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Professor Higgins's Problem Collection

Peter M. Higgins
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Megan Patnott
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Professor Higgins’s Problem Collection is, unsurprisingly, a collection of problems. It contains more than fifty problems and solutions, grouped into five themed parts and sorted, within each part, roughly according to difficulty. Each part begins with a brief overview of the mathematics required to solve its problems, and ends with more detailed notes on mathematical tools used by the solutions and brief comments on some of the problems. Odd-numbered pages contain a problem or, more often, a pair of related problems sharing a title, with the solution(s) appearing on the back. This is the nicest organization of problems and solutions that I’ve seen, as it makes it easy not to accidentally see the answer too early, but also doesn’t require bookmarking two spots in the book. American readers should be aware that the book has not been Americanized; in particular, it follows the British convention of using a vertically centered dot for a decimal point rather than a period.

Problems in the first part, “Number,” mostly ask the reader to play with integers. A reader with only a high school algebra background should be able to get started on all of these problems, although a few of them ask for proofs, which such a reader is likely to be uncomfortable with. The second part, “Algebra,” focuses on problems that require a clever solution to an equation. Although some of these problems require only a high school algebra background, two of them involve trigonometry and two involve both infinite series and integration. “Geometry” poses problems on areas, lengths, angles, volumes, etc. to the reader. Almost all of these problems can be solved algebraically, and the given solutions are algebraic; however, I found calculus to be a more natural way to tackle several of them. Counting and probability are the theme of part four, “Chance and Combinations.” With the exception of the last problem, which requires multivariable calculus, the problems in this part are more accessible than those in parts two and three. The final problems in the final part of the book, “Movement,” are all about figuring out how some object is moving, often in relation to another object. The first problems are very accessible, and require only high school algebra, but later problems require trigonometry and differential calculus.

The book claims, on the back, that most of its problems are accessible to most people, with only a few requiring “mathematics up to advanced secondary school level.” I think that it would likely be more accurate to say that some of the problems will be accessible to most people, but that most of the problems require advanced high school mathematics and a few require calculus. I suspect that this difference can be accounted for by differences between the British and American school systems (the author is British). Differences in school systems may also account for the fact that a larger proportion than intended of the problems seemed to me to be standard classroom exercises.

Overall, this is a nice collection of problems, with a good amount of variation in the problem types. It’s probably more useful as a source of enrichment problems for the classroom than as a purely recreational book, but an advanced high school student or first year undergraduate looking for a challenge may also enjoy it.

Megan Patnott is an Assistant Professor of Mathematics at Regis University in Denver, CO. Her training is in algebraic geometry and commutative algebra.

1. Number
2. Algebra
3. Geometry & Shape
4. Chance & Combinations
5. Movement