Problems from the Discrete to the Continuous

This is a collection of interesting problems that can be solved in several ways, and at least one of those ways belongs to discrete mathematics, and at least one of those ways belongs to a field that uses continuous tools, like analysis or probability.

A good example is a pair of facts from number theory. Select two integers from the set \(\{1,2,...n\}\) uniformly at random. What is the probability that they are relatively prime? Now select just one integer from the same set uniformly at random. What is the probability that it does not have a divisor that is larger than one and is a perfect square? It turns out that the answer to both questions is \(6/\pi^2\), and the author proves this fact in numerous ways, some of which are his original ideas.

Other chapters in the book discuss questions from graph theory, Ramsey theory, the enumerative combinatorics of permutations, and more questions from number theory. The problems are well chosen and interesting.

My only critical remark is that the book would have deserved better editing. Many sentences in the book are simply too long. Somebody could have told the author that it is bad style to start a theorem with a formula. And there are some weird statements, like when on page 6, the author talks about Herb Wilf's “little book.” That book, generatingfunctionology, is more than 50 percent longer than the book under review! It also deserved a proper citation of its title, not a misspelled one. These very avoidable flaws are an unnecessary distraction.

Miklós Bóna is Professor of Mathematics at the University of Florida.