(Un)Certainty

June 3, 2009

From John Derbyshire (best not to ask why I was reading this article) comes the following puzzle:

Ken and Bob find themselves in possession of three blank-sided dice. These are ordinary cubic dice, with six faces each.

Ken writes the numbers from 1 to 18 on the sides. No number is repeated. Each side of each of the three dice now shows a number from 1 to 18.

Bob then chooses one of the three dice. Ken chooses one of the other two. The third die is discarded.

The two men then play a game of dice war. The war consists of a hundred “rounds.” In each round, first Ken rolls his die, then Bob rolls his. The man with the highest number showing on the topmost face of his die, wins the round.

Whichever man wins the larger number of rounds, wins the war.

Question: If both men followed the strategy that gave them the best mathematical chance to win this war, what would the numbers on the dice look like?

I post it because it’s a mildly interesting puzzle, but mainly as an example of how even my strong intuition can be wrong. When I first read it, I was convinced that there was no way Ken could give himself an advantage. When I say convinced, that means that I would have been willing to bet something like $10 on it. Even as I was thinking that, though, it occurred to me that people generally don’t write puzzles to which there is no solution. So I kept working on it. I haven’t actually found the solution, but I’m somewhat convinced that the bulk of the interesting insights that can be gained are gained from thinking about the puzzle, not in actually finding the unique solution.

As far as I’m concerned, the true puzzle is this: What two-word phrase should you google to find a discussion of problems like this one?

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