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 it weren't the first one would be true. Making edward and Brian both as culprits. So I assume Edward and Brian didn't stole cake.

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.

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.