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The wall pieces can only touch 2 squares (not 3). For my purposes, there can be any number of walls allowed. (in the game, each player gets 10 walls and can either move or place a wall each turn)

My math question is how many possible paths are there:

a: That lead from one side to the other with the original 81 square board.

b: On an easier to physically construct 3x6 board.

c: How to find max paths for an m x n board