## Who is Guilty?

I am running a PRIMES-STEP program for middle school students, where we try to do research in mathematics. In the fall of 2015 we decided to study the following topic in logic.

Suppose there is an island where the following four types of people live:

**Absolute truth-tellers**always tell the truth.**Partial truth-tellers**tell the truth with one exception: if they are guilty, they will say that they are innocent.**Absolute liars**always lie.**Responsible liars**lie with one exception: if they are guilty, they will say that they are guilty.

See if you can solve this simple logic puzzle about people on this island.

It is known that exactly one person stole an expensive painting from an apartment. It is also known that only Alice or Bob could have done it. Here are their statements:

Alice: I am guilty. Bob is a truth-teller.

Bob: I am guilty. Alice stole it. Alice is the same type as me.

Who stole the painting and what types are Alice and Bob?

My students and I discovered a lot of interesting things about these four types of people and wrote a paper: *Who Is Guilty?*. This paper contains 11 cute logic puzzles designed by each of my 11 students.

I envied my students and decided to create two puzzles of my own. You have already solved the one above, so here is another, more difficult, puzzle:

A bank was robbed and a witness said that there was exactly one person who committed the robbery. Three suspects were apprehended. No one else could have participated in the robbery.

Alice: I am innocent. Bob committed the crime. Bob is a truth-teller.

Bob: I am innocent. Alice is guilty. Carol is a different type than me.

Carol: I am innocent. Alice is guilty.

Who robbed the bank and what types are the suspects?

## Liam Donovan:

Is “truth-teller” a class comprising both absolute and partial truth-tellers?

26 June 2016, 1:06 am## tanyakh:

Liam, yes, truth-tellers are both absolute and partial truth-tellers.

26 June 2016, 7:15 pm## harvey:

They’re all truth-tellers!

25 July 2016, 2:23 amI’m guilty of robbing that bank.

## A Guy Passing By:

It’s easy: Alice is an absolute liar and so is Bob; he just wants to watch the world burns, and our friend Carol is the robber, but she’s not a liar in the whole sense of the word, she’s a partially truth-teller. She’s just scared.

30 July 2016, 6:17 pm## weesh:

I don’t think the second one works.

In any case where one person is guilty, the other two people can’t be liars, or else they would claim guilt. Thus all the remaining statements from the two truth-tellers have to be true.

If Alice is guilty, then she is a liar claiming that bob is a truth-teller, so bob must be a liar, which can’t be true. Alice is innocent.

If Bob is guilty, then carol must be a liar since she accuses Alice, which can’t be true. Bob is innocent.

If Carol is guilty, then Alice and Bob accuse the wrong people, even though they can’t be liars. Caril is innocent.

18 August 2016, 6:17 pm## tanyakh:

weesh,

There is more to the puzzle than you noticed.

18 August 2016, 9:01 pm## weesh:

oh! how lovely!

19 August 2016, 10:25 amI think that puzzle is so masterful that I’m hesitant to make my answer to the puzzle public for fear that someone else would miss out on the discovery.

## Bruno:

The second puzzle is truly delightful. Thanks so much for sharing this!

30 August 2016, 5:21 pm