Rules of [M]

[M] Rules (Concise)

The game is played on an an empty Sudoku, typically n²(n²) using n² integers: 0 through n²-1. Players take positions on the gameboard by placing integers into empty cells.  The positions grant points equivalent to their integer value in their native Region.  If these positions border adjacent regions, orthogonally or diagonally, the positions grant points equal to 1/2 of the integer’s value in these adjacent Regions. When all positions that can be taken have been taken, or by mutual consent of all players, the gameboard is resolved.

Resolution begins in the Region(s) with the greatest disparity (Δ).  Regions are awarded to the player with most the points in that Region, known as the dominant player. When a Region is awarded, the dominant player is said to gain control, and all opposing integers in the Region flip to that player.  All points derived from the flipped integers are re-assigned to the controlling player, including points applied in adjacent Regions. When all regions have been resolved, the player controlling the most Regions wins.


The basic game involves two players, but there is no restriction.  Number of players is theoretically infinite, limited only by the gameboard size (number of positions in a given game.)

n is used for the grid size, which requires natural numbers, but negative integers may be used for placement on the gameboard.  Thus integers -i through n²-i-1, etc. may be utilized.  There is no requirement that placement integers be sequential, thus [-z², 1, 2, …, z²-1] where z=n.  There is no requirement that integers be fixed, thus-z² may increment each time it is utilized, ultimately resulting in a 0 value.

Opposite edges of the gameboard may connect, as in a torus.

There is no restriction to two spacial dimensions, thus the three dimensional game for humans, the n-dimensional game for automata.

Any Sudoku configuration will work, including Samurai Sudoku, HyperSudoku, SudokuX, Jigsaw Sudoku, and hybrids.

The method may be extended to Graeco-Latin squares.  Values in each layer of the combined orthogonal array modify each other.   Additional layers may be used for placement of integers, or may begin filled, granting bonuses or penalties.  Any number of additional layers may be applied.

Symbols may be applied in numerous ways:

  •  Integers may represent population, resource, or technological densities. (Castles, towns, mills, farms, mines, forests, etc.) (Grasses vs. Trees)
  • Additional layers may be applied to represent subterranean resources or topographical features. This concept may be extended to “orbital” layers, stationary and non-stationary.
  •  Regions may represent markets, with the integers as market positions. (For instance the center region may be the auto industry, with surrounding regions representing rubber, steel, plastics, petroleum, microprocessors, etc.)
  • Integers may represent information densities.  A political game may be expressed as a three dimensional game, where the first two dimensions represent geography and the third dimension represents demography.  In a computing context, integers may represent sectors of memory or other resources.

Symbolic applications are limited only by imagination.

For more detail on [M] extensions, see “Multiplayer Partisan Sudoku” [Patent Pending]