Christopher Baltus
A Truth Table on the
Island of Truthtellers
M
Students
look at the
implications
of the truth
tables that
they
construct
730
and Liars
y experience teaching truth tables left me looking
for problems that exercise students’ critical reasoning. This article grew from efforts to find such
problems.
After my students are familiar with the mechanics of building truth tables and their application in
identifying tautologies and in other standard exercises, I present the following story and exercises,
which involve statements made by truthtellers and
by liars. In addition to reinforcing basic skills, the
problems call for logical thinking and ask students
to look at the implications of the truth tables that
they construct. These problems demonstrate the
organizing power that a truth table can bring to a
confusing situation—evidence for the power of
mathematical thinking—and they furnish an
opportunity for students to explain their reasoning.
That any conclusion at all can be drawn from the
truth table may be the biggest surprise.
High school or college teachers of discrete mathematics should find that the problems challenge
their classes or at least offer enrichment to individual students. Readers who enjoy these puzzles
should add to them or create new scenarios and
problems.
Ellen decided that her best course was to reach
the capital. Walking from the beach, she came to a
road that split in two and saw two men working
nearby. After Ellen described her arrival on the
island, the first man told her, “The capital is in the
mountains, or the road to the right goes to the capital.” The second quickly said, “The capital is in the
mountains, and the road to the right goes to the
capital.” The first looked up and said, “That man is
a liar.” The second shrugged his shoulders and
added, “If the capital is in the mountains, then the
road on the right goes to the capital.”
Ellen took out a pencil, made a calculation on the
back of her guidebook, thanked the two men, and
then walked down the road to the left. Intrigued by
the astuteness of the visitor, the men asked to see
what she had written.
She showed them the truth table in figure 1.
After the first two columns, which list all possible
truth assignments to the “atomic” propositions c
and r, a column is devoted to each of the statements
made. The speaker is identified, and the statements
A VISIT TO THE ISLAND OF
TRUTHTELLERS AND LIARS
Faced with engine problems, Ellen Wright put her
small plane down on the beach of the Island of
Truthtellers and Liars. Her guidebook offered only
the briefest description:
Let n1 denote the first native, and n2, the second.
Let c be “The capital is in the mountains.”
Let r be “The road on the right goes to the capital.”
The symbol ∨ denotes or, and the symbol ∧ denotes
and.
statements that they make. The actual population
is unknown, since no census has proved reliable.
Island of Truthtellers and Liars
c
r
The island is fifty-six miles long and up to thirtythree miles wide. A range of inland mountains
runs the length of the island. The climate is
pleasant through the year. The island has only
one city, the capital, Quandary. The most singular feature is the character of the inhabitants.
The population is made up entirely of members
of two groups, the truthtellers and the liars. The
truthtellers utter only true sentences; and the
liars, only false sentences. Any inhabitant can
recognize whether another person is a truthteller
or a liar, but an outsider must judge them by the
T
T
F
F
T
F
T
F
n1: c ∨ r n2: c ∧ r n2: c → r
T
T
T
F
T
F
F
F
T
F
T
T
Fig. 1
The first truth table used by Ellen
Christopher Baltus, [email protected], teaches at State
University of New York—College at Oswego, Oswego, NY
13126. His interests include the history of mathematics
and preparing secondary school teachers.
MATHEMATICS TEACHER
Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
are listed in symbolic form: c ∨ r for “c or r,” c ∧ r for
“c and r,” and c → r for “c implies r.” The standard
rules are followed in assigning a truth value to each
“compound” sentence. These truth values are based
on the truth value of the atomic propositions. An
atomic proposition is one that asserts that a certain
thing has a certain quality, for example, “this ink is
black.”
Since the second man, n2, could not give both
true and false statements, only the first two lines of
the table could possibly fit the situation. Both indicate that the first man, n1, is a truthteller. Therefore, n2 is a liar, and the second line of the truth
table is based on truth. The capital is in the mountains, but the road on the right does not go there.
After an hour walking up the road to the left,
Ellen encountered a group of people gathered at
what she hoped was a bus stop. She approached
three women and asked them whether the road
went to the capital and whether this location was a
bus stop. She received three different answers:
“The road goes to the capital, and the bus stop is
not here”; “The road does not go to the capital, and
the bus stop is here”; and “The road does not go to
the capital, and the bus stop is not here.” Puzzled
for a moment, Ellen asked the women whether they
were truthtellers or liars. This time, the three
answers were all the same: “Two of us are
truthtellers, and one is a liar.”
Exercise 1
How many of the three women were truthtellers?
Does the road go to the capital? Is a bus stop at the
location where Ellen spoke to the women?
Ellen then noticed signs for buses B1, B2, and B3
and approached another trio. The following conversation took place:
Ellen. Where do the buses go?
The first. At least one of B1 and B2 goes to the
capital.
The second. B1 goes to the capital.
The third. B2 and B3 go to the capital.
The first. B3 goes to the beach.
The second. B2 and B3 go to the beach.
The third. B1 goes to the beach.
After some jotting on her guidebook, Ellen stepped
onto the right bus and was soon in the capital.
Exercise 2
Which bus did Ellen take?
On reaching the bus station in Quandary, the
capital, Ellen noticed three computer terminals.
Vol. 94, No. 9 • December 2001
She asked a young woman whether the computers
had Internet connections. The woman told her,
“Computer 1 is not connected to the Internet. Ask
that man; he is a truthteller.” When Ellen
approached the man, he told her, “Computer 2 has
an Internet connection, but computer 3 does not.” A
second man, who overheard the conversation, said,
“If computer 2 has an Internet connection, then so
does computer 1. Computer 3 is not connected to
the Internet.” With a little more work on the back
of her guidebook, Ellen moved to a computer and
ordered parts for her plane.
Exercise 3
Which computer had an Internet connection?
No speaker
can make
both true
and false
statements
SOLUTIONS
Exercise 1
Figure 2 shows the truth table for this exercise.
Since the three women all gave the same statement, “Two of us are truthtellers, and one is a liar,”
they must all be truthtellers or all must be liars.
The statement must then be false, so all are liars.
Alternatively, we observe that on no line of the
table do exactly two of the women make true statements, so the statement “Two of us are truthtellers,
and one is a liar” is false. Only in the first line of
the truth table do all three women give false statements. Therefore, g and b are both true statements:
The road does go to the capital, and a bus stop is at
the location.
Let w1, w2, and w3 denote the three women.
Let g be “The road goes to the capital.”
Let b be “The bus stop is here.”
The symbol ¬ represents negation.
g
b
T
T
F
F
T
F
T
F
w1: g ∧ ¬ b w2: ¬ g ∧ b w3: ¬ g ∧ ¬ b
F
T
F
F
F
F
T
F
F
F
F
T
Fig. 2
Truth table for exercise 1
Exercise 2
Figure 3 shows the truth table for this exercise.
Since we already know that the capital is in the
mountains, a bus that goes to the beach does not go
to the capital.
Since no speaker can make both true and false
statements, only the fourth line can describe the
actual situation. So Ellen takes bus B1.
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Fan B. You do not like the Mets. You like the
Dodgers.
Fan A. We both like the Dodgers.
Let p1, p2, and p3 denote the three people at the bus stop.
Let b1 be “Bus B1 goes to the capital.”
Let b2 be “Bus B2 goes to the capital.”
Let b3 be “Bus B3 goes to the capital.”
Line
b1
b2
b3
1
2
3
4
5
6
7
8
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
p1: b1 ∨ b2. ¬ b3
T
T
T
T
T
T
F
F
F
T
F
T
F
T
F
T
Additional exercise A
p1: b1. ¬ b2 ∧ ¬ b3
T
T
T
T
F
F
F
F
p3: b2 ∧ b3. ¬ b1
F
F
F
T
F
F
F
T
T
F
F
F
T
F
F
F
F
F
F
F
T
T
T
T
Exercise 3
In the truth table shown in figure 4, we can eliminate lines 1, 3, 6, and 7, since m2 cannot make both
true and false statements. Since w said that m1 is a
truthteller, if w’s statement, ¬ c1, is true, then the
statement of m1 must be true; and if w’s statement
is false, then the statement of m1 must be false. We
can then also eliminate lines 2, 5, and 8. So only
line 4 of the truth table is possible. Ellen must use
computer 1.
Let w, m1, and m2 denote the woman and two men at the bus terminal.
Let c1 be “Computer 1 is connected to the Internet.”
Let c2 be “Computer 2 is connected to the Internet.”
Let c3 be “Computer 3 is connected to the Internet.”
c1
c2
c3
w1: ¬ c1
m1: c2 ∧ ¬ c3
1
2
3
4
5
6
7
8
T
T
T
T
F
F
F
F
T
T
f
F
T
T
F
F
T
F
T
F
T
F
T
F
F
F
F
F
T
T
T
T
F
T
F
F
F
T
F
F
m2: c2 → c1. ¬ c3
T
T
T
T
F
F
T
T
F
T
F
T
F
T
F
T
Fig. 4
Truth table for exercise 3
ADDITIONAL EXERCISES
In the bus terminal, Ellen overheard this conversation.
Fan A. I like the Mets.
732
Solution: Only in line 6 of the truth table shown
in figure 5 do both fA and fB each make only true or
only false statements. Therefore, fan A does not
like the Mets, fan A does like the Dodgers, and fan
B does not like the Dodgers.
Let fA and fB denote fan A and fan B.
Let m be “Fan A likes the Mets,” let x be “Fan A
likes the Dodgers,” and let y be “Fan B likes the
Dodgers.”
Fig. 3
Truth table for exercise 2
Line
Does fan A like the Mets? Who likes the Dodgers?
Line
m
x
y
1
2
3
4
5
6
7
8
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
fA: m. x ∧ y fB: ¬ m.
T
T
T
T
F
F
F
F
T
F
F
F
T
F
F
F
F
F
F
F
T
T
T
T
x
T
T
F
F
T
T
F
F
Fig. 5
Truth table for additional exercise A
While she was in the capital, Ellen saw notices
from the national census bureau, seeking a list of
questions to send to each household in the country.
The census bureau wanted to learn the name of one
adult in each household and the number of people
in the household. Ellen submitted her suggestion.
Additional exercise B
Can you help the census bureau develop a list of
questions? Remember that the census bureau needs
to determine the correct information regardless of
whether the person answering the questions is a
truthteller or a liar.
Solution: The first question to ask might be, Are
you a truthteller and a liar? That question would
separate the truthtellers from the liars, and further
questions could proceed from that knowledge.
The parts for the plane arrived, Ellen repaired it,
and she returned home a few days later. Because
she always appreciates a puzzle, she granted the
author permission to report her story. ¿
MATHEMATICS TEACHER