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Here is a simplified version of the above game. The theory is to be written: help is welcome! It seems likely that every position can be brought to the initial one in not more than five moves. Any proof or disproof of this fact will be published here.
Here is a variant of the preceding one by using circular permutations.
Here is yet another game with permutations, like the two previous ones, but with no fixed points, which makes the theory rather different!