Review of "Mathematical Miniatures"

Mathematical Miniatures, by Svetoslav Savchev and Titu Andreescu. ISBN 0-88385-645-X.

This book consists of 50 short chapters, punctuated by 10 even shorter sets of "coffee break" problems and their solutions. The chapters too are concerned with mathematical problems, of Olympiad type: that is, short pithy problems with short pithy (and elementary, but not easy) solutions. Unless you are unusually good at such things, your coffee breaks may sometimes need to be long.

Each chapter deals with about three problems, related by a common theme or means of solution. The best way to give a feel for a book like this is to list some problems, so here are some, chosen at random.

Twelve painters live in 12 red and blue houses built around a circular lane. Each month one of them goes counterclockwise along the lane and, starting from his own house, repaints the houses according to the following rule: If a house is red, he paints it blue and passes to the next house; if a house is blue, he paints it red and goes home. Each painter does this once a year. Prove that if at least one of the houses is red, then a year later each house will have its initial colour.

Let a, b, c, d and e be integers satisfying 1 <= a < b < c < d < e. Prove that 1/[a,b] + 1/[b,c] + 1/[c,d] + 1/[d,e] <= 15/16, where [x,y] denotes the least common multiple of x and y.

Let 3n-2n be a power of a prime for some positive integer n. Prove that n is a prime.

This isn't a typical olympiad problem book; the idea isn't necessarily that you sit down and puzzle out each problem before turning to the solution. (If you have the necessary self-control and the time, that's probably the best way to read the book even so.) The point is as much to exhibit some beautiful mathematics as to provide the reader with a selection of challenging problems. The authors call it "something of a modest anthology of mathematical verse at a certain level": something of a modestly pretentious description to a certain extent, but not altogether wrong.

In the back of my copy, a piece of paper has been glued over one of the authors' biographies, with a slightly different version of the bio on it. It turns out that the original said

Not one young mind was influenced by the spirit of his seminars.