[M] is part of a special class of games, sometimes referred to as “non-chance games of perfect information.” Games in this class, which includes Chess, Checkers and Go, are extremely rare because the underlying mathematics are so unforgiving.
Attempts to create such games usually end in models that are either so imbalanced that one player will always win, or so balanced that the games always end in a stalemate or tie.
[M] games are interesting in that, in the most reduced form, a 2×2 grid of 2×2 regions, games will always result in a stalemate under optimal play because the second player can always mirror the starting players placements, a state of perfect equilibrium. This solution extends to infinity for all gameboards with even numbers of even numbered regions. Thus 16×16 (4×4 of 4×4), 64×64 (8×8 of 8×8), ect., will always result in a stalemate. (These configuration may be referred to as “supersymmetrical” in that they are comprised of gameboard with nxn regions comprised of nxn cells.) However, this “mirror move” solution may applied to any even configurations, such as a 4×4 grid of 2×2 regions or a 2×2 grid of 4×4 regions. These solutions can be demonstrated by symmetry alone, despite the computation intractability of higher order Latin and Sudoku squares.
Where it gets interesting is in the next step up from 2×2(2×2), which a 3×3 grid of 3×3 regions. This configuration still meets our definition of supersymmetry, but the odd configuration makes mirroring an opponent’s placements impossible. Because of the factorial nature of Latin squares, this incrementation by 1 in regard to regions (i.e. 2×2 to 3×3) results in a non-trivial variant. What non-triviality means is that, although the basic form of the 3×3(3×3) game is assumed to be solvable, the solution is difficult. As of today, the 3×3(3×3) game remains unsolved.
The asymmetry of the odd configuration is balanced by the complexity of the model. In other words, like Chess, Checkers and Go, the basic 3×3(3×3) game is complex enough that determining an optimal move at any given point can be very difficult, and placement constraint inherent to the Sudoku grid allow tactics based on superior pattern logic which can result in an advantage to the more skilled player. Thus, although the second player is assumed to be disadvantaged, (an assumption notably not yet proven,) the indeterminacy arising out of intractability takes the place of randomness in “leveling the playing field” for the presumed, disadvantaged player.