociraptor
(E) the iguanadon, the lambeosaur, the stegosaur, the ultrasaur,
the velociraptor
13. If the tyrannosaur is not included in the display, then the display
must contain each of the following EXCEPT:
(A)
(B)
(C)
(D)
(E)
a green iguanadon
a mauve velociraptor
a mauve lambeosaur
a mauve ultrasaur
a yellow plateosaur
14. Which one of the following is a pair of toys that could be included
in the display together?
(A)
(B)
(C)
(D)
(E)
a green lambeosaur and a mauve velociraptor
a green lambeosaur and a yellow tyrannosaur
a green lambeosaur and a yellow ultrasaur
a yellow tyrannosaur and a green ultrasaur
a yellow tyrannosaur and a red velociraptor
15. If the display includes a yellow tyrannosaur, then which one of
the following must be true?
(A)
(B)
(C)
(D)
(E)
The iguanadon is included in the display.
The plateosaur is not included in the display.
The display includes two yellow toy dinosaurs.
The display contains a green lambeosaur.
The display contains a mauve velociraptor.
16. If both the iguanadon and the ultrasaur are included in the display,
then the display must contain which one of the following?
(A)
(B)
(C)
(D)
(E)
a mauve tyrannosaur
a mauve ultrasaur
a yellow lambeosaur
a yellow plateosaur
a yellow ultrasaur
17. If the display includes two green toys, then which one of the
following could be true?
(A) There is exactly one yellow toy included in the display.
(B) The tyrannosaur is included in the display and it is green.
(C) Neither the lambeosaur nor the velociraptor is included in the
display.
(D) Neither the tyrannosaur nor the velociraptor is included in the
display.
(E) Neither the ultrasaur nor the velociraptor is included in the
display.
MAUVE DINOSAURS?
You have now experienced the pain felt by so many in the middle of 2009. It
doesn’t sound so terrible - everyone loves dinosaurs. But this game quickly
reveals an evil side. Regardless of how you fared, there are important lessons
to be learned.1 This game is a synthesis of many topics discussed in previous
chapters.
1. Setup
When you read the introduction, it’s crucial to search for the basic challenge.
There are seven dinosaurs.2 The task is to select five of them for a display.
Since you are selecting some of the dinosaurs, this is an In and Out
grouping game.
But that’s not all. Each of the
prehistoric beasts is a different
color. That introduces another
variable set into the equation.
How do you track a second
variable set? You build a second
tier. This a tiered grouping game.
Your setup needs an In group and
an Out group for the dinosaurs. In
addition, the In group needs a second tier to track the colors.
BP Minotaur: This game follows the same formula for
creating a difficult game. Tiers are normally found in
ordering games, but here a second tier is used to
complicate a grouping game.
The first challenge is building the correct setup. Most grouping games only
involve one variable set. Going forward, you should expect two things: (1)
prevalent grouping relationships, and (2) restrictions about the colors of the
dinosaurs that are selected.
2. Rules
The rules are another hurdle. It’s not always what the rules say, but rather
what they don’t say. In this game, it’s important to test the implications of
each rule and work slowly.
The first rule tells you that two mauve dinosaurs are selected. So two of
the five have to be mauve - that’s huge. Plug two mauves directly into
the setup.
This rule spawned the “mauve
dinosaurs” nickname for this game.
The whole game revolves around these
two mauve slots. When you work
through the rest of the rules, you have
to keep a checklist of the dinosaurs that
can’t be mauve.
Ninja Note:
The second rule is sweet - it fills another slot
for you. The stegosaur is selected and is red.
(That means it’s not mauve.)
Make sure to diagram the third and fourth rules
correctly. “Only if” introduces a necessary
condition. If the iguanadon is included, it must
be green. If a dinosaur is not green, then it’s
not the iguanadon. Also, if the plateosaur is
included, it must be yellow. If a dinosaur is not
yellow, it’s not the plateosaur. (That’s two
more dinosaurs that aren’t mauve. The list is
growing.)
The fifth rule gives a big grouping relationship. The velociraptor and the
ultrasaur can’t both be included.
The final rule is tricky - it should be diagrammed as a conditional. If
both the lambeosaur and the ultrasaur are included, then at least one of
them is not mauve. (Did someone say not mauve?)
Understandably, many students suffer from information overload right about
now. There’s a complicated setup and a slew of tough rules. It’s hard to see
where to go from here, so students jump into the questions. But that is
guaranteed self-destruction.
The only way through this game is to notice all of the
restrictions on the two mauve dinosaurs.
The first rule tells you there are two mauve dinosaurs. Many of the
subsequent rules directly or indirectly tell you various dinosaurs can’t be
mauve. If you notice that, it’s time to investigate...
3. Deductions
Two mauve dinosaurs are selected. Let’s make a quick list of the dinosaurs
that can and can’t be mauve.
The stegosaur is red (close to
mauve, but not quite the same).
The iguanadon can only be green.
The plateosaur can only be
yellow.
Either the lambeosaur or the ultrasaur can be mauve, but not both. Thus,
either the lambeosaur or the ultrasaur is ruled out as well.
This only leaves three options for the two mauve dinosaurs: the
tyrannosaur, the velociraptor, and either the ultrasaur or the
lambeosaur.
This is a huge moment. If you spot this deduction, you have significantly
improved your chances of success. However, you can go a bit further.
Choosing two out of three is a
common way to form scenarios. Watch
out for this type of deduction in all
types of games.
Ninja Note:
As soon as you recognize that the mauve slots are so constrained, it’s time to
think about scenarios. Officially, there are four dinosaurs that could be
mauve. But since the lambeosaur and the ultrasaur can’t both be mauve, there
aren’t many options.
The velociraptor and the ultrasaur can’t both be
included in the display.
There are only four options for the two mauve
dinosaurs: the tyrannosaur and the velociraptor; the
tyrannosaur and the lambeosaur; the tyrannosaur and
the ultrasaur; and the velociraptor and the lambeosaur.
Should you do scenarios? There are four clear scenarios for the game based
on the constrained mauve slots. Also, it’s very likely that further deductions
will come from working through these scenarios. That’s a long-winded way
of saying yes.
Scenario #1: The tyrannosaur and the velociraptor are
mauve.
Since the velociraptor is included,
the ultrasaur can’t be included.
Any two of the lambeosaur, the
iguanadon, and the plateosaur could
fill the last two spots.
Scenario #2: The tyrannosaur and the lambeosaur are
mauve.
The ultrasaur and the velociraptor
can’t both be included, so at least
one of them must be out.
Scenario #3: The tyrannosaur and the ultrasaur are
mauve.
Since the ultrasaur is included, the
velociraptor can’t be included.
Any two of the iguanadon, the
lambeosaur, and the plateosaur could
fill the last two spots.
Scenario #4: The velociraptor and the lambeosaur are
mauve.
The velociraptor is included, so the
ultrasaur is out.
The last two spots in the display
could be the iguanadon, the
plateosaur, or the tyrannosaur.
Four scenarios is a good deal of work, but you can work through them
quickly. Although we weren’t able to completely determine any of the four
scenarios, the remaining slots are easy to visualize. And, as you are about to
see, this gives us a huge head start on the questions.
1 First lesson: Mauve is an actual color. It’s a reddish-purple. We didn’t know either
until this game appeared.
2 Can we take a moment to discuss this ridiculous assortment of dinosaurs? An
iguanadon? An ultrasaur? It’s like they never went to kindergarten or saw Jurassic
Park. Can we buy a triceratops, please?
4. Questions
Question #12 (elimination, could be true)
The stegosaur is always included. Say no to (D).
The velociraptor and the ultrasaur can’t both be included. This kills (A)
and (E).
To satisfy the condition that two mauve dinosaurs are included, either
the tyrannosaur or the velociraptor (or both) must be included. (C) is out
because it has neither. In (C), only the lambeosaur or the ultrasaur (but
not both) could be mauve.
(B) is the champ.
Question #13 (conditional, could be false)
The tyrannosaur can only be out in
scenario 4.
If the tyrannosaur is out, then both
the iguanadon and the plateosaur
must be included.
The iguanadon must be green.
The plateosaur must be yellow.
There can’t be a mauve ultrasaur. (D) is the answer.
(D) must be false. Even though this is only a
could be false question, remember that a must be false
answer is acceptable.
BP Minotaur:
Question #14 (absolute, could be true)
This can be a very tricky and time-consuming
question. But it all comes back to the mauve
dinosaurs. If you keep that in mind, there is a common
theme to all of the incorrect answers.
(A) In scenario 1, a mauve velociraptor is included. A lambeosaur
could also be included and it could be any color, including a
nice shade of green. (A) could be true; we find the correct
answer quickly.
(B) A non-mauve tyrannosaur can only be included in scenario 4. But
in that case, the lambeosaur must be mauve.
(C) A yellow ultrasaur can only be included in scenario 2. But there’s
that damn mauve lambeosaur again. No good.
(D) The only time that you can have the joy of a non-mauve
tyrannosaur is in scenario 4, and you can’t have an ultrasaur at all.
(E) Another non-mauve tyrannosaur? Are you seeing a pattern here?
The yellow tyrannosaur points to scenario 4, but in that scenario
you must have a mauve (not red) velociraptor.
Question #15 (conditional, must be true)
A yellow tyrannosaur can only be
included in scenario 4.
In scenario 4, the display includes
a mauve velociraptor. Sounds
scary, but it makes (E) the right
answer.
Question #16 (conditional, must be true)
The iguanadon could be included in any of the four scenarios, but the
ultrasaur is more restricted. The ultrasaur is included in scenario 3 and it
could be included in scenario 2.
In both scenario 2 and scenario 3, the big, scary mauve tyrannosaur
is included. (A) looks great.
Question #17 (conditional, could be true)
The display could include two green
toys in any of the four scenarios.
Since no yellow toys are included, the
plateosaur must be Out.
The ultrasaur and the velociraptor
can’t both be selected, so the other spot in the Out group must be either
the ultrasaur or the velociraptor.
The tyrannosaur, the lambeosaur, and the iguanadon must be included.
The iguana don must be one of the green toys.
The velociraptor and the lambeosaur could be the two mauve
dinosaurs (scenario 4). If that occurs, the tyrannosaur could be the
second green dinosaur. (B) is correct.
Wow - that was quite a ride. The mauve dinosaurs game can be a lifechanging event. Here are some important lessons to take away from that
game:
Don’t hate the mauve dinosaurs. Learn from the mauve
dinosaurs.
1. The same scary-game recipe was used to create that prehistoric
nightmare. The LSAT takes two common games elements (In and
Out grouping; tiers) and throws them together unexpectedly.
Students freak out. The test wins.
2. If you don’t want to let them win, you have to stick with the same
strategies. For In and Out grouping games, the big grouping
relationships reign supreme. They were very helpful with the dinos.
In tiered games, scenarios are commonly found since there is so
much information to handle. Not surprisingly, scenarios were
crucial in this game.
3. Work slowly through the rules and test the implications of each one.
In this game, the big deductions about the mauve dinosaurs can be
found only if you see past simply what the rules say. It didn’t matter
that the iguanadon was green - it mattered that it wasn’t mauve.
4. If you ever have three options for two spots, think about making
scenarios. This is a great opportunity to form three scenarios.
This chapter is devoted to the worst games that you could encounter on the
LSAT. Even though it’s not a very fun experience, it’s always good to
practice against the best. You aren’t going to run into any games that are
more difficult than those.
Believe it or not, that was our last game together. Our journey is complete.
First, we learned the basic skills needed for games. Then, we worked through
ordering games, grouping games, and lovely combo games. After that, we
discussed some advanced strategies to take your game to the next level.
Finally, we looked at the most difficult games on the test.
It’s almost time to set you out into the world on your own. But there are a
few topics we would like to discuss first.
26/TIMING
FASTER, FASTER...
Now we’re going to discuss some testing strategies. To complete the Logic
Games section, you have a whopping 35 minutes. That’s not a lot of time.
That’s one rerun of Seinfeld with a snack break. In Los Angeles, that’s the
time it takes to complete a 1.7-mile commute. Shoot, you can waste that
much time choosing a filter for your latest Instagram.
There are two important parts of doing well on Logic Games: (1) You
have to be good, and (2) you have to be fast. We have spent the last 25
chapters discussing how to be good, but the second issue deserves some
discussion as well. Obviously, these two issues are related. As you
improve your skills, you will naturally get faster. But there are some other
tips we would like to share.
The best way to improve
your speed is to focus on deductions.
Rather than trying to rush through the
questions, deductions allow you to
spend less time on the questions.
BP Minotaur:
GAMES ARE NOT CREATED EQUAL
Your job is to complete four games in 35 minutes. With the aid of a fancy
calculating machine, we have discovered this leaves you with an average of
8:45 per game. For this reason, many students think success in a games
section looks like this:
Game 1
Game 2
Game 3
Game 4
8:45
8:45
8:45
8:45
This is actually a terrible plan. Not all games are created equal, and it’s
dangerous to think that they all should take the same amount of time.
The amount of time that should be allotted to a game is
influenced by two factors: (1) the difficulty of the
game, and (2) the number of questions it contains.
Turn the page for some interesting facts about the games section.
1. The games at the end of the section are more difficult.
There’s a clear pattern that emerges when you look at the difficulty of games
across the section. Put simply, they get progressively more difficult. Since
you are mentally fatigued and feeling the time crunch at the end of the
section, this is a troubling fact.
Blueprint rated the difficulty of the every game since 1991. The scale
ranged from 1 (your dog would have a decent shot at solving the game) to
5 (Einstein would have needed some help). Here are some of the findings:
The average for the first game is
pretty low - only 2.03. The second
game jumps up significantly to
2.98. The third and fourth games
are rated 3.38 and 3.55,
respectively. So they are more
difficult than the first two, but
pretty similar to each other.
The medians show the same
pattern. The median for the first
game is 2, the second game is 3, the
third game is also 3, and the fourth game is 4.
There are some outliers. All four games, including the fourth, had at
least one game rated as a 1. Also, all four games had at least one game
rated as a 5 (even the first one).
This graph should teach you two important lessons: (1) games get
progressively more difficult as you work through a section, and (2) the games
in the beginning of the section should take significantly less time than those
at the end.
2. Some types of games are harder than others.
During this adventure, you might have had more trouble with certain game
types. That wasn’t a coincidence. Some types of games tend to be on the
easier side. Other types of games are generally more difficult.
On the next page, you can see a chart that breaks down the relative difficulty
for the game types we have discussed. This is another factor that affects the
amount of time a game should take.
* Here are the real outliers. You gotta like your
chances.
Game Type
Avg.
Difficulty
Basic
Ordering
1.50
1:1 Ordering
1.81
Underbooked
Ordering
3.21
Overbooked
Ordering
3.47
Tiered
Ordering
3.00
In and Out
Grouping
3.64
Stable
Grouping
3.35
Unstable
Grouping
3.43
Profiling
3.45
Combo
3.63
Characteristic
Grid
2.33
Weird
Neither
Games
3.29
}
Here are the real outliers. You
gotta like your chances.
It’s interesting to see how close many of the difficulty ratings are to each
other. Most of the common categories range between 3.00 and 3.64. But there
are the two exceptions:
Basic ordering games (1.50) and 1:1 ordering games (1.81) tend to be
much easier than other game types.
You are likely to find simpler ordering games early in the section. When you
attack one of these games, it should be possible to complete it in as little as
five or six minutes.
There are twists in all game types, so you can’t perfectly diagnose the
difficulty of a game until you are in the thick of it. But these general
guidelines can help you shape your strategy. Now, let’s discuss some tips for
how to improve your timing.
QUESTIONS
The Blueprint strategy is largely designed around what happens before you
hit the questions. We urge you to search for deductions and slowly work
through the setup and rules. But once you complete those steps, you can
definitely save time when working through the questions. Here are some
strategies:
Don’t read all of the answers. This is a tough one. Everyone likes to
feel comfortable. You don’t just want to know that the right answer is
right, you also want to make sure that the wrong answers are wrong.
Unfortunately, that wastes valuable minutes. Suppose you are doing a
must be true question. If you read (A) and can see that it must be true,
don’t waste time checking that the other answers could be false.
Use elimination questions to your advantage. There are two things
you know about elimination questions. First, they are super easy. Don’t
ever leave an elimination question blank, even if you skip the rest of the
game. It’s a free point. Second, they give you a good chance to visualize
a game. If you want to gauge the difficulty of a game before fully
committing to it, just do the elimination question. You can answer the
question by simply reading the introduction and the rules. You don’t
even need to build a setup.
DEVELOP A GAME PLAN
To maximize your performance on games, you need to have a plan. Here’s
the dangerous mindset of most students walking into the LSAT:
“Boy, I sure hate those games. But I’ll give it my best.
I’ll work hard and try to finish as many as I can. And,
above all, I’m hoping for easy games.”
That’s not a plan. That’s just hoping for the best. Would you walk into an
important job interview without a plan? Would you go on a hot date without
thinking through a plan for the evening? Of course not. Logic Games should
be no different.
Before we discuss various plans for the section, we need to discuss
something. It’s a brutal truth that students don’t want to hear. For many
students, it’s a good idea to plan on skipping at least one of the games.
Frantically rushing through four games with low accuracy can be very
detrimental to your overall score.
Completing less games with
better accuracy can improve your
overall score. Rushing to do more can
be stressful and counterproductive.
BP Minotaur:
You don’t need to decide on a strategy right now. It’s too early for that.
You will continue to improve as you practice more games. But as the test
approaches, it’s vital to develop a game plan you are going to use to attack
the section. Here are three strategies that work well for students:
Plan 1: Push for four
If you are capable of completing all four games, that’s great. We are very
proud. But you still need to have a plan for managing your time
appropriately. If you spend too much time on the early games, you won’t
have enough time to complete the later, more difficult games.
Here is a rough idea of how the section should look:
Game 1
Game 2
Game 3
Game 4
7:00
8:00
10:00
10:00
The goal is to complete the first half of the section in roughly 15 minutes.
Push yourself to move quickly in order to reduce the stress and mistakes on
later games.
You don’t have to complete the four games in order. Odds are that the first
two games will be easier, and you have to play the odds. So always complete
the first two games. There could be a brutal game waiting for you as either
the third or fourth game. It’s great to preview both games and attack the
easier one first. Do the elimination questions for both the third and fourth
games. You don’t have to build a setup - just read through the introduction
and the rules. Choose the game that seems easier and attack that one first.
Plan 2: Aim for three
For many students, the best plan is to attempt to complete three games. It’s
important to do this in the right way. You always want to do the first game,
but you don’t always want to do the first three games. After you finish the
first game, do the elimination question for the second game. Unless it looks
like cruel and unusual punishment, work through the second game. Then, do
the elimination questions for both the third and fourth games and pick the one
that seems easier. Skip the other game altogether and know that you made the
right decision.
But there’s no way I can get a
good score if I skip a whole game. This
is crazy.
Ditz McGee:
Actually, you are mistaken. Let’s use an example. Assume that a games
section has 23 questions (five on the first game and six on the others). Betty
is an LSAT student. She can solve most games, but her accuracy suffers
when she is pressed for time. Here are two ways it could play out for Betty:
Option 1: Betty tries to rush through all four games, but her accuracy suffers.
She gets 100% on the first game but only averages 50% on the final three
games. Correct answers: 14
Option 2: Betty focuses on accuracy and slows down a bit. She still hits
100% on the first game. Betty only misses one question on the second game
and one on the third game. She skips the fourth game, but she gets the
elimination question and guesses correctly on one other question. Correct
answers: 17
Option 2 is clearly better for Betty, and it might be better for you.
Plan 3: Ace two
Some students simply can’t get through games quickly, no matter how much
they try. They can build the proper setup, represent the rules, spot deductions,
and find most of the correct answers. It just takes a while. When they try to
rush, everything falls apart.
If this happens to be you, you should aim to complete two games perfectly.
Complete the first game and then choose between the other three games.
Remember to answer the elimination questions for the remaining games.
Let’s see how Betty would fare using this strategy.
Option 3: Betty spends about 15 minutes and gets 100% on the first game.
She chooses to attempt the third game and only misses one question. Betty
completes the elimination questions on the second and fourth games and gets
two lucky guesses. Correct answers: 14
(Note that Betty’s score using this strategy is the exact same as her score
when she rushed through the section with lower accuracy.)
A true ninja learns through
trial and error. As the test approaches,
try these different strategies and
determine which one works best for
you.
Ninja Note:
Even though this chapter is devoted to timing, don’t incorporate timing into
your studies prematurely. While you are learning the techniques, it’s
important to work slowly and focus on the details. You need to develop the
correct approach; then you can work on speeding up.
As you progress, slowly introduce timing pressure into your practice. Finally,
as test day approaches, settle on a personal strategy for the section and
practice using that strategy repeatedly.
With proper techniques and a good plan, you can
accomplish great things in 35 minutes.
27/NEXT...
PERFECT PRACTICE MAKES PERFECT
The training wheels are now coming off. You have blossomed into a
beautiful butterfly and it’s time to set you out into the world. Okay - enough
cheesy metaphors.
What should you do now? You need more practice. But practice doesn’t make
perfect. Perfect practice makes perfect. In this book, we only use the real deal
- games from past tests. You can find replica games, but they tend to suck.
Make sure to only use 100% grade-A, genuine LSAT games when you study.
You can buy books filled with real exams. They are
available directly through LSAC or through small
websites like Amazon. All of the PrepTests are
helpful, but focus on the recent tests since the games
(and the rest of the material) will be more
representative of what you will see on game day.
When you continue to practice, don’t just start
taking full practice tests. There are a few stages you
should work through to maximize your score.
Here are a couple pointers for each stage:
Stage 1: Individual Game Types
In this book, we focus on individual types of ordering, grouping, and combo
games. You should continue to do the same on your own. By repeatedly
doing the same type of game, you will become more comfortable with the
important features and common deductions.
Of course, it would be annoying to search through endless tests to find
another In and Out grouping game to practice. So we’re here to help.
The appendix at the end of this book characterizes the
game type and important features of every game dating
back to PrepTest 1. It also includes a difficulty rating
for each game, and it tells you whether you should
make scenarios and/or play the numbers.
Don’t spend the same amount of practice time on all game types. Mapping
games do not require as much attention as tiered ordering games. You have to
play the odds. Spend your time mastering the types of games you are most
likely to see. Here’s a chart that shows the prevalence of each game type
dating to 1991 (and since 2004 for more recent numbers).
As you can see, certain game types are much more common than others. It’s
no coincidence that a large portion of this book focused on the most common
ones.
Here are more details:
1:1 ordering games are the most common game type. You are nearly
assured that one will pop up on your LSAT.
Underbooked ordering, overbooked ordering, tiered ordering, In and
Out grouping, and stable grouping games are also very common.
The games section has been less varied recently. Since 2004, common
game types have popped up more, and rare game types have nearly
disappeared.
Stage 2: Full Sections
When you feel comfortable with individual game types, it’s time to attack
entire sections. The new challenge will be jumping between game types.
Some sections will include four different types. Warning: Do not time
yourself yet. You can’t immediately go from 0 to 60. Give yourself plenty of
time to complete the section and search for deductions.
Stage 3: Timed Sections
When you feel comfortable working through sections, then it’s time to apply
the timing pressure. However, don’t think that you have to immediately crank
it to 35 minutes. It can be helpful to begin by giving yourself some extra time
(45 or 50 minutes is a good starting point) and then cutting the time down as
you get more comfortable.
Well, that’s all from us at Blueprint. After over 500 pages, there’s literally
nothing left to say about Logic Games. Except that we hope you kick some
LSAT butt on test day.
We hope you enjoyed the journey. At Blueprint, we strive to provide the
absolute best LSAT preparation that has ever been created. Hopefully, you
think we fulfilled that mission.
For more information about the LSAT, and for more wonderful resources,
just stop by our website. Best of luck from the Blueprint team. Sayonara.
www.blueprintlsat.com
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APPENDIX
PT = PrepTest; SC = Scenarios; PTN = Play the Numbers;
Diff = Difficulty (1 - 5)
(Blueprint favorites are highlighted. If you want the
best practice, focus on these games.)