In spite of the tremendous variety of puzzles and paradoxes that confront us in everyday life, many of them share a few underlying principles. For that reason, care must be taken to avoid redundancy in telling tales. The same puzzle can be done up in different garb and look new. It might even fool readers into thinking they have not seen that puzzle before. When I present my standard talk on decision theory and point out that hunting for terrorists is a lot like early detection of cancer, many in the audience look skeptical. They are considering only the outward form on the problem and not the underlying mechanisms.

So, from time to time, the puzzles presented here will resemble ones that have preceded them. If you recognize that, then you are thinking about the decision mechanism instead of the outward appearance.

Today I have two easy examples. One deals with probability in a straightforward way, and the other with a classic problem that has a few new twists.

Probability: This is similar to a problem from last week. In a tourbus, the guide wants to help pass the time, and so starts checking to see when the various passengers have birthdays. The bus holds 44 people. What is the probability that two people have the same birthday? What is the probability that one of the passengers on the bus has your birthday?

Classic Problem: This is another one that I probably first saw in Martin Gardener’s column in Scientific American, but I will give it a new twist that I have not seen elsewhere. Feedback on this is welcome. The problem is simple. You have twelve coins. Eleven of them are good, but one is counterfeit. The counterfeit one has a slightly different weight than the others. You have a standard two-pan balance like the kinds seen in old depictions of Liberty holding the scales of justice. You could weigh one coin against the others one at a time and eventually find the bad coin. However, a wise man tells you that you can unambiguously find the bad coin with only three weighings. How can this be?

So much for the classic problem. Now let’s ask it the other way around. How many coins can you test with only three weighings? Do you think twelve in the maximum?

From this, it is only a step to making a table of the maximum number of coins that can be separated as a function of the total number of weighings.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.