Ahmes' Legacy

This is a study of problem-solving, that is founded on general psychological principles and then specialized to mathematical puzzles. For the most part it’s not a study of how math students or mathematicians solve problems; it has little in common with books such as the several by Pólya on mathematical discovery.

The book makes a strong but not clearly-defined distinction between puzzles and problems. On p. 3 it says, “a problem presents information that leads directly to a solution; a puzzle, on the other hand, does not.” I think what this means is that a problem is something that can be solved by applying standard methods; in a math textbook this would be called a drill or an exercise. Puzzles on the other hand require some added insight or intuition. In math this would be a research problem. This book, however, focuses almost exclusively on puzzles within recreational mathematics, and surveys a wide variety of them.

The book’s title comes from the Ahmes Papyrus (also called the Rhind Papyrus), that contains some of the oldest known puzzles. One of the peculiarities of the problem-puzzle distinction is that is it time-dependent, because we develop new standard methods from working on particular puzzles. Thus many of the questions in the Ahmes Papyrus were puzzles when it was written, but are problems today.

The two main psychological concepts used here are abduction and archetypes. Abduction is another word for hunches, intuition, and sudden insight into a problem (also known today as the Aha effect). Archetypes are used in Jung’s sense as an explanation of why the same types of puzzles keep recurring, even in different cultures that have no contact with each other. The book’s focus is on the statements of the puzzles, in particular the structure of each one and in creating a rough taxonomy of their types. It gives very little attention to how they are solved, perhaps because by definition none of them can be solved systematically. The book presents a “cognitive flow” model (pp. 93, 105) in which we start our solution of a puzzle with intuition or imagination, and then use logic to generalize and theorize about a solution.

The mathematical material in the book is not very reliable. Unfortunately it is also not footnoted, so there’s no easy way to check it. The book uses the term “squaring the circle” on p. 16 (and later) to mean inscribing a square in a circle. There’s a good explanation of the Sieve of Eratosthenes on p. 19, but the figure to illustrate it is just a list of the first 100 integers (arranged in a grid) without anything sieved out. On pp. 19–20 the discussion of the Riemann zeta function and of its relation to the distribution of prime numbers is garbled. The book states on pp. 35–36 that Oswald Veblen was a popular re-worker of classic mathematical puzzles, a claim that is not footnoted and that I have not been able to find anywhere else (apart from one article of his on magic squares). On p. 41 Danesi has Gauss becoming interested in the Eight Queens problem in 1859, four years after his death (Gauss did work on the problem in 1850). On p. 81 the book explains that the golden ratio \(\phi\) is \(0.618\dots\) (actually this number is \(1/\phi\)), and on p. 84 it gives the Binet formula for the Fibonacci numbers, but neglects to write it in terms of the golden ratio. The description of constructibility and trisection on p. 86 is garbled. On p. 93 there’s a reasonable discussion of the St. Petersburg Paradox, including the fact that the expectation is infinite; but this is followed by an extremely confusing discussion that attempts to remove the paradox; I think it is based on Daniel Bernoulli’s idea of using the “moral expectation” instead of the mathematical expectation. On p. 96 it says that Gödel showed that “undecidability is a fact of the human brain”. On p. 147 the book misattributes the proof of Fermat’s Last Theorem for exponent 3 to Gauss (should be Euler).

Bottom line: Probably not very useful for mathematicians or students. The author’s speciality is semiotics and the scholarly works quoted are from semiotics; I am not able to judge how useful this book would be in semiotics. For mathematicians the works of Pólya are much more useful for studying problem solving.

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