The book is centered around Sudoku puzzles, using them as a jumping-off point "to discuss a broad spectrum of topics in mathematics".[1] In many cases these topics are presented through simplified examples which can be understood by hand calculation before extending them to Sudoku itself using computers.[3] The book also includes discussions on the nature of mathematics and the use of computers in mathematics.[4]
After an introductory chapter on Sudoku and its deductive puzzle-solving techniques[1] (also touching on Euler tours and Hamiltonian cycles),[5] the book has eight more chapters and an epilogue. Chapters two and three discuss Latin squares, the thirty-six officers problem, Leonhard Euler's incorrect conjecture on Graeco-Latin squares, and related topics.[1][4] Here, a Latin square is a grid of numbers with the same property as a Sudoku puzzle's solution of having each number appear once in each row and once in each column. They can be traced back to mathematics in medieval Islam, were studied recreationally by Benjamin Franklin, and have seen more serious application in the design of experiments and in error correction codes.[6] Sudoku puzzles also constrain square blocks of cells to contain each number once, making a restricted type of Latin square called a gerechte design.[1]
Chapters four and five concern the combinatorial enumeration of completed Sudoku puzzles, before and after factoring out the symmetries and equivalence classes of these puzzles using Burnside's lemma in group theory. Chapter six looks at combinatorial search techniques for finding small systems of givens that uniquely define a puzzle solution; soon after the book's publication, these methods were used to show that the minimum possible number of givens is 17.[1][4][5]
The next two chapters look at two different mathematical formalizations of the problem of going from a Sudoku problem to its solution, one involving graph coloring (more precisely, precoloring extension of the Sudoku graph) and another involving using the Gröbner basis method to solve systems of polynomial equations. The final chapter studies questions in extremal combinatorics motivated by Sudoku, and (although 76 Sudoku puzzles of various types are scattered throughout the earlier chapters) the epilogue presents a collection of 20 additional puzzles, in advanced variations of Sudoku.[1][4]
