Question

Solve the following logic problems. Remember, everyone you meet is either a knight or a knave, knights make true statements, and knaves make false statements. Give your reasoning for each problem.

a) You meet two residents, Alex and Bill. They say the following: Alex: I’m a knight. Bill: Alex is a knight, but I’m a knave. Is Alex a knight or a knave? Is Bill a knight or a knave?

b) You meet Clara and Davis, who are all like: Clara: One of us is a knight and the other is a knave. Davis: Clara is a knave. Is Clara a knight or a knave? Is Davis a knight or a knave?

c) You meet Edith and Frank, though only Edith speaks. Edith: Both Frank and I are knaves. Is Edith a knight or a knave? Is Frank a knight or a knave? (Note: Frank’s silence gives no indication of his type, but you can figure out from Edith’s statement.)

d) You meet Gina, Herbert, and Ichabod. Gina: Ichabod is a knave, if and only if I’m a knight. Herbert: Ichabod is a knight, if and only if I’m a knave. Ichabod: I like pudding. Does Ichabod like pudding?

Answer 1

(a)

Let

A - Alex

B - Bill

Let 1 represent truth and 0 represent false. Then a knight will be assigned the value 1 since knights make true statements and a knave will be assigned a value 0 since knaves make false statements.

Then, let

a - What Alex said which is "A=1"

b - What Bill said which is "A=1 B=0"

P - The situation where "A=a B=b "

We need to find the situation where P=1 and thus P is true because if P is true, then a knight will make a true statement and a knave will make a false statement which is as per the question.

We construct the truth table as follows:

A	B	a	b	P
0	0	0	0	1
0	1	0	0	0
1	0	1	1	0
1	1	1	0	0

We can see from the truth table that P=1 only when A=0, B=0.

Since A and B are both 0, so everyone makes false statements and thus everyone is a knave.

Hence, Alex is a knave and Bill is a knave.

(b)

Let

C - Clara

D - Davis

Let 1 represent truth and 0 represent false. Then a knight will be assigned the value 1 since knights make true statements and a knave will be assigned a value 0 since knaves make false statements.

Then, let

c - What Clara said which is "(C=1 D=0) (C=0 D=1)"

d - What Davis said which is "C=0"

P - The situation where "C=c D=d "

We need to find the situation where P=1 and thus P is true because if P is true, then a knight will make a true statement and a knave will make a false statement which is as per the question.

We construct the truth table as follows:

C	D	c	d	P
0	0	0	1	0
0	1	1	1	0
1	0	1	0	1
1	1	0	0	0

We can see from the truth table that P=1 only when C=1, D=0.

So Clara makes true statements and Davis makes false statements.

Hence, Clara is a knight and Davis is a knave.

(c)

Let

E - Edith

F - Frank

Let 1 represent truth and 0 represent false. Then a knight will be assigned the value 1 since knights make true statements and a knave will be assigned a value 0 since knaves make false statements.

Then, let

e - What Edith said which is "E=0 F=0"

P - The situation where "E=e "

We need to find the situation where P=1 and thus P is true because if P is true, then a knight will make a true statement and a knave will make a false statement which is as per the question.

We construct the truth table as follows:

E	F	e	P
0	0	1	0
0	1	0	1
1	0	0	0
1	1	0	0

We can see from the truth table that P=1 only when E=0, F=1.

So Edith makes false statements and Frank makes true statements.

Hence, Edith is a knave and Frank is a knight.

(d)

Let

G - Gina

H - Herbert

I - Ichabod

Let 1 represent truth and 0 represent false. Then a knight will be assigned the value 1 since knights make true statements and a knave will be assigned a value 0 since knaves make false statements.

Then, let

g - What Gina said which is "I=0 G=1"

h - What Herbert said which is "I=1 H=0"

P - The situation where "G=g H=h"

We need to find the situation where P=1 and thus P is true because if P is true, then a knight will make a true statement and a knave will make a false statement which is as per the question.

We construct the truth table as follows:

G	H	I	g	h	P
0	0	0	0	0	1
0	0	1	1	1	0
0	1	0	0	1	1
0	1	1	1	0	0
1	0	0	1	0	1
1	0	1	0	1	0
1	1	0	1	1	1
1	1	1	0	0	0

We can see from the truth table that P=1 in the following cases:

• G=0, H=0, I=0
• G=0, H=1, I=0
• G=1, H=0, I=0
• G=1, H=1, I=0

In all the cases, I=0.

So Ichabod makes false statements.

So Ichabod's statement "I like pudding" is false.

Hence, Ichabod does not like pudding.