Who stole the cake?

This is yet another question I found on the Internet.

During a recent police investigation, Chief Inspector Stone was interviewing five local villains to try and identify who stole Mrs Archer's cake from the mid-summers fayre. Below is a summary of their statements:

Arnold: it wasn't Edward. It was Brian

Brian: it wasn't Charles. It wasn't Edward

Charles: it was Edward. It wasn't Arnold

Derek: it was Charles. It was Brian

Edward: it was Derek. It wasn't Arnold

It was well known that each suspect told exactly one lie. Can you determine who stole the cake?

Brian's and Derek's statements give the answer

Charles

because

if exactly one of Brian's statements is a lie, then the culprit must be either Charles or Edward; if exactly one of Derek's statements is a lie, then the culprit must be either Charles or Brian.

So Arnold, Charles, and Edward's answers (we can check each one told exactly one lie) aren't actually necessary to solve the puzzle.

• Arnold lied saying it was Brian. Brian lied saying it was not Charles. Charles lied saying it was Edward. Derek lied saying it was Brian. Edward lied saying it was Derek. Jun 7, 2015 at 16:33
• @KritixiLithos Sorry! I got confused. Not sure what my problem was there. You're right of course. +1 to the question. Jun 7, 2015 at 16:36
• Sorry, I accidentally turned a comment into an answer. Jun 7, 2015 at 17:07
• We can also get the answer with just Arnold and Brian's clues.
– Carl
Jun 9, 2015 at 0:13
• We also can get the solution from Arnold, Charles and Edward statements. So we can do without Brian and Derek's statements. Since we also can do without Arnold, Charles and Edward's statements, well, we don't need any statement at all, right? :-) Sep 30 at 12:35

A) That the cake was stolen by exactly one person.
B) The thief was one of five people interviewed.
C) That the lies spoken are actually false statements.
D) That everyone that was interviewed knows who the thief is.

Three people (Arnold, Charles and Edward) each gave testimony of the form:

It was X. It wasn’t Y. (where X and Y are different people)

If X were the thief, then both statements are true.
If Y were the thief, then both statements are false.

So from testimony of this form, we can eliminate both X and Y as thieves since it was given that each suspect told exactly one lie.

Jointly the statements from Arnold, Charles and Edward eliminate all suspects except Charles who is therefore the thief.

• Nothing says there is only one thief. Sep 29 at 7:18

First two statements give the answer i.e. Arnold's and Brian's.

Arnold: it wasn't Edward. It was Brian.

Looking at Arnold's statement one can be say that second part of his statement is lie ,because if it wasn't, the first one would be true, which would make Edward and Brian both culprits. Both Edward and Brian couldn't have stolen the cake. Therefore, "It wasn't Edward" is true.

Brian: it wasn't Charles. It wasn't Edward.

Coming to Brian's statement. We know Edward isn't culprit, so secont part is true making first statement a lie. This means Charles is the thief.

That's it.

• Good explanation! Jun 20, 2015 at 7:00

If one of Brian's statements was a lie, than either Charles or Edward did the crime. If one of Derek's statements was a lie, that either Charles or Brian did the crime. Charles is included in both statements, proving that he is the one who committed the crime.

It could be anyone.

Because

As Alois Christen says nothing says there is only one thief.

With that in mind:

Let us list the true facts for each person's statements, for the two cases whether the first or second statement is true.
For example, A1: If Arnold's statements are true then false, A2: If Arnold's statement are false, then true.

A1: It wasn't Edward. It wasn't Brian.
A2: It was Edward. It was Brian.

B1: It wasn't Charles. It was Edward.
B2: It was Charles. It wasn't Edward.

C1: It was Edward. It was Arnold.
C2: It wasn't Edward. It wasn't Arnold.

D1: It was Charles. It wasn't Brian.
D2: It wasn't Charles. It was Brian.

E1: It was Derek. It was Arnold.
E2: It wasn't Derek. It wasn't Arnold.

You can see that A1 is compatible with B2 and C2, while A2 is compatible with B1 and C1, because of what they say about Edward.
This allows for two possibilities for A, B and C: A1+B2+C2 and A2+B1+C1.
Similarily, about Charles, D1 goes with B2 and D2 goes with B1. And about Arnold, E1 goes with C1 and E2 goes with C2.

This resolves in two possible groups: A1+B2+C2+D1+E2 and A2+B1+C1+D2+E1. You can verify that both groups yield compatible statements.

In the first case it was Charles.
In the second case it was Arnold, Brian, Derek and Edward.