Mathematical frontiers

Press, 15 April 1978, Page 17

Mathematical frontiers

Further Mathematical Diversions. By Martin Gardiner. Penguin. 255 pp. $2.50. Martin Gardiner is well known to readers of “Scientific American” as the indefatigable author of a column on mathematical problems. He is extremely adept at introducing new and esoteric fields of mathematics in a very illuminating way, building up from examples and simple problems to the frontiers of discovery. Gardiner also has a flair for personalising problem solving: he tabulates known solutions and reviews the history of false starts and crazy theories so that we learn easily from others’ errors. If the first or best solution of a particular problem was discovered by an out-of-work treacle refiner in the backwoods of Venezuala, you may be sure Gardiner will have had correspondence with him and given credit where it is due. Pure mathematics as a discipline has no boundaries which cut it off from recreational problem solving, except perhaps the ones concerning formal conventions in demonstrating proofs, and so the gifted amateur has often made contributions to match those of the professionals. The chapters, as might be expected, are strong on Gardiner’s favourite areas of topography, graph theory and probability, but codes and deliberately silly problems all get their share. The one puzzle to which this edition offers no solution is why the book is called, from front to back inside the cover, “The Unexpected Hanging.” This famous problem will be familiar to some as that of the unexpected

examination whose date cannot be known in advance. Gardiner summarises both the valid and invalid

proofs that such an event respectively can or cannot happen, but is a bit fuzzy; he is at his sharpest on numbers and tessellations. There are plenty of problems, with answers, which are medium to advanced; the bright 7th Former and above should enjoy them and Gardiner is generous with references of where to go for more details. — R. A. Mi GREGSON.