This book is packed with interesting brain teasers and thought provoking questions about variations on the humble 9x9 sudoku that we are all now familiar with. The book starts with a very clear exposition of the problem and the basics of solution. It then takes us on a tour through Eulerian and Hamiltonian circuits, Latin and Greco-Latin squares amongst other tricks of the maths trade. The reader is taught the thought processes of a mathematician and, more importantly perhaps, why these processes are so valuable.

It becomes apparent early on that a few simple rules can help to break most conventional Sudoku patterns and the reader is soon au fait with the terminology of the ‘sport’: forced cells, twins, Ariadne’s thread, X-wings and Swordfish. The authors then introduce variants of the basic puzzle and show how these can continue to stretch the logical skills of the puzzle solver beyond repetition of a few now established rules.

Later chapters return to the standard set-up to address questions such as: How many Sudoku squares exist? (approximately 6.67 sextillion – established via computer search) How many fundamentally different Sudoku squares are there? (5,472,730,538) How do we make a Sudoku puzzle from a Sudoku square? What is the smallest number of starting entries needed to form a unique solution? What is the maximum number of starting entries that yield a unique solution? How is difficulty assessed?

All this is done in a way that does not expect the reader to have any prior mathematical knowledge as all concepts are beautifully explained using vibrant examples to keep the reader’s interest. Even the simpler concepts such as basic probability and factorials are explained where needed. Colour is used cleverly throughout to help draw the readers attention to important features when giving explanations and also to provide added interest in more complex Sudoku variants. Throughout, the authors draw on well known historical problems such as “The seven bridges of Konigsburg”, Newton and cannonballs, Euler’s combinatorical problem of 36 officers of 6 different ranks introduced in a paper in 1782 and Guthrie’s 1852 four colour mapping problem..

Newer variants of Sudoku continue to be introduced along the way and the final 20 pages are packed with a gallery of novel Sudoku variations waited to be solved. (It should perhaps be stated that all solutions are given at the back of the book for those that require them.)

In conclusion, I thoroughly enjoyed this book and do not have any criticisms to make. The authors have produced a lovely addition to any budding or practiced mathematician’s bookcase. Well-presented and readable for both the novice and the maths expert, which is an admirable feat, this book is for anyone with an interest, no matter how vague or intense, in Sudoku.

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