Who Stole the Tarts? Part VIII

"Here is some more salt, so now you can make the tarts," said the King.

"Can't," said the Queen. "Somebody stole my baking pan."

"Baking pan!" shouted the King. "Well, of course we'll have to get that back!"

This time the search was narrowed down to the Frog-Footman, the Fish-Footman, and the Knave of Hearts. They made the following statements at the trial:

FROG-FOOTMAN: It was stolen by the Fish-Footman.
FISH-FOOTMAN: Your Majesty, I never stole it!
KNAVE OF HEARTS: I stole it!

"A fine help you are!" shouted the King to the Knave. "You usually lie through your teeth!"

Well, as it happened, at most one of them lied.

Who stole the baking pan?

(Source: Alice in Puzzle Land: A Carrollian Tale for Children Under Eighty by Raymond Smullyan)

There are four possibilities here: first, that all told the truth and, in addition, the three possibilities of each of the accused lying in turn. The first can be ruled out immediately because the Frog-Footman claimed that the Fish-Footman stole the baking pan while the Fish-Footman disavowed any responsibility.

But what if the Frog-Footman lied? Then the baking pan was not in fact stolen by the Fish-Footman. The Fish-Footman can claim, truthfully, that he did not steal the pan. And the Knave of Hearts can claim to have stolen it if he did. So it looks like an answer has already been attained!

To be sure, though, check the other possibilities. If the Fish-Footman lied, then he stole the baking pan. So the Knave of Hearts must also be lying when he said that he stole it, which is impermissible. If the Knave of Hearts was lying, then he didn't steal the baking pan, in which case the same contradiction arises as when all are assumed to have told the truth.

So the Knave's confession is genuine beyond any doubt.