Morse arrested Burglar Bill. Assume that Slick Jim's statement that it wasn't him is false, so he is the burglar. His second statement must be true so Burglar bill didn't do it. In that case both the statements that Burglar Bill made, that it wasn't him and it wasn't Dodgy Dick, were both true, which is not possible. Assume that Slick Jim's first statement is true. So Burglar Bill did it. Burglar Bill's first statement, that it wasn't him, is therefore false, and his second statement, that it wasn't Dodgy Dick, is true. Dodgy Dick's first statement that it wasn't him is true, and his statement that they were all at the races is false. So Burglar Bill is the culprit.
Morse arrested Burglar Bill. Assume that Slick Jim's statement that it wasn't him is false, so he is the burglar. His second statement must be true so Burglar bill didn't do it. In that case both the statements that Burglar Bill made, that it wasn't him and it wasn't Dodgy Dick, were both true, which is not possible. Assume that Slick Jim's first statement is true. So Burglar Bill did it. Burglar Bill's first statement, that it wasn't him, is therefore false, and his second statement, that it wasn't Dodgy Dick, is true. Dodgy Dick's first statement that it wasn't him is true, and his statement that they were all at the races is false. So Burglar Bill is the culprit.