Who Stole the Tarts? Part XI

"Well, here is your cookbook back again," said the King, "so you now have the recipe. So make me the tarts!"

"Without milk, butter, or eggs?"

"Oh, me!" cried the King. "This is too much!"

"And this time I know it was the March Hare, the Mad Hatter, and the Dormouse," shouted the Queen, stamping her feet in rage. "I actually saw them sneaking out of the window when I came into the kitchen. Each was carrying something, but I couldn't tell who was carrying what."

"We'll soon settle that!" roared the King.

Well, the ingredients were all found at the house of the March
Hare, Mad Hatter, and Dormouse. The three were tried and made the following statements at the trial:

MARCH HARE: The Hatter stole the butter.
HATTER: The Dormouse stole the eggs.
DORMOUSE: I stole the milk.

As it happens,the one who stole the butter told the truth and the one who stole the eggs lied. Who stole what?

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

Here, things get a little bit more complicated. Each of the three accused stole one item and there are \(3!\) (or six) ways to pair off the accused with those items. But these can also be narrowed down fairly quickly.

First, consider the issue of the butter theft. First, assume that the March Hare stole the butter. If he stole it, then he wouldn't have said the Hatter stole it, because the butter thief tells the truth. So that can be ruled out. The Hatter's statement is at least consistent with stealing the butter. And the suspicion that the Hatter stole the butter is confirmed by the impossibility of the Dormouse being the thief; he would not have said he stole the milk. So the Hatter stole the butter.

Now the issue of the theft of the eggs can be considered. Culpability has already been assigned to the Hatter. That leaves the March Hare and the Dormouse. It can be known now that the March Hare told the truth, as he said that the Hatter stole the butter. The only option remaining is the Dormouse. And, indeed, the Dormouse's statement is a lie, given this inference. Then the Dormouse stole the eggs.

So, all told, the Hatter stole the butter, the Dormouse stole the eggs, and the March Hare stole the only remaining item: the milk.