Mathematics of Sudoku

The class of Sudoku puzzles consists of a partially completed row-column grid of cells partitioned into N regions each of size N cells, to be filled in using a prescribed set of N distinct symbols (typically the numbers {1, ..., N}), so that each row, column and region contains exactly one of each element of the set. The puzzle can be investigated using mathematics.

The mathematical analysis of Sudoku falls into two main areas: analyzing the properties of a) completed grids and b) puzzles. Grid analysis has largely focused on counting (enumerating) possible solutions for different variants. Puzzle analysis centers on the initial given values. The techniques used in either are largely the same: combinatorics and permutation group theory, augmented by the application of programming tools.

There are many Sudoku variants, (partially) characterized by the size (N) and shape of their regions. For classic Sudoku, N=9 and the regions are 3x3 squares (called blocks or boxes). A rectangular Sudoku uses rectangular regions of row-column dimension R×C. For R×1 (and 1×C), i.e. where the region is a row or column, Sudoku becomes a Latin square.

Other Sudoku variants also exist, such as those with irregularly-shaped regions or with additional constraints (hypercube) or different (Samunamupure) constraint types. See Sudoku – Variants for a discussion of variants and Sudoku terms and jargon for an expanded listing.

The mathematics of Sudoku is a relatively new area of exploration, mirroring the popularity of Sudoku itself. NP-completeness was documented late 2002. Published enumeration results began appearing in 2004.

In contrast with the two main mathematical approaches of Sudoku mentioned above, an approach resting on mathematical logic and dealing with the resolution of the puzzles from the viewpoint of a player has recently been proposed in Denis Berthier's book "The Hidden Logic of Sudoku". This formalizes certain mathematical symmetries of the game and elicits resolution rules based on them, such as "hidden xy-chains".

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