Graduate Student Outreach Project
Puzzles and Games of Chance Explained
I.
Abstract
Puzzles are fascinating games of mind that intrigue and equally entertain kids and wise people. Historically,
the study of puzzles initiated some of the most fundamental mathematical theories. To get insight into these
complex theories one can start by studying puzzles. By understanding puzzles and the way people solve
them, one comprehends the motivation of formal study of probability, graph theory, recursion and logical
reasoning.
The goal of this course is to present a selection of some of the most famous puzzles and to challenge the
students. Solutions are built interactively and principles of probability and graph theory are explained by
means of example. The aim is to motivate deep mathematical concepts by playing games of mind.
II.
Course Description
Introduction
This course is intended to motivate abstract thinking while playing games of mind. The focus is on
understanding probability and graph theory by means of challenging and entertaining examples. A more
rigorous presentation of the theory most solutions rely on should make the subject of a follow-up class.
Several games of chance are discussed, such as playing cards, the Monty Hall puzzle, figuring odds of
winning the lottery. Principles of graph theory are explained while solving the 7 bridge problem, finding
shortest routes from a departure city to a holiday resort, or organizing a tour of Paris without visiting the
same place twice. The difference between paradox and mere coincidence is illustrated by the birthday
problem. Payoff games are played giving everybody a chance to win, whether optimist, pessimist, or
somewhere in between. Basic problem solving techniques, such as recursion, are acquired while solving the
towers of Hanoi problem and its bicolor version.
Goals
More specifically, puzzles were selected in an attempt to reach the following goals:
a. practice logical reasoning
b. understand the concept of random choice and correctly compute probabilities of different events
b. understand the difference between independent and correlated events
c. recognize hidden assumptions and extra information
d. think about expected outcomes as opposed to particular outcomes
e. be familiar with generalization techniques: extrapolating a result easily proved for a simple case
to a more general case
f. understand how to model a problem in terms of graphs and the motivation for graph theory
g. realize when two apparently different problems are actually special cases of the same problem and
how in some cases the similarity becomes clear when problems are modeled using graph theory
h. compute expected wins or losses for payoff games and find best strategies depending on your
personality: optimist, pessimist or somewhere in between.
Suggestions for teachers
This mini-course is suited for high-school students as it builds on basic understanding of probability and
requires some familiarity with more abstract modeling of a problem and generalizations techniques.
However, the course is designed to be entertaining and gradually familiarize students with more abstract
concepts by playing puzzles. The course stresses the importance of logical thinking and I believe that if the
accent is placed on presenting puzzles and less on graph theory as a tool for solving more intricate puzzles,
the material is also accessible to students in grades 6-8.
III.
Graduate Student Biography
My name is Sabina Petride and I grew up in Bucharest, Romania. There I earned a Bs. degree in
Mathematics and Computer Science, and I am currently a Ph. D. student in the Computer Science
department, Cornell University. I am interested mainly in logical reasoning and applications. I believe this
is mostly because I like puzzles and I had the chance to take good courses as a student.
I was a teaching assistant for several courses here at Cornell University and I always enjoyed
communicating with students. I believe one of the best ways to learn something is without even realizing
you are studying, and this why I find teaching about puzzles very entertaining and rewarding.
IV.
Individual Session Descriptions
Session 1: Playing 20 Questions
Learning outcomes
Upon completion of this session, students will:
1. review the concept of probability and chance
2. be reminded simple laws of probability
3. be familiar with simple probability computations
4. understand the difference between rare events and likely events and the concept of expected
outcome
5. understand how to generalize their answers and apply problem solving techniques used for simple
problems to new puzzles
6. have a better understanding of general problems that involve minimizing the number of steps or
questions needed for finding a solution.
Duration: 1 hour
Activities and procedures
Start with the following puzzle:
I pick at random one number out of 1, 2,…, 10. You have to guess it by asking me questions of the
form: Is the number 3? Or Is the number greater than 6? I can only answer “Yes” or “No” and I always
say the truth.
If you are lucky, you can guess it with one question; but this happens very rarely. You want to make
sure that, on average, you ask as few questions as possible to find the number I picked.
What questions should you ask? How many questions do you have to ask?
How many questions you have to ask if the number is any of 1, 2, …, 1 million?
Present a similar puzzle involving cards this time:
I shuffle two packs of cards and pick 1 ace, 2 deuces, 3 threes, 4 queens, 5 kings and 6 tens.
I hide from you and pick one card of these 21 cards. You have to guess what card I picked: an ace, a
deuce, a three, a queen, a king, or a ten? You can only ask questions of the form: “Is the card either a
two or a queen or a king?” and I can only answer “Yes” or “No”. You can not ask more than 20
questions!
I never lie!!!
Which questions should you ask so that you guess the card I picked asking as few questions as
possible? Think that if you ask too many questions in one or two games is ok, but if we play this game
many times you should ask as few questions as possible.
Play the game a number of times and explain that, while it is possible for someone to guess the card from
the first try, asking same question might be inefficient if other cards are drawn. The cards game has the
advantage that it can be played a number of times and is entertaining. It can also be repeated for different
choices of cards.
The goal of minimizing the expected number of questions needed to find the truth is similar to the problem
of guessing a ball using as few weightings as possible:
Puzzle: Find the Odd Ball
If you have two identical balls, one heavier than the other, you can easily determine which is heavier
than the other by putting them on opposite pans of a balance scale. If there are four balls, all the same
weight except for one heavier one, you can find the heavier one in two weightings.
Suppose you have nine identically looking balls, one of which is heavier than the others. What is the
smallest number of weightings needed for positively identifying the odd ball?
The goal of the first version of the puzzle is to remind the student simple laws of probability and the
difference between rare events and very likely events. Students usually give correct answers for the version
of the puzzle when there are only 10 numbers. The goal is to show how to generalize their reasoning to
solve the “1 million” version of the puzzle.
Once students find the correct answer it is interesting to ask what happens if the dice is biased, for example
it has two faces showing 6 and no face showing 5.
Homework problem: The puzzle of Crazy Cut (see Appendices), where students are supposed
to make a single cut and divide the shape in two identical parts.
Materials
A pack of cards, dices, and balls of different colors are needed to play these games. The homework
problem can be found in the Appendices.
Background information
The puzzle “Playing 20 questions” is described in Martin Gardner’s book “My Best Mathematical And
Logic Puzzles” (see Appendices). The games of guessing a number is also mentioned in the book. I chose
one particular case of the cards game, and the game can be repeated with different cards. The “Crazy cut”
puzzle is described in the same book and is only meant to entertain and raise students’ curiosity in the
subject matter.
Suggestions
It is important for this first session to remind students basic concepts of probability and to develop an
understanding of their level of comfort with probability. The concepts needed throughout this mini-course
are mostly covered by the curriculum for the 12th grade.
The puzzles presented in this session were specifically chosen as they do not involve complicated
computations and rely on common sense understanding of rare or very likely events and expected outcome.
As for all other sessions, I find it very important to constantly communicate with student, play the games
with them and ask questions.
Session 2: Travel Puzzles. Introduction to Graph Theory by Examples
Learning outcomes
Upon completion of this session, students will:
1. understand how to model a problem in terms of graph theory
2. understand the motivation behind graph theory
3. learn how to compute shortest routes between two cities and understand why modeling this
problem as finding shortest paths in a graph helps them find a general solution and not focus on
details
4.
5.
6.
be familiar with the famous puzzle of 7 bridges of Königsberg
be familiar with the salesman problem and be able to identify the problem when presented as the
museum tour problem
more generally, develop an understanding of commonalities in problems and how graph theory
helps us focus on essential aspects of problems.
Duration: 1 hour
Activities and procedures
Travel problem: Finding Shortest Routes from Your Home City to Holiday Paradise (Dijkstra Algorithm
by Example):
Start by printing maps with possible routes from Ithaca to Miami Beach and ask students to choose a
route they think it is the shortest (spring break is only a week, so there is no time to waste while
driving from Ithaca to sunny places). Usually the answers to do coincide, which raises the question of
developing a method that is guaranteed to find the shortest route. Some students start drawing
sketches of the map, which is the best motivation to actually drawing the most important aspects of the
map as a graph.
Let students discover Dijkstra’s algorithm or gradually explain it. There is no need to write the
algorithm in detail.
The 7 Bridges Problem (Eulerian Graphs):
Look at the drawing of the town Königsberg and its seven bridges. The goal is to answer the following
question: Is it possible to cross the seven bridges in a continuous walk without crossing any bridge
twice (multiple times)?
The City Tour Problem (Hamiltonian Graphs):
Draw a schematic map of a town you wish to visit during spring break. Can you find a way to visit
each important place of this town without coming in the same place twice?
A Museum Puzzle:
You are in a museum and you have a booklet with the building plan and the numbering of doors and
exhibits. Is it possible for you to see every exhibit in the museum, but none of them twice? How is this
puzzle different than the city tour problem?
Materials
Blackboard and chalk, printed maps, the picture of the town Königsberg and its seven bridges, and article
“Atlas Shrugged” (see Appendices), a schematic plan of a museum.
Background information
Finding shortest routes, Hamiltonian and Eulerian graphs are studied in any class or book on graph theory.
The picture of Könisgberg is a reproduction of the one presented in Gary Chartrand’s book “Introductory
Graph Theory”. However, for the purpose of this mini-course and especially for this introductory session
on graph theory, these problems are not presented in their full generality, and only a number of examples
are given.
Suggestions
I found it useful to present students an article titled “Atlas Shrugged” (see Appendices) that explains how
modern days map servers still use Dijkstra’s algorithm for finding shortest paths in a graph. This gives
extra motivation and reassures the student that the theory is well-grounded.
Backboard and chalk are enough for graph problems, but it is also useful to actually print the Mapquest
directions and maps for a route from Ithaca to Miami Beach and ask students to guess, looking at the map,
how to find the shortest route. Next show them a much simpler map, for example one of the surroundings
of the school, and ask them to find again a short route from the school to the coffee shop. Modeling these
problems afterwards as graphs and finding shortest paths will make clear why it makes sense to think at this
level and make abstraction of details.
Session 3: Games of Chance and a Nobel Prize Winner’s Joke
Learning outcomes
Upon completion of this session, students will:
1. have a better understanding of what independent and correlated events are
2. be familiar with computations involving conditional probability
3. be more aware of ambiguities in some puzzles
4. start to have a feeling for false premises.
Duration: 1 hour
Activities and procedures
Playing with Balls of Two Colors
In a bag there are W white balls and B black balls.
1. I take one ball from the bag. What is the probability it is a white ball? What is the probability it is a
black ball?
2. I put the ball back and pick again a ball. What is the probability this ball is a white ? What is the
probability it is a black ball?
3. Call this ball B1. I show you B1 and don’t put it back in the bag. I pick another ball from the bag;
call this ball B2. What is the probability it is a white ball? What is the probability it is a black ball?
4. This time I do not show you the ball I picked and I don’t put it back in the bag. I extract another ball
from the bag, say B3. What is now the probability the ball is white? What about black?
Lewis Carroll’s Problem
A bag contains a ball, known to be either white or black. A white ball is put in, the bag is shaken, and a
ball is drawn out, which proves to be white. What is now the chance of drawing a white ball?
Suppose I tell you that initially there is a 2 in 3 chance that the bag has a white ball and not black.
How would your answer change?
Three Pancakes
There are 3 pancakes in your friend’s fridge: one is golden on both sides, one is brown on both sides,
and one is golden on one side and brown on the other. You want to eat a pancake, and not to be unfair,
open the fridge with closed eyes. You take one pancake, look at the facing side, and see that it is brown.
What is the probability that the other side is brown?
The Monty Hall Problem
You are on a game show, and you are given the choice of 3 doors. Behind one door is a car, behind the
others, goats. You pick a door, say number 1, and the host, who knows what’s behind the doors, opens
another door, say number 3, which has a goat. He says to you: “Do you want to pick door number 2?” Is
it in your advantage to switch your choice of doors?
Nobel Prize Joke
When Dr. Linus Pauling received his second Nobel prize, he remarked that, while the chances of any
person in the world receiving his first Nobel prize were one in several billion (the population of the
world), the chances of receiving the second Nobel prize were one in several hundred (the total number of
living people who have received the prize in the past) and that therefore it was less remarkable to receive
one’s second prize than one’s first.
Do you agree? Why?
Materials
Handouts with the description of the puzzles. For the first problem solutions are also included.
Background information
The only background needed for this session is the first session. All computations involved in these puzzles
are straightforward. The challenge is in understanding the concept of conditional probability and in
discovering false premises and ambiguous statements.
Some of these puzzles are well-known and it is difficult to say who exactly invented them. A very good
source of puzzle can be fond on Internet at http://www.cut-the-knot.org/probability.shtml.
Suggestions
I found it useful to play a couple of times the Monty Hall puzzle. This made the solution to the puzzle clear.
If students are asked to write on a piece of paper their answers and then confront their answers,
conversation starts and this leads to finding the source of ambiguity in the problem.
Session 4: Probability and Graphs Revisited
Learning outcomes
Upon completion of this session, students will:
1. be able to give qualitative descriptions of probabilities of different events without computing the
exact probabilities; this is useful in being able to quickly compare two events in terms of their
likelihood
2. learn an interesting application of graph theory to derive solutions for a well-known puzzle
involving glasses of different capacities
3. practice their geometrical skills and see a nice example of the interplay between geometry and
probability
4. understand the difference between an ad-hoc solution and a method.
Duration: 1 hour
Activities and procedures
8 Drawers (or The Problem of Diminishing Hopes)
John has a table with 8 drawers, where he randomly (but with equal probability) stores his documents.
John is a very busy person, and 2 out of 10 cases he simply forgets to store the document, and the paper
gets lost.
When John wants to look for a document, he starts with the first drawer, then the second, and goes like
this until he either finds the document or discovers that he lost it. John is now looking for his tax report:
1.
2.
3.
John checked the first drawer and did not find the tax report. What is the probability that John
has not lost the document?
John checked and did not find the tax report in any of the first 4 drawers. What is now the
probability that John has not lost the document?
John checked all first 7 drawers, but his tax report is not there. What is the probability that
John has to do his tax report again?
The Problem of 3 Glasses
You have 3 glasses: one of 8 quarts full of water, and two empty glasses of 3 and 5 quarts capacity.
You want to pour half of the water in the 5 quarts glass. How can you do it?
Can you think of this problem in terms of graphs? If so, how would you find a solution?
Geometry Puzzle: The Random Triangle Problem
You select at random 2 distinct points on a 1 inch straight line. 3 segments are formed. What is the
probability that you can construct a triangle using these segments?
Materials
Handouts with the above puzzles. A printed solution with the graph solution for the problem of glasses of
different capacities is also useful, since deriving the complete solution is somehow repetitive and takes too
much time to be done in the class. Instead, showing just the first steps in writing the adjacency matrix
suffices, and then students can look at the complete solution and interpret the results.
Background information
The puzzle of diminishing hopes and hints for a graph-based solution for the problem of three glasses can
be found on the site http://www.cuttheknot.org/probability.shtml. The problem of three glasses is wellknown.
Suggestions
For the first problem, I suggest first presenting the three scenarios and ask students to compare the chances
that John has not lost the file in all these scenarios. This exercise should test their ability to asses qualitative
information. Then the exact probabilities can be computed.
After students discover a solution for the problem of three glasses I suggest asking the same question but
for much larger numbers, and alternatively for more than three glasses. The difficulty of the more general
problem is a very good motivation for searching for less ad-hoc methods to solve the problem. The graphbased solution amounts to associating to each possible configuration (i.e. how much water is in each glass)
a node in a graph, and to connect two nodes exactly when it is possible with only one move to change one
configuration into another. Then the adjacency matrix is computed and the problem amounts to finding a
path from the initial configuration to the desired final one. This shows the difference between an ad-hoc
solution and a method and proves the usefulness of the more general graph-based solution.
Session 5: Recursion
Learning outcomes
Upon completion of this session, students will:
1. have an understanding of the general technique of breaking a problem into smaller subproblems
2. practice solving problems recursively.
Duration: 1 hour
Activities and procedures
The Tower of Hanoi
On a table there are three identical pegs and on the first peg there are disks stacked one on top of each
other forming a tower: no larger disk is placed on top of a smaller disk. Can you move the tower on the
third peg, temporarily placing disks on any of the pegs, such that at no moment a larger disk is on top
of a smaller disk?
Try to solve the problem for 5 disks. Next try for 6 disks. How can you use your first solution to
quickly solve the problem for 6 disks?
Chess Games: Knight’s Tour
Is it possible for a Knight to start in one corner of the chess board and cover the whole board without
coming in the same position twice? What about 3 by 4, or 4 by 6 boards?
Bicolor Towers of Hanoi
Suppose that the disks forming the tower are colored in red and green and that for each size there are
exactly two disks: one red and one green. We start with towers on the first two pegs, and in each tower
colors are alternating. The goal is to build the red and green towers separately. It is also required to
swap the largest disks.
Materials
Handouts with puzzles and the article on knight’s tour included in the Appendices.
Background information
The tower of Hanoi puzzle is a classic introduction to recursion theory. An article on the Knight’s tour is
included in the Appendices.
Suggestions
I suggest focusing on particular instances of these problems and let students come up with possibly more
solutions for the more complicated cases like tower of Hanoi for 6 disks or the 8 by 8 chess board. Most of
the solutions proposed by students are actually mimicking recursive solutions and offer good motivation for
the general technique.
Session 6: Luck, Coincidences, and Games of Chance
Learning outcomes
Upon completion of this session, students will:
1. be able to correctly evaluate the probability of winning when playing games of chance
2. give rational explanations for statistical facts falsely described as miracles or paradoxes
3. be familiar with simple decision rules for payoff games.
Duration: 1 hour
Activities and procedures
Chance of Winning When Playing At Casino
If I have been playing same game at the casino 99 times and I have always lost, does it mean that I
“exhausted” my bad luck and so if I play one more time I have great chances to win?
Lies, Lies, Lies: The Puzzle of Unreliable Witness
I am new in a town famous for the fact that half of its people are always telling lies and the other half
are always honest. I meet two people and one of them tells me: ”Big storm is coming tomorrow!”. The
other one jumps into conversation: “Don’t listen to him. He is a liar!”. Whom should I believe?
The puzzle of identical twins
I have two good friends and they are twins. He only problem is that one of them is always telling jokes
and lies, but the other one is always honest. They look so much alike that I cannot tell which is which.
One day I meet them in the street and I am determined to find which one is the jokester, once for all.
What question(s) should I ask to be sure whom I am talking to?
Unsolved Mysteries or Simple Coincidence? The Birthday Problem
If a school has same number of students in each class, how many classes are enough so that in at least
half of them there are at least two students with the same birthday? Surprisingly, only 23. However, if
we do the math, this is not so surprising after all.
Optimists, Pessimists and in Between: Payoff Games and Decision Rules
Play a number of payoff games. For each game ask students to decide on a strategy. Discuss all
possibilities. Discover based on the rules they follow if they tend to be rather optimistic than pessimistic.
Materials
Handouts with the above puzzles.
Background information
Good introductory books on strategic games are “An introduction to game theory” by Martin J. Osborne
and “Games of strategy” by Avinash Dixit and Susan Skeath.
Suggestions
There are many variations in the “lies” puzzles and the birthday problem and it is interesting to ask students
whether they know similar problems to the puzzles above.
V.
Resource Materials
Most of the puzzles are well-known are extensively studied in school. The towers of Hanoi and the
knight’s tour are classical introductions to recursion theory; see one additional article on the knight’s tour
included in Appendices.
The tour of Paris and museum puzzle are instances of the more general problem of finding a Hamiltonian
cycle in a graph. The problem of 7 bridges laid the foundation of Eulerian graph theory. Dijkstra’ s
algorithm for finding shortest path in a graph is basically the subject of any introductory course in graph
theory. In the appendices I have included an article that mentions the applicability of Dijkstra’s algorithm
for online maps systems.
Playing 20 questions (when probabilities are know), the puzzle of odd ball and the crazy cut problem are
described in Martin Gardner’s book “My Best Mathematical And Logical Puzzles” (Dover Publications,
New York, 1994). These puzzles were initially presented by Gardner in the Mathematical Games column
for Scientific American.
Acknowledgements: I would like to thank Veronica Vazquez for offering me the opportunity to teach
about puzzles, for creating a very friendly atmosphere and for many interesting discussions. I would also
like to thank the GSSOP organizers for their support, and to the high-school students of Cascadilla school
for their interest and participation during the sessions.
VI.
Appendices
Crazy Cut
Make one cut –it needn’t be straight- that will divide the figure into two identical parts.
The Seven Bridges of Königsberg
C
D
A
B