“Mathematics and Chess” by Petkovic – Reviewed by David Davis

MATHEMATICS AND CHESS
Miodrag Petkovic
Dover Publications, 1997
Limp cover

Reviewed by Dave Davis

This is a collection of 110 problems in geometry, algebra, and combinations based on the moves of the chess pieces. The only chess knowledge required is the rules of the game. The amount of mathematical skill needed is considerable.

The preface states: “Almost none of the problems and puzzles, from the very old ones to the newest ones, involving computer procedures, exceed a high school level of difficulty; advanced mathematics is excluded”.

This is certainly-not true of an American High School education. The author is Yugoslavian and this may be so over there, where a “High” school is equivalent to a Junior College in the United States.

The problems are divided into six chapters, with solutions at the end of each chapter. The first chapter contains almost nothing directly related to chess, but calls on the reader to solve recurrence relations, add probabilities, understand the binomial distribution, supply proof by mathematical induction, and evaluate 8 x 8 determinants.

The subject level is that of a course in Foundations of Mathematics as studied by Math Majors, or Discrete Structures as studied by computer science majors. If these are not your fields, you will get very little from these problems.

The second chapter involves a little more chess. Problem 2.3 requires White, with King and Queen, to checkmate the lone Black King without moving White’s King.

This is a good exercise for novices because it requires visualization of mating patterns and using the Queen alone to drive the Black King to a square where it will be checkmated. However, this is the only problem I could find of any value to a chess player.

Many problems are ill-posed. Problem 2.20 involves an interesting piece dalled a “Destroyer”, but it never tells you that a Destroyer moves like a King. Why not use a King?

Problem 2.14 goes to great length explaining hos a “Beetle” moves, but it actually simply moves like a Bishop.

Chapter tnree gives a wonderful description of the Knight’s Tour problem. (A Knight is moved so that it lands on each of the 64 squares only once.), and it mentions that the famous mathematicien Euler was interested in this problem, but I am disappointed that his solution is not given.

Chapters four, five and six are all geometry and pencil and paper type problems and most of them can be solved by any avid puzzle fan, but the problems in chapter four require that you remember the formulas from geometry and trigonometry.

The solutions rarely give details of what mathematics is needed to reach them, and four problems have computer programs written in four different programming languages. These seem pointless because anyone who understands them will code them in Visual Basic, or C++, or some other currently fashionable platform.

Some of these problems may be interesting for players interested in Fairy chess with unusual pieces, and there is one fine Retrograde chess problem (2.2), but this book was clearly written for the mathematical enthusiast and not for chess players.

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_ _ K _ _ _ _ _ 3
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_ Q _ _ _ _ _ _ 1
a b c d e f g h

Problem 2.3.
White checkmates without moving White King.

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a b c d e f g h i

Problem. 2.6.
Five Queens command every square on a 9 x 9 board. Note that four Queens are a Knight’s move from the central Queen: A typical solution for that type of problem.

Copyright 2000 – David Davis – USA

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