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Ingenious Mathematical Problems and Methods

Louis A. Graham
Dover Publications
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Underwood Dudley
, on

This is a collection of one hundred problems in recreational mathematics (57 pages) and their solutions (180 pages). Its publisher, Dover Publications, mostly puts out reprints, but this volume is an original. It first appeared in 1959 and is now being reissued.

Its contents, expanded for the book, first appeared in a periodical, the Graham Dial, now defunct, from around 1941 to 1958. Today’s most powerful search engines could turn up no more information about it. Likewise, its author, Louis A. Graham, has disappeared into the mists of history. In the 1959 edition he is identified as “Engineering executive and founder, Graham Transmissions, Inc.” but that is all that I know about him.

His periodical circulated “to over 25,000 engineers and production executives.” It contained a “private corner for mathematicians” that he edited, with problems and solutions contributed by readers. Mathematicians seem to have been mostly unaware of it because the affiliations of solvers are given as Bundy Tubing, National Forge & Ordnance, Onsrud Machine Works, Anglo-Canadian Pulp and Paper Mills, and like places. The companies have gone away, taking their employees with them. Do engineers who work on recreational problems and write up their solutions still exist? Perhaps not. Well, times change.

The problems, which are in the tradition of H. E. Dudeney and Sam Loyd, are mostly drawn, as would be expected, from geometry and number theory, but there are others from probability, electricity, and mechanics. A few, but not many, involve calculus.

Some of the problems are classics. The four fours problem is here, with a representation of 71 that not many would think of: 4 degrees and 44 minutes divided by 4 is 71 minutes. The problem of the one counterfeit coin among twelve, to be discovered with three weighings is also — it may have appeared in print for the first time in the Dial in 1945. The truck getting across the desert by caching gasoline is motoring along, but most of the problems will be new to readers, even those learned in recreational mathematics. Many problems have more than one solution presented, a good feature.

For good measure, the author includes some “mathematical nursery rhymes”, along the lines of

            A mathematician named Ray
            Says extraction of cubes is child’s play.
            You don’t need equations
            Or long calculations
            Just hot water to run on the tray.

Though this is not the way that most ice cubes are gotten today, the book’s contents have dated hardly at all. The solutions are clear and well written. Dover is to be congratulated for keeping the book available.

Woody Dudley finished in 104th place in the 1956 Putnam examination, and his problem-solving ability has been going downhill ever since.

The table of contents is not available.