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March 14, 2003

Martian Knightlife Card Puzzle

Doing It' is Sometimes the Best Way

A scene in Martian Knightlife features a classic puzzle often known as the "three door problem," but in this case illustrated with regular playing cards. I present you with three cards face down, one of which is an Ace, say. I know which it is. You are invited to guess. After you have done so, I turn over one of the three to show that it isn't the Ace. The question now is, for the best chance of picking it from the two that are left, should you (a) change your choice, (b) keep it the same, or (c) it makes no difference? The correct answer is (a), which doubles your chances. However, this is highly counter-intuitive to many people and can provoke heated arguments against (interestingly, often from technical and mathematical people).

I got a letter from a college student by the name of Benny Tsai, insisting that with two choices the odds had to be simply 50-50, and asking me to explain if not, why not. Well, I thought I had in the book. But it is a fact of this particular puzzle that many people for whom the verbal arguments don't work see the logic instantly when they try actually working it through. I suggested getting together with a friend and recording the results for sticking with the original choice 10 or 20 times in a row, and then doing the same thing but this time changing the choice. I wrote: You'll find that the second time around, you score about double what you did the first time. But more important, I would bet that long before you finish, something "clicks" and you'll suddenly say, "Right! I get it. I can see what's happening."

Well, due to the holiday schedule Benny didn't have any friends available just at the moment, so he wrote a Java program to simulate the situation instead, and it turned out to be even more revealing. In 10,000 runs, the original guess was right 3,330 times, while switching it produced the right answer 6,670 times. Even I was impressed. Anyone curious to try it for themselves, or who has an interest in such puzzles that they'd like to share, is welcome to contact Benny directly.

The late Julian Simon showed me this puzzle over dinner one night in Washington as an illustration of the effectiveness in education of working problems through as opposed to listening to theory.

 
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