Apples

Difficulty:
★★★☆☆

Problem:

Al, Ben, Chris, and Don each took a number of apples from a bucket of 11 apples. Each person took at least one apple.

Al: Ben, did you take more apples than I did?
Ben: I don't know. Chris, did you take more apples than I did?
Chris: I don't know.

Having heard this conversation, Don was able to figure out exactly how many apples each person took. How many apples did Don take?


Solution:

Let a, b, c, d denote the number of apples that Al, Ben, Chris, and Don took, respectively.

Think about what information is known to everyone after each step. When Al asks Ben whether he took more apples, everyone learns that a ≤ 4. This is because if a ≥ 5, the largest that b can possibly be is 4 in order to make the total 11 (since c and d need to be at least 1), and so Al should already know that a is the greatest. The fact that Al is asking the question means a ≤ 4. So at this point, everyone knows a ∈ {1, 2, 3, 4}.

Then Ben tells Al that he doesn't know the answer to his question. This implies that b ∈ {2, 3, 4}, because:

• If b ≥ 5, Ben would have answered yes.
• If b = 1, Ben would have answered no.

By similar logic, when Chris says he doesn't know whether he took more apples than Ben did, everyone realizes c ∈ {3, 4}.

Finally, Don takes all of this information and figures out what a, b, c are. The only value for d that makes this possible is d = 5.

• If d = 4, there are multiple combinations of a, b, c such that a + b + c + d = 11. (1, 2, 4) and (1, 3, 3) are two of such examples. By the same logic, d cannot be less than 4.
• If d ≥ 6, there are no combinations of a, b, c that satisfy a + b + c + d = 11.

Thus, d = 5, and a, b, and c must take their respective minimum possible values: 1, 2, and 3.