The Grand Master paints dots on three logicians' heads, where there are at most five dots between any two logicians. Each logician cannot see his own forehead but can see the others'. The Master announces, "At least one of you has an odd number of dots. I painted 5 or 6 dots total. The first to correctly announce the number of dots on his forehead passes." After a brief silence, one shouts out the correct answer. How many dots could there be on his forehead? How did he know?

Solution

How many dots could there be on his forehead?

The logician could have 1 or 3 dots.

How did he know?

Each logician can see one of three scenarios:

• both even
• one odd, one even
• both odd

If one logician sees both even, he must be odd. There are thus 5 dots total, and since no logician has 0 dots, the two events must have 2 dots each. This leaves the last logician with 1 dot. This is immediately obvious, but since there exists a brief pause, no logician sees two evens.

If one logician sees one odd, one even, he is either even and the sum is 5, or he is odd and the sum is 6. If the former, one logician would have already shouted the answer. This is not the case, so he must be odd and the sum is 6. Either logician with an odd number could have shouted the answer. Since one even exists, one logician has 1 dot and the other has 3.

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