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If player A, who has a bare king, plays against player B who has around 9 pieces left on the board, and player B timer runs out (let's say player A's timer has 3 seconds remaining), who wins the match? Theoretically speaking, player A has no way to win the game but by player B's timer running out.

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It's a draw. FIDE rule 6.9 states (emphasis mine):

Except where one of Articles 5.1.1, 5.1.2, 5.2.1, 5.2.2, 5.2.3 applies, if a player does not complete the prescribed number of moves in the allotted time, the game is lost by that 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.

Since the player with the bare king cannot checkmate the opponent by any legal sequence of moves, the game is drawn if his opponent runs out of time.

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