We consider games of conquering regions, e.g., generalizations of the game of go 囲碁, perhaps higher dimensions or on a general graph. Any connected group (a chain) has a volume (number of stones) and surface area (neighboring points not part of the group). What else? Much graph theory: diameter, vertex connectivity, edge connectivity.
Between two disconnected groups of the same color, the difficulty of cutting them or joining them. Min-cut, max-flow.
Inspired by the difficulty of depicting on a 2D screen who owns what within a 3D space conquest game. The graph theory statistics listed above could be provided to the player by a computer UI. The game could be designed so that those graph theory statistics are relevant.
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