Combinatorial Explorations in Su-Doku

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. Su-Doku is a well-known logic-based number placement puzzle. The objective is to fill a 9x9 square grid so that each line, each column or file, and each of the nine 3x3 blocks contains exclusively the digits 1 to 9, only once each. A puzzle is a partially completed grid.