# An Orange Consumption Problem

K Sengupta
Four perfect logicians sat around a table that had a dish with 11 oranges in it. The chat was intense, and they ended up eating all of the oranges. Everybody had at least one orange, and everyone knew that fact, and each logician knew the number of oranges that he ate. They didn't know how many oranges each of the other ate, though. They agreed to ask only questions that they didn't know the answers to.

Their queries are as follows:

Addison:: Did you eat more oranges that I did, Braxton?

Braxton:: I don't know. Did you, Carlos, eat more oranges than I did?

Carlos: I don't know.

Darryl : Aha!

Darryl figured out how many oranges each person ate.

Can you also deduce the number of oranges consumed by each logician?

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Jimmy Snyder

Braxton must have had at least 2 since if he only had 1, he would know the answer to the question posed him. He must have had less than 5 or he would not be able to ask the question he asked.
Carlos knew that Braxton must have had at least 2 so he must have had at least 3, or he would know the answer to the question posed him. He must also have had less than 6 or he would have known the answer to the question.
Darryl must have had 5 because in any other case, he couldn't disambiguate between possible cases. For instance, if Darryl had 4 oranges, there are at least two possibilites:
A B C D
2 2 3 4
1 2 4 4

A B C D
3 2 3 3
2 2 4 3

A B C D
3 2 4 2
2 3 4 2

A B C D
3 3 4 1
2 4 4 1

Since Darryl knew, Darryl must have had 5 oranges
A B C D
1 2 3 5

eom.

davee123
Can you also deduce the number of oranges consumed by each logician?

Addison ate 1, Braxton ate 2, Carlos ate 2, Darryl ate 6

Darryl knows he ate 6, so there are only 5 left to have been eaten between the other 3. Initially, Darryl knows the possibilities could be (in order of Addison, Braxton, Carlos):

1, 1, 3
1, 3, 1
3, 1, 1
1, 2, 2
2, 1, 2
2, 2, 1

When Braxton states that he doesn't know if he ate more than Addison, this proves that Braxton ate at *least* 2, because otherwise Braxton could definitively say that he did NOT eat more than Addison. Hence, Darryl knows the possibilities are:

1, 3, 1
1, 2, 2
2, 2, 1

When Carlos states that he doesn't know if he ate more than Braxton, this shows (again) that Carlos ate at *least* 2. So Darryl is left with only one possibility:

1, 2, 2

DaveE

davee123

Since Darryl knew, Darryl must have had 5 oranges
A B C D
1 2 3 5

I think that's not quite right-- if Darryl had 5 oranges, he's left with the possible situations of:

A B C D
1 2 3 5
1 3 2 5
2 2 2 5

He can't determine which of those is correct.

DaveE

Jimmy Snyder
I think that's not quite right
Carlos knew that Braxton had at least 2 for the same reason that you do. Therefore he must have had at least 3.
eom

davee123
Carlos knew that Braxton had at least 2 for the same reason that you do. Therefore he must have had at least 3.
eom

Ahhh, true! Good catch!

DaveE