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Suppose that in a KQ vs. K endgame (or in an endgame where Black can't possibly mate the other side), White makes two illegal moves. Can Black claim a win?

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No, the game is drawn, as per article 7.5.5 of FIDE Laws of Chess (emphasis mine):

7.5.5 After the action taken under Article 7.5.1, 7.5.2, 7.5.3 or 7.5.4 for the first completed illegal move by a player, the arbiter shall give two minutes extra time to his opponent; for the second completed illegal move by the same player the arbiter shall declare the game lost by this player. However, the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves.

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  • weird wording by fide. it should be like 'for the second completed illegal move by the same player the arbiter shall declare the game lost by this player unless the position (...) legal moves, in w/c case the game is drawn' i think
    – BCLC
    Apr 7 at 15:25
  • As far as I know, there is exactly one way for the player who can't checkmate anymore to still win -- if the player who can still checkmate resigns (see chess.stackexchange.com/questions/9312/… ). Other than that, even if the stronger side forfeits for using their phone or so, it'd still be 1/2 - 0 instead of a win. Apr 7 at 20:23
  • @RemcoGerlich: How about a team match with fixed ranking numbers which were botched by the team captain, or a player that hasn't an allowance? (OK, technically, both cases still count as win for ELO computation, but list as lose for the tournament, if I recall correctly.) Apr 8 at 16:07

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