Players: Black and White.
A crosscut is a 2x2 subgrid with two white and two black stones, in which like-colored stones are diagonally adjacent (and opposite-colored stones are orthogonally adjacent).
To swap is to exchange locations of two stones.
Start: Fill the squares on a 8x8 grid with any configuration of 32 white stones and 32 black stones.
Play: Each turn, the player selects a piece of their color in a crosscut. The player swaps the selected piece with different opponent stones sharing a crosscut with it, until it is no longer in a crosscut. During this process, each swapped enemy stone must be different.
Goal: If, at any time during your turn or your opponent's turn, there is an orthogonal path of stones connecting your goal edges (for Black it is North-South, for White it is East-West), you win.
There are cooperative cycles in this game, which seem to always arise from a single 3x3 pattern. However, for every one of those known cooperative cycles on the 5x5 grid, continuing the cycle appears to be a game-losing mistake. Therefore, I am interested in whether forced cycles might exist (a situation in which it is in both players interest to continue the cycle).
I think I have narrowed it down to the idea that a forced cycle, if it existed, must happen because one corner of the 3x3 cyclic template would somehow be important for making a winning connection. However, no matter which way I try, it is always possible for players to make a winning connection without using that corner.
However, multiple of these 3x3 cyclic templates could fit on a board, complicating the issue.
