Solution to the "What's That on My Head?" problem


From Ann’s answer, you know that your total is either 17, the same as Bill’s or 11, the same as Charlie’s.

Bill says that he sees all the even numbers. You can see that he sees every even number but the 6; so you must have a 6 on your head.

Ann now says she knows what’s on her head. Here’s where things get tricky. Ann would know what she has only if you yourself do not have a 2, a 4, or an 8. So, now all you have to do is come up with three numbers that total either 11 or 17, have a 6, and do not have 2, 4, or 8.

Maybe you could work this out by trial and error. But notice that either total you must reach is an odd number. You know you have a 6, an even number. The only way to get an odd total is for one of the other numbers to be odd and the other be even. Hmm. Since the other even number can’t be a 2, a 4, or an 8, it must also be a 6. So you have two 6s. The third number must then be a 5, which will bring the total to 17. So the answer is 6 6 5.

Just as chess puzzles aren’t much like chess, this “What’s That on My Head?” puzzle isn’t much like “What’s That on My Head?” When playing the game, you would hardly ever take into account the reasoning of other players. But it could happen.

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