Cross Purposes - Futility Closet

Three men, A, B, and C, are given a test in quick thinking. Each man’s forehead is marked with either a blue or a white cross, and they’re put into an empty room. None of the three can see the color of his own cross, and they aren’t allowed to communicate in any way. Each is told that he can leave the room if he either sees two white crosses or can correctly deduce the color of his own cross.

The men know each other well, and A knows he’s just a bit more alert than the others. He sees that both B and C have blue crosses, and after a moment’s thought he’s able to leave the room, having correctly named the color of his own cross. What was the color, and how did he deduce it?

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His cross was blue. He thought, “If my cross were white, then B would see one blue and one white cross. In that event B would immediately know that his own cross could not be white, because if it were then C would see two white crosses and immediately leave the room. Similarly, if my cross were white then C would know that his own cross must be blue, since otherwise B would have gone out. Since neither B nor C has moved, my cross must be blue.”

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