3 Player Chess

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For the sake of argument, let's increase the size of the chess board by double. So how does it play? You have player 1 & 2 with their own chess pieces and player 3 who has double the chess pieces except for the king and queen. Player 3's queen requires two moves to take down and the king requires three moves. When a move is made against Player 3's king and queen, Player 1 or 2's chess piece that struck them must go back to it's previous position on the board. The game begins with an enemy of my enemy is my friend so player 1 & 2 must work together to defeat player 3. When player 3 is defeated, player 1 & 2 are given a choice: Honor a truce or turn against each other. If a truce is honored, the game ends. If not, player 1 & 2 continue the game with their remaining pieces until one of them wins.

That's one way you could do it, but there's several others: https://en.wikipedia.org/wiki/Three-player_chess

Just going with traditional games, I know that 4-player chess games have been invented independantly in India (Chaturaji) and Spain (Four Seasons chess). Chaturaji used to involve 4-sided dice. It is still played to this very day, without the dice. Neither of those have the exact same piece movements as modern Western chess. There are also some multi-player versions of Xiangqi (Chinese Chess). The pieces are placed on the points a grid of intersecting lines, rather than on squares, which gives more options for mutating the board. So there are 3-player, 4-player and even 7-player variants.

But do they have a serpent and an old woman?

https://bigbangtheory.fandom.com/wiki/Three-Person_Chess

https://en.wikipedia.org/wiki/Three-sided_football

I think three-sided hockey might be interesting.