# This module contains Cython code for a backtracking algorithm to solve Sudoku puzzles.#

Once Cython implements closures and the yield keyword is possible, this can be moved into the sage.games.sudoku module, as part of the Sudoku.backtrack method, and this module can be banned.

sage.games.sudoku_backtrack.backtrack_all(n, puzzle)#

A routine to compute all the solutions to a Sudoku puzzle.

INPUT:

• n - the size of the puzzle, where the array is an $$n^2\times n^2$$ grid

• puzzle - a list of the entries of the puzzle (1-based), in row-major order

OUTPUT:

A list of solutions, where each solution is a (1-based) list similar to puzzle.

ALGORITHM:

We traverse a search tree depth-first. Each level of the tree corresponds to a location in the grid, listed in row-major order. At each location we maintain a list of the symbols which may be used in that location as follows.

A location has “peers”, which are the locations in the same row, column or box (sub-grid). As symbols are chosen (or fixed initially) at a location, they become ineligible for use at a peer. We track this in the available array where at each location each symbol has a count of how many times it has been made ineligible. As this counter transitions between 0 and 1, the number of eligible symbols at a location is tracked in the card array. When the number of eligible symbols at any location becomes 1, then we know that must be the symbol employed in that location. This then allows us to further update the eligible symbols at the peers of that location. When the number of the eligible symbols at any location becomes 0, then we know that we can prune the search tree.

So at each new level of the search tree, we propagate as many fixed symbols as we can, placing them into a two-ended queue (fixed and fixed_symbol) that we process until it is empty or we need to prune. All this recording of ineligible symbols and numbers of eligible symbols has to be unwound as we backup the tree, though.

The notion of propagating singleton cells forward comes from an essay by Peter Norvig [sudoku:norvig].