Solution to Problem:

Dick did it!

You have one of two ways to approach this problem.

One way is to assume that one of the statements is true and that the others are false, and see what you can conclude from that. For example, suppose that Dick's statement is true. Then, Jack's is false, but Sally's is true, which contradicts the information that only one statement is true. So some other statement must be the true one, and Dick's is false. So, Dick committed the crime!

Another way to attack the problem is to assume that one of the suspects committed the crime and determine which statements are true and false. Suppose Dick was the criminal. Then Dick was lying. So his statement was false, Jack's was true, Sally's was false, and Jimmy's was false. Since there are three false statements and one true, this fits and Dick was the criminal! (Of course, you should check the other possibilities in case there were two criminals!)