**[M] Rules (Advanced)**

The game is played on empty Sudoku, typically m^{n}(m^{n}) using integers 0 through m^{n}-1. Players alternate taking positions on the gameboard by placing integers into empty cells. The positions grant points equivalent to the integer value in their native region. If positions border adjacent regions, orthogonally or diagonally, these positions grant 1/2 of the integer’s value in each adjacent region. When all desired positions have been taken, 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 defect 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.

**STRENGTH OF VICTORY
**[M] games are zero-sum and utilize the concept of strength of victory. In a game with 9 regions, 5/4 represents the narrowest margin of victory. Stalemates are rare but possible.

In tournament play, the player awarded the most regions over a series of games is the victor. Where tie-breaking is desirable, the victor is the player who expended the least amount of points.

The concept of “efficiency” can be extended so that victory is a measure of points expended to regions won. Thus the presumed advantaged player may win 5/4, but have a lower efficiency rating than player who “lost” 4/5.

**Visit: mbranegame.com to try the mechanics for basic, non-trivial [M] on the Mbrane app**.

*(The app is free, ad-free, contains no purchases and collects no data.)*

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**BASIC EXTENSIONS:
**Basic [M] extensions are those that do not require additional mechanics.

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.)

Opposite edges and corners of the gameboard may connect, as on a torus. Influence carries across these connections. This has the effect of eliminating a true center.

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

Play is not restricted to the supersymmetrical 3^{2}(3^{2}) “classic” Sudoku. Any configuration will work, including Samurai Sudoku, HyperSudoku, SudokuX, Jigsaw Sudoku, and hybrids, such as Samurai-Jigsaw and Samurai-X.

Alternate board configurations such as m^{n}(j^{n}) may be employed, using either a set of integers j^{n} or m^{n}. These configurations do not have to produce completely filled in Sudoku. Thus 3^{2}(2^{2}) using integers 0 through 3 results in a sparser game, and integers 0 through 8 results in a denser game.

Basic gameboards are comprised of *cells* (positions) and *megacells* (regions). These constitute a *gigacell*—a group of regions comprising a basic Sudoku. *Teracells* may be formed with groups of gigacells, which may contain overlapping regions, as in Samurai Sudoku.

Any set of integers may be used for placement. Thus 1 through 9 on 3²(3²) instead of 0 through 8.

Negative integers may be utilized. Thus integers -z through m^{n}-z-1. There is no requirement that integers be sequential, thus [-m^{n}-1, 1, 2, …, m^{n}-1].

There is no restriction to integers, and any type of number may be utilized for placement.

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**GENERAL [M] EXTENSIONS:
**General [M] extensions are those which add mechanics. General extensions are nearly limitless.

Resolution sequence may be based on the *ratio of the deltas*. Thus a region where the point differential is 10/1 will resolve before a region with a differential of 20/10. (i.e. “ten to one” is less stable than “two to one”.)

Linking and pattern bonuses may be applied.

Gameboards may start with preset “deadcells”, cells in which no integer may be played. (These deadcells change the mathematical equilibria of a given board configuration, and the possible number combinations are vast.) These unplayable cells may have different qualities, for instance “mountains” which create blockages by removing influence positions, or “seas” which transfer influence to to all connected regions.

Imperfect information may be utilized. For instance, player’s can only see positions in regions where they have tokens, or adjacent to where they have tokens.

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.

There is no requirement that integers be fixed, thus in a non-sequential set utilizing negative integers, -m^{n} may increment each time it is utilized, ultimately resulting in a 0 value. A -❋ token may be introduced, where the playable negative is the max integer with the min instances +1. (Thus, if an 8 is played on the first turn, 7 is the maximum integers with the minimum instances on the gameboard. ❋ connotes the chaos symbol, distinct from stochasticity in that chaotic systems can arise in purely deterministic models as a function of complexity exclusively.)

Random factors can be introduced. For instance, at the beginning of each set of turns, an integer is selected randomly from the set of remaining, placeable integers, and both players must place that integer on that turn.

These represent only a small subset of the possible [M] extensions. See “Multiplayer Partisan Sudoku” [Patent Pending | US 15/273,708 | PCT/US2016/053211] for more details.

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**SEMIOTIC EXTENSIONS:
**Semiotic extensions involve symbols applied to game elements. [M] games have numerous symbolic applications.

- Integers may represent population, resource, or technological densities. (Castles, towns, mills, farms, mines, forests, etc.) (Grasses vs. Trees)
- Additional layers may 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 can represent sectors of memory or other resources, as in grid network.

Symbolic applications are limited only by imagination.