# Half a Century of Pythagoras Magazine

This is the English translation of the book *De Pythagoras Code* that appeared in Dutch in 2011. It brings an anthology of puzzles, and papers that appeared in *Pythagoras*, a magazine appearing in The Netherlands since 1960 and that wants to bring recreational and popular mathematics to youngsters (and of course for anyone casually or professionally interested in mathematics).

There are of course many books on the market today that bring a mixture of puzzles, history, and art all related to mathematics. Many of the items discussed in these books are circulated and iterated so that they became a part of mathematical folklore. This book is essentially different in that most of the items are original contributions that were harvested from the issues published in half a century of the magazine's existence. And there are some real gems among them. The offer is also very diverse, both in type of contribution, in complexity and in skill required to find the solution. The problems are often challenging and their solutions are surprising. Enough properties to attract any passionate puzzler.

The book starts off with a hundred brainteasers. Short problems, formulated in only a few lines, and occasionally a plot. For example how to cut the ying-yang symbol into two congruent pieces or how to generate all numbers from 2 to 20 in the simplest possible way using only four 4's combined in brackets and algebraic operations.

A second chapter has more complex problems, like card tricks, board and other games (and their winning strategies), cutting puzzles, knot puzzles, tangrams, logic puzzles, etc., they are all there.

Some text related to art and mathematics are collected in the next chapter. Of course, there is Escher and impossible objects (Bruno Ernst, Escher's biographer was one of the founders of the magazine). Artist Popke Bakker's raw material are wooden beams with a square cross section. He cuts them under certain angles and glues the pieces back together after rotation, which gives artistic skeleton structures. Many art forms are represented such as perspective drawing, architecture, but also the writings of Raymond Queneau and others.

There is a rather extensive chapter four devoted to geometry, plane geometry as well as three-dimensional geometry. Here mainly classical problems and theorems are discussed. For example how one may compute the diameter of the Earth. The reader is amazed by the surprising height above the ground that is obtained when you enlarge a rope around the Earth by one meter and then lift it uniformly above from the ground. But also classical theorems in triangles, Platonic solids, Euler's formula for polyhedra, Penrose tiling, etc.

Numbers is the title of chapter five. We meet series and sequences, of course pi and e, the abc conjecture (an as yet unproved conjecture that claims that if a and b are relative prime positive integers with sum equal to c, and r is the product of the prime factors of a,b, and c, then c is smaller than the square of r), etc.

The last chapter collects 50 of the original puzzles that Dion Gijswijt provided for the problem section of the magazine. These are somewhat like the brainteasers of the first chapter, but more challenging. If ever, you do not succeed in finding the solution for yourself, (and I believe there is a reasonable chance that this will happen for some of these problems), then you may look up a solution for these (and for all the other problems) at the end of the book.

Every puzzle lover will enjoy this book and find many original gems, unless he or she had a subscription to the magazine, but then the Dutch edition of this book would have been a previous occasion to acquire this collection. They may still be happy with this English translation because it is an excellent give away present if they have non-Dutch speaking friends. For anyone who does not read Dutch, this is a marvelous occasion to have the best of the best of the magazine available to them too.

A last word about the marvelous cover of the book. Fritz Beukers plotted the interval [0,1] by covering every rational number p/q in the interval by a disk with of radius depending on q and using different colors. This does not only give a nice picture, but there is also a surprising theorem attached to it: if the radius of the disk covering p/q is c/q² then the interval is completely covered for c > (3-√5)/2 but for 1/3 < c < (3-√5)/2 there is a finite number of points left uncovered.

**Submitted by Adhemar Bultheel |

**2 / May / 2016