During a game of chess, after several moves. Is it an automatic win as soon as you get your king all the way across the board?
I was told this is a tournament rule.
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No, this is not a rule in chess. There are plenty of games where one king reaches its 8th rank, and the game continues, with all three results still possible. Here are the Fide rules
As Evargalo notes in their answer, that is not part of the official rules of chess. However, Lichess.com has an online, playable chess variant called "Racing Kings," you in which your mentioned rule holds true.