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This has been invented before:
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From the <u>Encyclopedia of Chess Variants</u>:
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<b>Check Chess</b> aka Presto Chess. Lawrence Crane (1916).
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According to the entry the game was shown to be win for White with an opening of (1 Nc3). Various improvements were to move the Pawns to the 3rd/6th rank, require the checking piece be unattacked, and to limit the Queen to moving 1 or 2 squares; however, White seems to always have the advantage. Apparently Three-Check Chess works better.