## 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:

Share: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?