The Chinese Go game has attracted the world’s attention recently when computer beat the world human champion. It also got me thinking about a higher dimensional version.
Essentially the Go game is a graph whose vertices starts with transparent color but may become black or white when stones are placed in the vertex as the game progresses. Once colored, a vertex can only become transparent again when the stone is removed due to loss of liberties. A stone’s liberty is the number of uncolored neighboring vertices on the graph and it becomes captured when its group has zero liberties. Captured vertex immediately return to transparent. Connected stones having the same color is a group and combines their liberties. All liberties of a group must be taken before any stone of the group becomes captured.
Players take turn placing stones on vertices. Stones having no liberty are removed by the board. Game ends when at least one player resigns or when there no play remaining that would not reduce players’ stone count plus liberties. Winner is the player having most stones on the graph at the end.
The traditional go board is such a game played on a 19×19 planar graph located in the square from origin to (18,18) having nodes at grid points and edges between closest neighbors. Internal nodes have 4 neighbors. Edge nodes has 3 neighbors each and corner nodes has two neighbors each.
Therefore, we shall disregard planarity and uniformity. Consider, for example, the game played on a graph that looks like a donut when laid out in 3-d. Internal verteces have 6 neighbors, external verteces has 5-neighbors.
Shall we play the game?