Problem:

There are three monks on a silent retreat. They can only speak once during the entire retreat and can only say one sentence when they do. Each monk wears a hat that is either white or black. They can see each other’s hats but not their own. One day, they are seated in a triangle so that each monk can see the others’ hats. The rule is if a monk sees two hats of the same color, he must declare the color of his own hat. One morning, all three monks suddenly declare the color of their own hats. What colors are the hats?

Scroll to the bottom of the page for the answer

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Solution:

For all monks to declare the color of their hats, they must each see two hats of the same color. If any monk had seen two hats of different colors, he would have been unable to declare his own hat's color confidently. Since each monk successfully declares his hat color, all three monks must be wearing hats of the same color. Since they all declared correctly without contradiction, they can only be seeing two hats of the color they are wearing, confirming all hats are the same color.