In conversion Go, when a group has all of its liberties occupied by the opponent, it is immediately "converted" --- that is, it is removed from the board and replaced by stones of the opponent's color. (Flippable pieces, such as those used in Othello, are ideal for this game.) The key rule here is that if a group touches at least two of the board's four sides, it is immune from this conversion process. So touching two sides is like having two real eyes in regular Go --- the group is unconditionally alive.
Initially called Conversion Orbit, this game was described in Nov.2001 Games. I've since renamed it conversion Go because it is really quite similar to the ancient Chinese game, and as such is a Go variant. There is no arising of "shared territory," the mechanism of liberties is used, and the related phenomenon seki is possible. For these reasons Go players may feel more comfortable giving this one a try.
It's worth noting, BTW, that conversion Go can be neatly played with pen-and-paper, despite the fact that pieces are flipped and re-flipped. The reason is that an "x"ed group is always known to be the color of the living formation that ultimately encloses it.
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