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David, I'm not sure how you'd combine checkmate and a rule that moving out
of check wins. It seems to me once check is made, the game is over the
next turn, since the checked King's player either moves out of check, and
wins, or is checkmated, and loses. Now, if a player won, once in check, by
<em>starting</em> a turn out of check, the game would turn into a desperate
series of checks at that point. Of course, this could easily turn into
perpetual check.
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A more dynamic name for Ralph's proposed game might be <u>King's
Escape</u>.
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I threw together a quick ZRF of this game by hacking up the Anti-King
Chess ZRF. Too quick -- it didn't work quite right. However, in the
process I came up with a mild variant. A player won, if after the movement
of their <i>King</i>, their King was not under attack. This had the
interesting effect that a player could leave their opponent's King
unattacked as long as it had no move that would move it to an unattacked
square.