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.

Who Stole the Tarts? Part X

Shortly after the cookbook was returned to the Queen, it was stolen a second time—again by either the Duchess, the Cook, or the Cheshire Cat.

At the trial they made exactly the same statements as at the last trial. Only this time, the thief lied and the other two either both lied or both told the truth.

Who stole the cookbook this time?

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

Recall that these were the statements from the last trial:

"The Cheshire Cat stole it!" said the Duchess at the trial.
"Oh, yes, I stole it!" said the Cheshire Cat with a grin.
"I didn't steal it!" said the Cook.

The most straightforward way is to consider whether each individual in turn might be the thief.

If the Duchess stole the cookbook, then her statement that the Cheshire Cat stole it is false, which is good so far because it is required that the thief lied. Then the Cheshire Cat is also lying because he claimed to have stolen it. But the Cook is telling the truth when she says she didn't steal it, and it is required that the two non-thieves either both told the truth or both lied. So the Duchess is not the thief.

If the Cheshire Cat stole the cookbook, then the other two are both telling the truth, which is good. But because it is required that the thief lied, the Cheshire Cat would have had to say he did not steal the cookbook. The Cheshire Cat is now also exculpated.

The only remaining option is the Cook. But to verify, it checks out that she lied, because she said she didn't steal it. And everyone else here has also perjured themselves so the criterion that the others must either both lie or both tell the truth is also met.

Who Stole the Tarts? Part IX

"Here is the baking pan," said the King, "so now you can make me the tarts."

"Without a recipe?" inquired the Queen.

"Use your usual recipe," cried the King impatiently, "last time your tarts were delicious!"

"Can't," said the Queen. "The recipe is in my cookbook, and the cookbook has just been stolen!"

Well, the most likely suspect was the Duchess's Cook, and the cookbook was indeed found in the Duchess's kitchen. The only possible suspects were the Cook, the Duchess, and the Cheshire Cat.

"The Cheshire Cat stole it!" said the Duchess at the trial.

"Oh, yes, I stole it!" said the Cheshire Cat with a grin.

"I didn't steal it!" said the Cook.

As it turned out, the thief had lied and at least one of the others had told the truth.

Who stole the cookbook?

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

As in part VII, an initially larger number of possibilities is constrained by the content of the statements to only two possibilities:

• Duchess is truthful; Cheshire Cat is truthful; Cook is lying
• Duchess is lying; Cheshire Cat is lying; Cook is truthful

In the first possible world, the Duchess and Cheshire Cat jointly confirmed that the latter stole the cookbook. But the Cook is lying and therefore also stole the cookbook, resulting in a contradiction. (It seems to be implied here that there was only one thief, which makes sense.) The only other alternative is the latter: the Cook is truthful, and did not steal the cookbook, and both the Duchess and the Cheshire Cat are lying about the guilty party. The only remaining party to take the blame is the Duchess.

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.

Who Stole the Tarts? Part VII

"Well," said the King, "here is your sugar, so you can make me the tarts."

"Without salt?" asked the Queen.

So! The salt had also been stolen! Well, this time it was found that the culprit was either the Caterpillar, Bill the Lizard, or the Cheshire Cat. (One of them had come into the kitchen and eaten up all the salt; the container wasn't missing.) The three were tried and made the following statements in court:

CATERPILLAR: Bill the Lizard ate the salt.
BILL THE LIZARD: That is true!
CHESHIRE CAT: I never ate the salt!

As it happened, at least one of them lied and at least one told the truth.
Who stole the salt?

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

Initially, it seems like there could be a total of \(\dbinom{3}{1}\) ways that one truth-teller could be assigned to a group of two liars and, then, respectively, \( \dbinom{3}{2}\) ways two truth-tellers could be assigned to a liar, to create the all the groups with compositions acceptable by the last stipulation in the story. Both of these combinatorial expressions evaluate to three and so six possibilities all told would appear to need consideration.

But wait! Before I enumerate them all! Know that things are actually more closely constrained because of the agreement between the Caterpillar and Bill the Lizard. There are indeed only two possibilities, namely:

• Caterpillar is truthful; Bill the Lizard is truthful; Cheshire Cat is lying
• Caterpillar is lying; Bill the Lizard is lying; Cheshire Cat is truthful

In the first instance, Bill the Lizard ate the salt. But then it is the case that the Cheshire Cat is not lying. Only the second option is therefore tenable and, in this instance, the Caterpillar and Bill the Lizard are both lying and the Cheshire Cat is truthful. From this it can be inferred that neither the Cheshire Cat nor Bill the Lizard ate up all of the salt, and the only remaining option is that the Caterpillar did it.

Who Stole the Tarts? Part VI

N.b. another part intervened between what was solved in the last post and the current one but it did not concern logic as such, but an incongruity with one of the characters used in story. As such I have skipped over it:

"That certainly cost me a lot of work finding the pepper," said the King angrily, "and I doubt that the tarts will be all that much the better for it! Pepper indeed!" continued the King. "Why don't you use blotting paper while you're at it?" he added sarcastically.

"I do," replied the Queen, "but not much."

"Very funny!" said the King. "Anyway, now you have your pepper back, so will you please make me the tarts?"

"Without sugar?" said the Queen.

"What's the matter, isn't the jam sweet enough?" asked the King impatiently.

"I need sugar for the dough, and my sugar has been stolen!"

"Oh, not again!" said the King wearily. "These tarts will never get made!"

Well, recovering the sugar turned out to be a relatively simple affair. The sugar was found in the house of the Duchess, and as events proved, it was stolen by either the Duchess or the Cook, but not both. They made the following statements at the trial:

DUCHESS: The cook did not steal the sugar.
COOK: The Duchess stole the sugar.

The one who stole the sugar was lying. (It is not given whether the other one lied or told the truth.)

Which one stole the sugar? Also, was the other one lying or telling the truth?

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

The simplest way to tackle this is to consider the implications of guilt for both the Duchess and the Cook. If the Duchess stole the sugar, then it really is the case that the cook did not steal the sugar, but who stole the sugar has to be lying. The only alternative is that the Cook stole the sugar. And, of course, for the second part, anyone in Wonderland might be talking nonsense, even not to their own advantage: the Duchess was also lying.

Who Stole the Tarts? Part V

Then who did steal the pepper? "My, my, this is really a difficult case!" said the King.

The next suspects, curiously enough, were the Gryphon, the Mock Turtle, and the Lobster. At the trial, the Gryphon said that the Mock Turtle was innocent and the Mock Turtle said that the Lobster was guilty.

Again, no innocent one lied and no guilty one told the truth.

Who stole the pepper?

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

As in the prior case, one individual is claiming the other is innocent and this instance is formally identical to that other one. This means that Gryphon and Mock Turtle are both innocent and truthful. Accordingly, the Lobster is guilty.