## Monday, May 05, 2008

### How Many Apples Does it Take

Four wise men sat at a table eating apples during a dinner party. When they started, there were 11 apples. By the end of the party, all 11 were eaten. Each man ate at least one apple, and everyone knew that. Each man also knew how many he had eaten, but did not know how many the others had eaten.

The first wise man, let's call him Alfred, asked the second, let's call him Bert, "Did you eat more apples than I did, Bert?"

Bert answered, "I don't know. Did you, Carl, eat more apples than I did?"

Carl responded, "I don't know."

David immediately said, "Aha!"

David knows how many apples each person ate. Do you?

When you answer, lay out your logic for the rest of us! You must assume that each wise man is perfectly logical and can think his way through the puzzle.

1. Ok, I'm going to take a shot at this one...

First, we'll assume that when a wise man asks a question, he cannot figure out the answer with logic. This means that when a wise man asks another wise man if he ate more apples, the one asking the question cannot have more than 4 apples (because if he had 5 or more, he would necessarily have the most apples of anyone and wouldn't need to ask). Also, if the one answering says, "I don't know," he also must have 4 apples or fewer, by the same logic.

This means that Alfred has between 1 and 4 apples. Then, Bert must have between 2 and 4 apples, since he would have just said "no" to Alfred's question if had 1 apple. Then, Carl must have either 3 or 4 apples, since he would be able to figure out the same things we've laid out here, and he would have just said "no" to Bert's question if he had 1 or 2 apples. Therefore, the only way David would have enough information to know everyone's apple counts is if he had 5 apples, which would only leave one possibility for the apple counts of the other three members (the fewest possible for each one -- there is only one way to make 6 apples to round out the 11 total apples).

The final totals would be:
Alfred -- 1 apple
Bert -- 2 apples
Carl -- 3 apples
David -- 5 apples

2. Nice work as always Abe. Alfred had 1, Bert had 2, Carl had 3 and David had 5. Alfred could not have had 5 or more. Bert could not have eaten only 1. Neither could he have eaten 5 or more.

So Carl knows Bert had 2, 3 or 4 apples. But he still doesn't know if Bert had more than he did. That means he must have had 3 or 4.

The only way David would know everyone else's totals is if he had had 5. He knows the others must have eaten 1, 2 and 3.

3. David can only know how many apples each person ate if he ate 8 apples. This would leave 1 for each of the remaining three people.

The reason is simple. You only know that each person ate 1 apple and that all 11 apples have been consumed.

Answering "I don't know" doesn't mean that they ate more or less than you. It just means that they didn't eat more than 50% of the 11 apples.

4. 2
2
3
4

David has told 'Aha'. It meant he has eaten maximum apples. He could not have eaten 5 or more apple as in that case he would know in advance which is not the case.

So if david has eaten 4 apples. Then there are 7 apples left.
1 has been eaten by remaining 3 for sure.
So now 4 apples are left.
Since Albert has eaten more than 1 apple that is why he is asking a question to others whether they have eaten more than him. If he would hv eaten only 1 apple, then he would not hv asked this question as others would have obviously eaten more than him.
Berk does not know the answer as he has also eaten more than 1 apple.

Carl knows that

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