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Challenging Mathematical Problems with Elementary Solutions, Vol. 1: Combinatorial Analysis and Probability Theory

A. M. Yaglom and I. M. Yaglom
Dover Publications
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Problem Book
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a two-volume collection of 174 problems, with hints and complete solutions, aimed at high-school and early college students. The “elementary solutions” in the title means that no techniques are used that would be unfamiliar to advanced high-school students. The emphasis is on ingenuity and insight rather than mathematical knowledge.

The problems are divided into two volumes. Volume I has all the problems on combinatorics, along with a lot of discrete probability. This volume makes heavy use of the binomial theorem and the inclusion-exclusion principle. Volume II has everything else, and is quite varied, but has a lot of elementary analysis, including some expressions for \(\pi\), some limits for \(e\), a little bit of asymptotics, and the beginnings of the distribution of prime numbers. There is a mysterious chapter on Chebyshev polynomials that develops their properties but doesn’t do anything with them. Most problems are quite reasonable in difficulty, and the more difficult problems are marked with one to three asterisks. Framing the items as problems is something of a gimmick, to encourage students to work on their own; the authors seem to have picked the proofs they wanted to show, and then invented problems to do so. Some of the solutions are quite intricate, and I doubt that the target audience would be able to work out the solutions without a lot of coaching.

For the most part the restriction to elementary methods does not distort the presentation; we would have done them this way even with more advanced knowledge. There are several places where areas are calculated by a method of exhaustion, and it would have been very handy to have the definite integral here. In the combinatorial part there are several places where combinations are worked out laboriously, mimicking what generating functions would do for you with much less effort.

The present volumes are based on a Russian-language work published in 1954. The problems were rearranged a little, and everyday Russian terminology converted into everyday English terminology. The English-language volumes were published by Holden-Day in 1964 and 1967, and reprinted without change by Dover in 1987. These books are often cited in connection with the Moscow Mathematical Olympiads, but although a few problems are taken from that source, most of the results are familiar pieces of mathematics and not problems cooked up for a competition.

The page counts are about 20% problem statements, 5% hints, and 75% complete solutions. The solutions are very good: they are clear, and they take enough time to explain all the details. The hints are extremely brief and I didn’t think they would be very useful. One confusing aspect of the book is that it uses both natural and common logarithms, with the former being denoted ln (good) and the latter log (confusing).

I have mixed feelings about the problem-book approach here. My gold standard for problem books is Pólya & Szego’s Problems and Theorems in Analysis, one of whose key features is that problems always appear in a sequence, where you learn methods (not tricks) and apply them to related problems. The present book is slanted much more to isolated tricks, although there are some runs of problems where the same ideas are used again. On the plus side, most of the problems are interesting, and this format should entice students into struggling with the problems rather than skimming the proofs and solutions.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.


Introductory problems
II. The representation of integers as sums and products
III. Combinatorial problems on the chessboard
IV. Geometric problems on combinatorial analysis
VI. Problems on computing probabilities
V. Problems on the binomial coefficients
VII. Experiments with infinitely many possible outcomes
VIII. Experiments with a continuum of possible outcomes