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You Can Count on Monsters

Richard Evan Schwartz
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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This delightful book is the result of the author’s desire to teach his daughters about primes and factorization. Apart from an introduction and some explanatory material in the back, it consists of one hundred double pages: on the left page is a number and that many dots, arranged into clusters that display its factorization. On the right page is a picture that represents the same information using the author’s “monsters,” which represent the prime numbers.

It’s easier to show than to tell! Here is the double page for five:

The page for Five.

On the left is the number five and five dots, arranged symmetrically. On the right is the five-monster. As is only natural, some of its features occur five times.

Now here is the double page for ten:

The page for Ten.

On the left is the number ten, its factorization, and ten dots, arranged as five groups of two. On the right is the representation of ten in terms of the two monsters involved: the five-monster reappears, and one can also see the two-monster.

More examples can be seen at the author’s web site for the book.

As the numbers become bigger, one gets either bigger monsters (just imagine what 59 looks like!) or more complicated factorizations (think of 60). The images on the right get very complicated when many monsters are involved. The one for 60, for example, has to represent an interaction between a five-monster, a three-monster, and two two-monsters. They all appear, mixed up together in a complicated way that is a lot of fun to untangle. (Luckily, few numbers between 1 and 100 require complex interactions of complex monsters. I don’t think I could easily disentangle an image representing 17×59.)

The introduction is a short explanation of primes and factoring that assumes only that the reader knows how to multiply small numbers. It also explains how the book works and what the images represent. One nice feature is an explanation of why 1 is not a prime. There is a one-monster, but it is “disappointed because it doesn’t get to interact with any of the other monsters.”

At the back of the book is an explanation of Eratosthenes’ Sieve. There is also an account of Euclid’s proof that there are infinitely many monsters. (Unusually, this is actually done as Euclid did it, i.e., not as a proof by contradiction.)

The whole thing is a lot of fun. The book is well produced and nice to look at. The publisher was wise to do it as a small size paperback, since otherwise, with its 200+ pages, it would be too big and heavy for a child to handle. At the Joint Mathematics Meetings, A K Peters was offering refrigerator magnets with the smaller monsters; I don’t see these for sale on their web site, which is too bad.

You Can Count on Monsters is intended as a book for children, and it seems quite likely that an intelligent child would be both amused and instructed by it. Having no small child handy to test this theory, I have to admit that I’m not 100% sure whether a child would enjoy this book. But I certainly do.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. His sons are all grown up, but he can always hope for grandchildren.

The table of contents is not available.