In the strictest sense, games require multiple participants.
While many puzzles are considered games, particularly in the electronic context (Tetris, Candy Crush, etc.) they are nevertheless solo endeavors.
[M] games are doubly interesting because they present a competitive endeavor of pure strategic skill, and maintain the pattern-recognition puzzle qualities of Sudoku.
The transformation of Sudoku into a game expands its dimensionality in a profound way, the introduction of sequence as the dimension of time.
In the Sudoku puzzle, the order the integers are filled in has no significance. In [M] games, the sequence in which the integers are placed is of utmost importance.
While the number of strategically viable choices is greatly limited in [M], reducing the number of possible games, it is factorially expanded by the number of possible sequences in which the gameboard may be filled.
Because a game is a contest against an opponent, scoring may be utilized, and it becomes possibly to apply values to the integers which are merely symbols in the Sudoku puzzle. The application of a quantitative value in place of a one merely qualitative has the added benefit of eliminating reduction by relabelling, which maintains the size of the game tree.
The application of value, or magnitude, in a second method in which [M] games expands the dimensionality of the Sudoku construct.