[M] games provide a rich combinatorial structure upon which simple functions may be applied to create chaotic systems efficiently.
Latin squares and Sudoku are complexity engines par excellence. Their structure is intrinsic, and they may be understood as differentiation matrices.
The goal of chaos functions ❋ is to simulate randomness.
Advanced [M] incorporates negative integers. These integers may be static, as in (mn)n|[-j,0,…,mn-j] or more simply, (3×3)2|[-1,0,…,7] and (3×3)2|[-4,…,0,…,4]
Alternately, the negative integer may be a variable j that begins at 1-mn and increments with each use, to a max value of 0.
Although all of these methods increase the complexity of the game and change the equilibria, they are predictable and non-chaotic.
The integer ❋ takes the inverse value of the max least_placed integer. In basic non-trivial [M], using integers [❋,1,…,8] on the first turn, ❋ = -8. If the first player places an 8, ❋ = -7. This makes the value erratic, difficult to predict, and increase in game complexity is an intrinsic function of the board state.
A second ❋ function is called “Mountains and Seas” after the classic of the same name. In this variant, the final board state of the previous game creates a map for the subsequent game—the dead sectors from the previous game are converted to mountains, which remove a position and block influence, or seas, which remove a position but transfer influence. A mountains/seas ratio may be set to produce maps with greater or fewer influence vectors, where the dead sectors with the greatest value in surrounding integers are converted to mountains and those with the least value in surrounding integers are converted to seas, and visa versa. (in the first variant, “hotspots” from the previous game are suppressed, in the second variant, the “hotspots” undergo expansion. An additional ❋ function based on parity of cardinality of the final set of placed integers, or the cardinality of the set of dead sectors, may determine via which method the dead sectors are converted.
The ❋ function becomes an engine to generate variation, in a manner which cannot be predicted, and has the same effect as randomness.