This chapter is devoted to an extended exercise in the use of lists to solve problems, and in the use of equational reasoning to reason about them and to improve efficiency.

The game of Sudoku is played on a 9 by 9 grid, though other sizes are also possible. Given a matrix, such as that in Figure 5.1, the idea is to fill in the empty cells with the digits 1 to 9 so that each row, column and 3 × 3 box contains the numbers 1 to 9. In general there may be any number of solutions, though in a good Sudoku puzzle there should always be a unique solution. Our aim is to construct a program to solve Sudoku puzzles. Specifically, we will define a function solve for computing a list of all the ways a given grid may be completed. If only one solution is wanted, then we can take the head of the list. Lazy evaluation means that only the first result will then be computed.