Can you find the optimum strategy for guessing the highest number?

The radio show Car Talk features a weekly puzzler, and I found this week’s one to be pretty good. It inspired me to come up with a variation.

The problem given was the following. A number is written on each of 3 pieces of paper and the papers are turned over. You start by turning over the first piece of paper. If you think that it has the largest number then that is your guess. Otherwise, go on to the next piece of paper and select the number on it if you think it is the largest. Otherwise you are stuck with the number on the third piece of paper.

I welcome you to find the best strategy for choosing the largest number and to compute the probability that it will in fact choose the largest number.

After solving this problem, anyone with an interest in math knows the next step is to try to generalize it. In this case, one obvious generalization is to extend the problem to n numbers. I thought about it long enough to realize that finding a simple solution was beyond my meager abilities. To see what is involved, consider the case where n=5. What is the best strategy for guessing the largest number and what is the probability that it works.

When I get a chance, I will be writing a computer program to investigate other values of n to see if there is a simple pattern.