Here, once again, is the question:
Three perfect logicians, A, B, and C, sit around a circular table. There is a set of stamps, four red and four green. Each logician has had two stamps pasted on his forehead; he doesn’t know which they are, he can see the other ones’ stamps. They are asked, in rotation, if they know which stamps (red-red, red-green, or green-green) are on their own forehead.
What stamps does B have? (Easy.) And how does he know it? (Hard.)
And here’s the answer.
First, the easy part. The situation is completely symmetrical in red and green. That is, if there were a way to conclude “N greens”, you could run the same solution with red and green switched and get “N reds”. So the solution has to be also be symmetrical, and thus is “one red, one green”.
Now, the hard part.
- First, assume B has two reds.
- If A had two reds, C would see all four reds and would have known at his first turn to speak that he had two greens. So A does not have two reds.
- For the same reason, C does not have two reds.
- Suppose A has two greens. Then C will know he doesn’t have two greens (because B would have seen all four greens) or two reds (because A would have seen all four reds). So C would know he has one of each when he first speaks. Thus A does not have two greens.
- For the same reason, C does not have two greens.
- Thus B knows that A and C both have one of each.
- But consider A’s second turn to speak. He knows he can’t have two reds (or C would have known he had two greens himself) or two greens (because C would have known he had one of each himself). Since A said “no”, this is impossible.
- Thus B does not have two reds.
- By the same logic, B does not have two greens.
- Thus B has one of each.
There are these possibilities:
(Updated from here on, thanks to Alan Scott.)
- A has two greens, B one of each, C two reds
- A has two reds, B one of each, C two greens
- A and B both have one of each, C has 2 of the same color
- B and C both have one of each, A has 2 of the same color
- All three have one of each
In the first two, B knows what he has as soon as C says “no”. In the rest he doesn’t know until A says “no” the second time. There isn’t enough information to determine which is true.