This is an efficient Sudoku solving algorithm using constraint programming, written in Python. It is very much inspired by (but different from) Peter Norvig's Solving Every Sudoku Puzzle. It generates quickly all solutions to a given Sudoku (usually in less than 100ms).
The basic idea is to keep track of all the candidates (possible digits) which can go into a square that is not filled yet. When only one candidate is left, place it (this is called a naked single). Otherwise, find a square that has the least number of candidates (in practice, that number is usually 1 or 2) and try them one after another. Every time a digit is set, it is removed as a candidate from all of its peers - these are the squares that are in the same unit (row, column or box). Recursively apply this procedure until all squares are filled.
The solver produces a generator containing all solutions. When a contradiction has been found (that is, 0 candidates were left in a square), we do not require backtracking explicitly: it just means that the current search branch did not yield any new solution, and we just continue with the next one.
To make the algorithm even faster, the hidden single strategy has been implemented. A hidden single is a digit that can only go in one square of a unit. In this case, the square is filled with that digit and we continue.