Charlie committed the terrible crime. The way to solve this puzzle is to look at each clue. We know that exactly one of each person's statements is true. Looking at Arnold's statements, let's check to see 'it was Brian' is true? If 'it was Brian' is true, then we know the other statement is false, therefore it was Edward. This is a contradiction. Hence we now know it wasn't Brian, nor Edward (as 'it wasn't Edward' must be the true statement). Looking at Charlie's statement, we can similarly determine that it wasn't Arnold either. Edward's statement gives us that it wasn't Derek, which leaves only Charlie as the culprit. It is always wise to check this against the other clues.