Liars and Truth-teller puzzles.
January 20.

Math Explorer's Club
.

Module on Puzzles

Ithaca High School.

The setting for these problems is the following. There is this island in the middle of the ocean where there are two kinds of people: the liars and the truth-tellers. The liars always lie. Any question you ask them will be answered with a lie. The truth-tellers always answer the truth.

1. You're walking in this island and you meet a native, who could be either a liar or a truth-teller, and you want to know which kind of person he is. So you ask him
- are you a truth-teller?
When he is answering, the volcano makes a loud noise and you cannot hear the answer. So you ask him again
- Excuse me, I couldn't hear what you said, did you say you were a truth-teller? - and he answers
- no, I didn't say that, I said I was a liar.
Is the native a liar or a truth-teller?
(Hint: Think about what the native's could have answered to the first question, first assuming he's a truth-teller and then assuming he's a liar.)

2. Suppose now that you visit a second island close by. In this second island the people is even more weird. They lie on Tuesdays, Thursdays and Saturdays and they tell the truth the other days. After spending so many days in these islands you forgot which day it is. So you ask one of the islanders:
- Excuse me sir, what day is today?
- And, what day is it going to be tomorrow?
- Wednesday.
Can you deduce what day is today?

3.   Now suppose you meet three people, A, B and C form the first island (so they are either liars or truth-tellers). You ask A:
- How many truth-tellers are among you guys?
Again, the volcano makes a loud noise when A answers and you can't hear the answer. So you ask B:
- What did A say? -  and B answers
- A said that one of  us is a truth-teller and two of us are liars - and then C adds
- Don't believe what B said, he's lying.
What class of people are B and C?

4. Suppose again that you meet three people A, B and C from the island. You ask A:
- Are you a liar?
Again, the volcano makes a loud noise when A answers and you can't hear the answer. So you ask B:
- What did A say? -  and B answers
- A said he is a liar - and then C adds
- Don't believe what B said, he's lying.
What class of people are B and C?

5. This time you meet two people A and B.  A  says
-
Either I am a liar, or B is a truth-teller.
What class of person is B.

6. Once I met two people in the island and I asked one of them:
- Is any of you a truth-teller?
With his answer I was able to know the solution to my question. What class of people were they?

7. Three people, A, B and C are being judged for a crime. Each of these people could or could not come from the island. Recall that everyone that comes from the island is either a liar or a truth-teller. People that do not come form the island are called normal and might or might not tell the truth.
It is known that the crime was committed by one of the three suspects. It is also known that the person  who committed the crime is a truth-teller, and is the only truth-teller among the three suspects. The suspects said:
A: "I am innocent."
B: "That is correct."
C: "B comes from the island."
Who is the guilty one?

8. Somewhere in the island you find three boxes. Each box has two balls. One box has two white balls, another one has two black balls and the other one has a white ball and a black ball, but you don't know which one has what. Each box has a label saying what kind of balls it has inside. One label says "TWO WHITE", another label says "TWO BLACK", and the other label says "ONE WHITE, ONE BLACK". But the person who put the labels in the boxes was a liar, so all the labels are all wrong! So you want to change the labels and put them in the right boxes. You are only allowed to open one of the three boxes, take one ball out of it and look at its color. Which box should you open?

Antonio Montalbán.
Department of Mathematics.
Cornell University.
Ithaca, NY 14853.
e-mail: antonio at math.cornell.edu