Who Stole the Tarts? Part XII

"Well, here is your butter, eggs, and milk all back," said the King, "and I see you have your jam, flour, sugar, salt, baking pan, and cookbook—even your pepper. Now you can surely make me the tarts!"

Well, the Queen made a marvelous batch of tarts. "These are even better than last time," said the Queen to herself. "I'm sure the King will be delighted!"

The Queen went up to the Royal Chamber to announce to the King that the tarts were ready. Arm in arm they went down together to the kitchen, but when they got there, they found the table empty—the whole platter of tarts was clean gone!

"Now this has gone too far!" cried the King, paling with rage. "Who sneaks into my house like this? I've half a mind to really execute the culprit!"

Well, needless to say, the culprit did not really get executed, but he was caught, and the tarts were fully recovered. That ends my story.

"What do you mean, that ends your story?" asked (the real) Alice excitedly. "You haven't told us who stole the tarts, nor whether there was a trial, and if there was, what happened at the trial—you haven't told us anything!"

"Well, there was a trial," I added, "but it was a very complicated one, and for you to figure out who was guilty involves solving a complicated logic puzzle, so I think I'll wait a few years until you're all grown up, and then I'll tell you what happened."

"No, we want to know what happened!" said Tony.

"I'll let you know what happened," I replied, "but in a few more years when you're all grown up."

"No, no, we want to know now!" they all shouted.

"All right," I replied, "but you won't blame me if I give you a very complicated logic puzzle?"

"We won't blame you—really we won't. Only stop keeping us in suspense—tell us what happened!"

So I continued—

Well, as I said, the trial was quite a complicated one. The first suspect was the Knave of Hearts, but circumstantial evidence was brought forth which established beyond any reasonable doubt that the Knave of Hearts couldn't have stolen the tarts. The next suspect was the Dormouse. However, several reliable witnesses testified that the Dormouse was fast asleep at the time of the robbery, hence it couldn't have been the Dormouse. At this point the trial came to a dead standstill.

Suddenly the door of the courtroom burst open, and the White Rabbit proudly entered, bearing the tray of tarts. Behind him came the soldiers, dragging in the Gryphon and the Mock Turtle in chains.
"The tarts were found on the beach," explained the White Rabbit. "The Gryphon and the Mock Turtle were just about to eat them when the soldiers happened to come by and put them in custody."

"That proves their guilt without any question of doubt," shouted the Queen, "so off with their heads immediately!"

"Now, now," said the King, "we must give them a fair trial, you know!"

Well, events happened which proved that the Gryphon and the Mock Turtle were not both guilty—the questions that remained were whether either one was guilty, and if so which one; or whether someone else was guilty: Was it a mere coincidence that the tarts were found by the Gryphon and the Mock Turtle? No; evidence was soon produced that conclusively proved that either the Gryphon or the Mock Turtle was guilty (but not both), but the court could see no way to decide which one it was. It seemed that no further progress could be made, but quite suddenly a whole medley of witnesses came up, making various statements.

"The Gryphon never stole the tarts," said the Duchess.

"But he has stolen other things in the past," said the Cook.

"The Mock Turtle has stolen things in the past," said the Cheshire Cat.

"The Cheshire Cat has stolen things in the past," said the Caterpillar.

"The Cook and the Cheshire Cat are both right," said the March Hare.

"The Cook and the Caterpillar are both right," said the Dormouse.

"Either the Cheshire Cat or the Caterpillar is right—and maybe both," said the Hatter.

"Either the March Hare or the Dormouse is right—and maybe both," said Bill the Lizard.

"The Cook and the Hatter are both right," said the Knave of Hearts.

"Bill the Lizard is right and the Knave of Hearts is wrong," said the White Rabbit.

There was a dead silence.

"All this proves nothing!" roared the King. "Just words, words, words—all useless words!"

"Not so useless, your Majesty," said Alice, rising from the jury. "It so happens that the White Rabbit and the Duchess made statements which are either both true or both false."

All eyes turned eagerly to Alice. Now, everyone knew that Alice makes only true statements, and subsequent investigation showed that this statement was no exception. Moreover, this statement solved the entire mystery.

Who stole the tarts?

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

This is the final challenge and there is quite a lot to unpack here. However, what Alice said is enough to unravel the entire thing: the statements of the White Rabbit and the Duchess are either both true or both false and so the implications of only two alternatives need to be worked out.

Consider the first alternative: that the White Rabbit and the Duchess both told the truth. What did the Duchess say? She said that the Gryphon didn't steal the tarts. What did the White Rabbit say? He said that Bill the Lizard is right and the Knave of Hearts is wrong. (I am going to assume that the amended statement the White Rabbit made about the simultaneous guilt of the Gryphon and the Mock Turtle is to be left out of this.)

Now we've reached the bottom with the Duchess, but there is a sort of recursion that needs to be done with the White Rabbit's statement. What did Bill the Lizard say? He said that "either the March Hare or the Dormouse is right—and maybe both" (inclusive or, in other words). What did the Knave of Hearts say? He said that the Cook and the Hatter are both right.

Here again, there is another layer of recursion. What did the March Hare say? He said that the Cook and the Cheshire Cat are both right. What did the Dormouse say? He said that the Cook and the Caterpillar are both right. What did the Hatter say? He said that the Cheshire Cat or (in the inclusive sense) the Caterpillar is right.

With the statements of the Cook, the Cheshire Cat and the Caterpillar, we have finally reached bottom. The Cook said the Gryphon has stolen in the past; the Cheshire Cat said that the Mock Turtle has stolen in the past; and, lastly, the Caterpillar said that the Cheshire Cat has stolen in the past. Reviewing all previous puzzles, the Gryphon and Mock Turtle were both exculpated the only time they were accused. The Cheshire Cat was accused in part VII, part IX, part X, but exculpated in every instance. So the statements of the Cook, the Cheshire Cat and the Caterpillar are all false.

Now, working backwards, the statements of the March Hare, the Dormouse and the Hatter are all false. At the next level up, the statements of Bill the Lizard and the Knave of Hearts are also false. (Seems like a pattern is emerging here.) And at the highest level, the statement of the White Rabbit is false, because its truth is contingent on the falsity of the statement of the Knave of Hearts, which he has, but also on the truth of the statement of Bill the Lizard, which he does not have.

So the statement of the White Rabbit is false. Remember that the very first assumption was that the statements of both the Duchess and the White Rabbit were true. This assumption has now been demonstrated false. Remember also that it was either this, or both of their statements being false, which we must now discard. The only remaining option, though it disappoints us in the two, is that neither the Duchess nor the White Rabbit told the truth. Accordingly, both statements are false, including the one the Duchess made in which she said the Gryphon was innocent of the crime. But it's awful hard to behead someone with the body of a lion who can just fly away, isn't it?