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If you know the name of the chess variant I'm trying to describe, please leave it in an answer.

My friend and I have conceived an (most likely already thought up, nevertheless very fun) idea of chess where 2 moves take place at a time. The main rule we have that concerns checkmating in this variant is that you can move through check so long as you don't end up in check at the end of the second move. (This also means you can move another piece before getting your king out of check)

How would one checkmate in this situation? An even situation would be optimal for the response, I know how one could checkmate with an inordinate amount of pieces.

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    I guess you don't let a player give check with the first of his two moves and capture the king with the second, right? If you did that, and had White make just one move at the start of the game, you'd be playing Doublemove Chess. Without allowing the king to be captured, what you've got is a mixture of Doublemove and Marsellais Chess which is probably not playable; it's too hard to checkmate the king.
    – bof
    Commented Nov 15, 2017 at 4:47

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This is basically Marseillais Chess, the main difference between it and your version is that a player can't move into and then out of, check. When a player gives check on the first move, he then loses his second move.

Additionally, the most commonly played version is "balanced", with white only making one move on the first turn, so as to not give White too much of an advantage.

One would checkmate just as in regular chess.

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