Combinatorial Explorations in Su-Doku

TL;DR

This paper investigates combinatorial aspects of Su-Doku puzzles, exploring their solution space, difficulty estimation, and the role of the alldifferent constraint in solving and analyzing these puzzles.

Contribution

It introduces new insights into Su-Doku's combinatorial properties and compares various algorithmic approaches based on different CSP formulations.

Findings

Number of distinct Su-Doku grids estimated

Comparison of algorithmic approaches for solving Su-Doku

Analysis of puzzle difficulty and minimal clues

Abstract

Su-Doku, a popular combinatorial puzzle, provides an excellent testbench for heuristic explorations. Several interesting questions arise from its deceptively simple set of rules. How many distinct Su-Doku grids are there? How to find a solution to a Su-Doku puzzle? Is there a unique solution to a given Su-Doku puzzle? What is a good estimation of a puzzle's difficulty? What is the minimum puzzle size (the number of "givens")? This paper explores how these questions are related to the well-known alldifferent constraint which emerges in a wide variety of Constraint Satisfaction Problems (CSP) and compares various algorithmic approaches based on different formulations of Su-Doku.