Thursday, August 24, 2006

The King's Wise Men

It's been a while. But it's mid-day puzzle time (no cheating):

The King called the three wisest men in the country to his court to decide who would become his new advisor. He placed a hat on each of their heads, such that each wise man could see all of the other hats, but none of them could see their own. Each hat was either white or blue. The king gave his word to the wise men that at least one of them was wearing a blue hat - in other words, there could be zero, one, or two white hats, but not three. The king also announced that the contest would be fair to all three men. The wise men were also forbidden to speak to each other. The king declared that whichever man stood up first and announced the colour of his own hat would become his new advisor. The wise men sat for a very long time before one stood up and correctly announced the answer. What did he say, and how did he work it out?

Click on the Comments section for the answer...

1 comment: said...


Suppose that you are one of the wise men. Looking at the other wise men, you see they are both wearing white hats. Since the king specified that there were at most two white hats, you would immediately know that your own hat must be blue.

Now suppose that you see the other wise men, and one is wearing a white hat and the other is wearing a blue hat. If your own hat was white, then the man you can see wearing the blue hat would be himself seeing two white hats and would - by the logic above - have immediately declared his hat colour. If he doesn't do this, it can only be because your hat isn't white, therefore it must be blue.

Now suppose that you see the other wise men and both are wearing blue hats. You can't work anything out from this. However, if your own hat was white, then one of the two other wise men would be seeing a blue and a white hat, and would have declared his hat colour by the rule above. Thus, if he hasn't done so, he must also be seeing two blue hats and thus your hat must be blue.

Note also that, in all the rules above, it is only possible for a person wearing a blue hat to win. Thus, since the king said that the contest would be fair to all three men, that must be the configuration in use - and the wise man announced that his hat was blue.