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It is important to note that having multiple royal pieces on the board makes "forking" a valid way of mating. If you threaten two royal pieces of opponent with one of your pieces, the only way to end all the threats in one move is to capture the threatening piece. See the diagram above for an example (assuming both black pieces are the last of their kind ie royal).
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Another way of mating that we do not see in regular chess is when a royal piece has to move to get away from being threatened, but this would cause another royal piece to come under threat. See the diagram above for an example (again, assuming both black pieces to be the last of their kind).
I think this would make Extinction Chess more in line with the original (FIDE) chess. The rules change would not matter for 99.9% of the games (except that noone would lose for forgetting their king under check). The remaining 0.1% of games would involve stalemates, for which under current rules the side who has to move their pieces into threat would lose the game. See the diagram below for an example of a stalemate. Black, to move, has no legal moves, and since white is not giving any check, the position would be a stalemate, ie a draw.
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