Three-person chess, a game mentioned on by Sheldon on The Big Bang Theory, is now a reality. It requires a specialized board, but other than that, you can just follow the rules at the link. Here’s an overview from the creators:
The only changes from conventional chess are some protocol issues that must be followed to maintain order where the teams border each other, which is simple and necessary. Also, please notice that the trajectory lines orienting from the outer rank, are simply visual aids to help guide diagonal moves passing through the center. If the path is clear, a diagonal move starting from the outer rank can pass through the center and sweep back around to where it originated. The complexities of the third player are infinite. Your threatened piece may be allowed to maintain occupancy as your position is beneficial to the threatening player.
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