Logic Problem: Figuring Out How Many Apples Each Man Ate

[MeesaWorldWide]

TL;DR

Intriguing logic problem that has me stumped

Hey, I'm new to these forums, so thanks in advance for any help I get! :D

4 men sat around a table that had a dish with 11 apples in it. By the time their discussion was over, all the apples had been eaten. Each man had at least one apple, and each man knew that fact. Each man knew the number of apples that he ate, but not how many apples each of the other men ate. They needed to figure out how many apples each of them ate by only asking questions that they didn't know the answers to.
Alonso asks "Bert, did you eat more apples than I did?"
Bert responds "I don't know. George, did you eat more apples than I did?"
George responds "I don't know".
Kurt suddenly cries "Aha" because he figured out how many apples each man ate. Can you also figure that out?

The answer is 1 (Alonso), 2 (Bert), 3 (George), 5 (Kurt), but I cannot figure out how one would logic their way to that answer.

 

[Dale]

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

If Bert had eaten only one apple then he would have known that he did not eat more apples than Alonso. So Bert must have eaten at least two.

Bert responds "I don't know. George, did you eat more apples than I did?"
George responds "I don't know".

If George had eaten only one or two apples then he would have known that he did not eat more apples than Bert. Therefore George must have eaten at least three apples.

Kurt suddenly cries "Aha" because he figured out how many apples each man ate.

So at this point everyone knows Alonso ate at least one, Bert ate at least two, and George ate at least three.

Kurt can only know how much everyone else ate if knowledge of the number of apples he ate lets him uniquely determine how many apples everyone else ate. So he must have eaten five. That allows the solution given above.

If Kurt ate six or more there would not have been enough apples to go around, and if he ate four or fewer there would not have been a unique solution.

 

[DaveC426913]

Wut Dale sed. Me to.

 

[WWGD]

TL;DR Summary: Intriguing logic problem that has me stumped

Hey, I'm new to these forums, so thanks in advance for any help I get! :D

4 men sat around a table that had a dish with 11 apples in it. By the time their discussion was over, all the apples had been eaten. Each man had at least one apple, and each man knew that fact. Each man knew the number of apples that he ate, but not how many apples each of the other men ate. They needed to figure out how many apples each of them ate by only asking questions that they didn't know the answers to.
Alonso asks "Bert, did you eat more apples than I did?"
Bert responds "I don't know. George, did you eat more apples than I did?"
George responds "I don't know".
Kurt suddenly cries "Aha" because he figured out how many apples each man ate. Can you also figure that out?

The answer is 1 (Alonso), 2 (Bert), 3 (George), 5 (Kurt), but I cannot figure out how one would logic their way to that answer.

Welcome to PF, MeesaWorldWide!

 

[PeroK]

Five apples are a lot to eat at one sitting.

 

[bahamagreen]

If Bert had eaten only one apple then he would have known that he did not eat more apples than Alonso. So Bert must have eaten at least two.

If George had eaten only one or two apples then he would have known that he did not eat more apples than Bert. Therefore George must have eaten at least three apples.

So at this point everyone knows Alonso ate at least one, Bert ate at least two, and George ate at least three.

Kurt can only know how much everyone else ate if knowledge of the number of apples he ate lets him uniquely determine how many apples everyone else ate. So he must have eaten five. That allows the solution given above.

If Kurt ate six or more there would not have been enough apples to go around, and if he ate four or fewer there would not have been a unique solution.

Hi Dale, what are your thoughts on this?
It is based on the stipulation
"...by only asking questions that they didn't know the answers to."

Alonso asks "Bert, did you eat more apples than I did?"

Alonso can only ask this question not knowing the answer only if he did not eat more than 4 apples. Of the 11 apples, each ate at least 1, so 7 apples remain to be accounted for. If Alonzo ate 5 apples, the most anyone else could had eaten would be 4.
So Alonso ate no more than 4 apples. Everyone else now knows this.

Bert responds "I don't know"

Bert knows at most Alonso ate 4 apples. To reply that he does not know means Bert did not eat 5 apples (that is more, so reply would be"yes") or 4 apples (that is a tie, so reply would be "no"). Bert ate no more than 3 apples. Everyone else now knows this, too.

Bert asks George, "Did you eat more apples than I did?"

George knows at most Bert ate 3 apples.

George responds "I don't know"

To reply that he does not know means George did not eat more than 2 apples. Everyone else now knows this, too.

Kurt, knowing the two unique possibilities...

Alonso Bert George Kurt
4 3 2 2
3 2 1 5

...Kurt knows he either ate 2 or 5 apples, and then knows how many the others ate.

Kurt suddenly cries "Aha" ...

 

[Dale]

That is interesting because there isn't a unique solution, but whatever solution it is, he knows. Actually, wouldn't all four say "Aha" at the same time?

 

[bahamagreen]

That is interesting because there isn't a unique solution, but whatever solution it is, he knows. Actually, wouldn't all four say "Aha" at the same time?

Thinking about the "Aha" question has made me believe Kurt is the first to know unless George ate 2 apples, then George is the first to know!

Alonso's question reveals he either ate 1, 2, 3, or 4 apples
Bert's reply reveals he either ate 1, 2, or 3 apples
George's reply reveals he either ate 1 or 2 apples
The possible solutions known to all at that point are:

Alonso Bert George Kurt
4 3 2 2
4 3 1 3
4 2 1 4
3 2 1 5

The sequence of replies requires a strictly monotonic descending number of apples prefix for the solutions. George knows how many he ate, so if he ate 2, then he says "Aha" first because Kurt's values (2, 3, 4, 5) are unique across the four solutions.

 

[PeroK]

Thinking about the "Aha" question has made me believe Kurt is the first to know unless George ate 2 apples, then George is the first to know!

Alonso's question reveals he either ate 1, 2, 3, or 4 apples
Bert's reply reveals he either ate 1, 2, or 3 apples
George's reply reveals he either ate 1 or 2 apples
The possible solutions known to all at that point are:

Alonso Bert George Kurt
4 3 2 2
4 3 1 3
4 2 1 4
3 2 1 5

The sequence of replies requires a strictly monotonic descending number of apples prefix for the solutions. George knows how many he ate, so if he ate 2, then he says "Aha" first because Kurt's values (2, 3, 4, 5) are unique across the four solutions.

I don't agree with this. If Bert ate only 1 apple he would have answered no, so Bert ate 2,3 or 4 apples. If George ate only 2 apples, then he would have replied no. So, George ate 3 or 4.

At this point, none of Alonso, Bert or George can conclude anything. The status is:

A: 1,2,3 or 4
B: 2,3 or 4
G: 3 or 4

Which means they ate 6 apples or more between them. Unless Kurt ate 5, there is no unique solution. Only if Kurt ate 5 can he solve the problem.

 

[bahamagreen]

I don't agree with this. If Bert ate only 1 apple he would have answered no, so Bert ate 2,3 or 4 apples. If George ate on 2 apples, then he would have replied no. So, George ate 3 or 4.

At this point, none of Alonso, Bert or George can conclude anything. The status is:

A: 1,2,3 or 4
B: 2,3 or 4
G: 3 or 4

Which means they are 6 apples or more between them. Unless Kurt ate 5, there is no unique solution. Only if Kurt ate 5 can he solve the problem.

You're right... I see 20 coherent solutions that follow the replies and sum to 11.
Only if Kurt ate 5 will present a unique solution...

A B G K
========
1 2 3 5 <------- unique solution for Kurt
1 2 4 4
1 3 3 4
1 3 4 3
1 4 3 3
1 4 4 2
2 2 3 4
2 2 4 3
2 3 3 3
2 3 4 2
2 4 3 2
2 4 4 1
3 2 3 3
3 2 4 2
3 3 3 2
3 3 4 1
3 4 3 1
4 2 3 2
4 2 4 1
4 3 3 1