Extinction chess (go to itsyourturn.com to learn about or play it) is a fine variant as it is, however I couldn't help but try a few variations on it.
Variant 1: Extinction chess with the following starting position, where each major piece is reduced to one in number:
Variant 2: Usual chess starting position. First player who captures a non-pawn piece wins the game. King is just an ordinary piece.
Variant 3: This is a cross between three checks chess and extinction chess: first player who captures 3 nonpawn pieces wins the game. King is just an ordinary piece.
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