Logical Reasoning
A Lesson in the “Math + Fun!” Series
Jan. 2005
Logical Reasoning
Slide 1
About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during the 2003-04 and 2004-05 school years. The slides can
be used freely in teaching and in other educational settings.
Unauthorized uses are strictly prohibited. © Behrooz Parhami
Jan. 2005
Edition
Released
First
Jan. 2005
Revised
Logical Reasoning
Revised
Slide 2
Aging Three Years in One Year?
Amy says that she was 10 years old the day before yesterday
and that she will turn 13 next year. How is this possible?
(Think for a few minutes before you advance to the next slide.)
The day before yesterday
Yesterday
Today
2004
2005
Jan. 1
Age:
Dec. 31
10 11
Jan. 2005
Dec. 31
2006
This year
Next year
12
Logical Reasoning
13
Slide 3
Mislabeled Bags
We have three bags, each holding two balls.
One bag has two black balls. One bag has two white balls.
One bag has one black and one white ball.
The bags are labeled black, white, and mixed.
None of the labels is correct, however.
Correct the labeling by drawing a ball from only one bag.
Extra challenge:
B
B
W
W
M
M
Jan. 2005
W
M
B
M
W
B
M
W
M
B
B
W
Logical Reasoning
The number of balls in bags
is not known, but they are
all black, all white, or mixed.
How do you solve the puzzle
In this case?
Minesweeper: Example
of a reasoning game
Slide 4
Making Change for a Dollar
Can you have more than one dollar in change and still
not be able to give exact change for one dollar?
Extra challenge: What is the most amount of change that you
can have and still not be able to give exact change for one dollar?
Jan. 2005
Logical Reasoning
Slide 5
River-Crossing Problems
There are people, animals, and things on one side of a river
They must be transported to the other side
There is a boat that cannot carry all of them at once
There are restrictions on who can be left alone with whom or what
Example: Person, fox, duck, corn must cross
Boat can carry the person plus one animal/thing
Fox and duck should not be left together
Duck and corn should not be left together
P
P
F
D
C
Jan. 2005
F
C
P
D
P
F D
C
C
P
F
D
P
D F
C
Logical Reasoning
P
D C
F
F
P C
D F
D
C
P
D
C
F
Slide 6
Graphs to Solve River Crossings
Example: Person, fox, duck, corn must cross
Boat can carry the person plus one animal/thing
Fox and duck should not be left together
Duck and corn should not be left together
P
P
F
D
C
F
C
P
D
FC/PD
PDC/F
PFC/D
C
Our previous solution
D/PCF
F/PCD
Jan. 2005
D
From each state, there are
a number of legal “moves”
P
F D
C
C/PFD
PFDC/-
F
PDF/C
Logical Reasoning
PD/CF
-/PDCF
A different solution
Slide 7
Activity 1 – River Crossing
Three soldiers get to a river they must cross; there’s no bridge in sight.
They spot two boys playing in a tiny boat.
The soldiers ask the boys if they can use the boat to cross the river.
The boat is so tiny that it can carry only one soldier or the two boys.
How can the three soldiers cross the river?
S1
S2
S3
B1
B2
Jan. 2005
Logical Reasoning
Slide 8
Activity 2 – Bridge Crossing
2 min
1 min
Flashlight
9 min
5 min
Four people must cross a bridge
in 16 minutes or less.
Individually, they can cross in
1, 2, 5, and 9 minutes.
It is dark, so a flashlight is needed.
They have only one flashlight.
At most two can cross together.
They must walk at the pace of the
slower person.
No other equipment is available.
P1
P2
P5
P9
Jan. 2005
Logical Reasoning
Slide 9
Three Heads and Five Hats
Amy, Bert, and Cindy are arrested in Fantasyland and taken to an ogre.
They are shown five hats, two white and three black.
They are lined up, Amy in front, Bert in the middle, and Cindy last.
They are blindfolded and one of the five hats is placed on each one’s head.
The blindfolds are removed, but each one can see only those in front of him/her.
The ogre says that if any of them correctly guesses the color of his or her hat
in less than one minute, all three will be set free; otherwise, . . .
No one says anything for 58 seconds
On the 59th second, Amy shouts “black”, and they are set free. How did she know?
Black!
?
Black!
?
?
Black!
Cindy
Bert
Jan. 2005
Amy
Logical Reasoning
Slide 10
Activity 3: Blue or Red Spot?
Angelica, Bob, and Chuckie go to a job interview.
The manager says that she will put a red or blue spot on each forehead.
Each will see the spots on the other foreheads, but not the one on his/her own.
There are no mirrors in the room.
The manager puts red spots on all three foreheads.
She tells them that if they see a red spot, they should raise their right hand; as soon
as a person knows the color of his/her spot, he/she should lower his/her hand
After a minute or so, Angelica lowers her hand and says “I must be red.” Why?
Red!
Angelica
Chuckie
Bob
Jan. 2005
Logical Reasoning
Slide 11
Which Switch Turns the Light on?
There are three light switches in one room and a lightbulb in an adjacent room.
You know that the lightbulb is off now.
You are allowed to go or look into the other room only once.
How can you tell which light switch turns the lightbulb on?
Flipping one switch and checking on the lightbulb may not provide the answer.
Flipping two switches and checking on the lightbulb may not provide the answer.
Hint: How can you tell a lightbulb that has just been turned on
from one that has been on for a while?
Solution:
Flip the first switch; wait for a few minutes; flip the first switch again.
Flip the second switch and go into the next room.
If the lightbulb is off, but warm to the touch, the first switch controls it; if it is on,
the second switch controls it; if it is off, but cold, the third switch controls it.
Jan. 2005
Logical Reasoning
Slide 12
Truth Tellers and Liars
Two tribes live on the LiaTru island.
Members of one tribe always tell the truth; members of the other tribe always lie.
For example, if you ask a boy on the LiaTru island “Are you a boy?”
he will answer “Yes” if he is a truth teller and “No” if he is a liar.
If a girl on the LiaTru island is 10 years old and you ask her “Are you 9 years old?”
she will answer “No” if she is a truth teller and “Yes” if she is a liar.
Members of the two tribes can recognize each other, but you can’t tell them apart.
Puzzle 1: On your way to GFS,
you arrive at a fork in the road.
Someone appears, but you don’t
know whether he is a truth teller
or a liar. How can you find out
which way leads to GFS?
Which way
to GFS?
That way!
Puzzle 2: How can you discover
whether a person you meet is a
truth teller or a liar?
Jan. 2005
Logical Reasoning
Slide 13
Activity 4: Truth Tellers and Liars
Puzzle 1: You meet two people,
A and B. A says, “Both of us are
from the liars tribe.” Which tribe
is A from? What about B?
Puzzle 2: You meet two people,
C and D. C says, “Exactly one of
us is from the liars tribe.” Which
tribe is D from?
Puzzle 3: You meet two people,
E and F. F says, “It is not the
case that both of us are from the
truth tellers tribe.” Which tribe is
E from? What about F?
Jan. 2005
You
A
B
Puzzle 4: You meet two people,
G and H. Each of the two makes
a statement:
G: “We are from different tribes.”
H: “G is from the liars tribe.”
Which tribes are G and H from?
Logical Reasoning
Slide 14
Truth Tellers, Liars, and Randoms
Three tribes live on the LiaRanTru island.
Members of one tribe always tell the truth; members of the second tribe always lie;
members of the third tribe answer completely at random.
For example, if you ask a boy on the LiaRanTru island “Are you a boy?”
he will answer “Yes” if he is a truth teller, “No” if he is a liar, and “Yes” or “No”
(unpredictable) if he is a random.
Members of the tribes can recognize each other, but you can’t tell them apart.
Puzzle : Three people from the
LiaRanTru island, one representing
each tribe, come to town. You must
identify who is from which tribe by
asking only three yes/no questions.
Each question must be directed to
only one person, but you can ask
the same person multiple questions.
P1
P2
P3
Hint : There are 6 possibilities for P1, P2, P3: LRT, LTR, RLT, RTL, TLR, TRL
Jan. 2005
Logical Reasoning
Slide 15
Activity 5: Other Logical Puzzles
Puzzle 1: Twelve black socks and 12
white socks are mixed up in a drawer.
How many socks must you pull out,
blindly, to get a matching pair for sure?
Puzzle 2: Three travelers are told that
their hotel room costs $30, so they pay
the clerk $10 each. The clerk later
realizes that he made a mistake and
should have only charged them $25.
He gives the bellboy $5 to return to the
travelers. The bellboy is dishonest and
gives each of them only $1, keeping $2
for himself. So the travelers actually
spent $27 and the bellboy pocketed $2.
What happened to the other dollar of
the original $30?
Jan. 2005
Puzzle 3: Two mathematicians
are walking down the street.
The first says to the second, "I
know you have three children.
What are their ages?" The
second replies, “The product of
their ages is 36." The first says,
"I can't tell their ages from
that." The second says, “Well,
the sum of their ages is the
same as that address across
the street." The first says, "I still
can't tell." The second says,
"The oldest is visiting his
grandfather today." The first
says, "Now I know their ages."
Can you tell their ages?
Logical Reasoning
Slide 16
Next Lesson
Thursday, February 3, 2005
A problem to think about:
How many squares can you find
that have dots in their corners?
Jan. 2005
Logical Reasoning
Slide 17