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Peter: your 'EITHER-OR' checkmate proposal raises many interesting possibilities beyond this page. Since games will often be won by forking your opponent's two royal pieces, you can consider variants where both royal pieces are powerful. For example, remove the Kings from FIDE chess and give two royal Queens to White and two royal Chancellors (R+N) to Black. A royal Chancellor may turn out to be slightly stronger than a royal Queen in this game.
<p>This sort of rule is completely different from the 'checkmate the last surviving royal piece' rules in variants like Triumvirate Chess.