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The most obvious difference with Cheshire Cat Chess is when n>1. But also the elimination of draw rules makes it interesting, IMO. An insufficient material endgame, for example, becomes a puzzle similar to Joust Chess.
I've added a tag square-removal to track similar games.
This sounds like a reinvention of Parton's Chesire Cat Chess.
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Okay, I didn't notice that detail. Each space has a number of lives it can go through before disappearing. That makes it appreciably different from Chesire Cat Chess.