## Friday, September 24, 2004

### The Three-Sided Coin

I thought up the following puzzle today. Since I came up with it myself, I don’t know whether it’s difficult or not, so I’m posting it here to find out. My solution is not the only solution, but I suspect that all viable solutions will rely on the same basic principle. Here goes:
Using only a coin, how can you generate a probability equal to exactly one-third?
Here’s more context, if you feel the need for it:
Suppose you want to vacation at the beach, mountains, or desert. You want to choose your destination randomly, with equal chances for all three outcomes. Your only randomization device is a coin. What should you do?
If you think you have the answer, please email it to me at the address listed in the right-hand column. Also, give me an estimate of how long you had to think about it (be honest!). PLEASE DON’T POST YOUR ANSWER IN THE COMMENTS SECTION. I will post an update later with the (or a) solution, along with the name of the first person who got it. I’ll post multiple solutions if they rely on different principles, though I don’t think that will happen.

Let me head off one obvious, but incorrect, solution. “Flip one coin. If it comes up heads, go to the beach. If it comes up tails, flip the coin again, and go to the mountains if it’s heads and the desert if it’s tails.” This solution fails, because it generates one-half probability of beach, and one-quarter probability (each) for mountains and desert.

UPDATE: I've made a new post with the answers. Thanks to everyone for their responses. Please, no more emails!