I'm re-reading Mosteller's Fifty Challenging Problems in Probability With Solutions while traveling in Japan. This is absolutely one of my favourite books (not least because it is such a tiny volume). Fredrick Mosteller was "one of the most eminent statisticians of the 20th century" (according to Wikipedia), and a statistician who was passionate about education. The book actually has 56 problems, generally simply stated, like the following:
A drawer contains red socks and black socks. When two socks are drawn at random, the probability that they are red is 1/2. (a) How small can the number of socks in the drawer be? (b) How small if the number of black socks is even?
Reading the problems in this book lead me to thinking about what constitutes a 'good' puzzle. Personally, I like those problems that have one or more of the following qualities:
- Counter-intuitive: the birthday problem is like this, 23 sounds like a low number.
- Relies on basic number theory: Mosteller's problems require things like working with geometric series, binomial coefficents, etc.
- Motivates you to think probablistically about things that you don't generally consider in that way: for me, those problems involving randomly throwing sticks on a table are fun (but I do remember Johnny Ball performing this experiment live once on the BBC...)
- Problems that seem recursive but can be solved simply.
- Entertainingly cast as a brief story (e.g., a three person dual).
Intruigingly, Mosteller's solutions, while nicely written with an engaging informality, still require concentration, especially when they skip a few steps on route to the prize. Generally, he demonstrates a great way to attack the problems by first playing with a few examples, and then running through one or more full solutions (often employing induction and negation tactics).
If you are looking for other good sources of puzzles, Car Talk (on NPR, Saturday at 9) offers not just amusing car problems, but also the occasional Mosteller like puzzler...
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