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According to my understanding of the history of weiqi it originated during the Zhou Dynasty in the Wei River Valley. Did the *Wei* River influence the name of the game?
I need to reread some documents but I think I read somewhere that this river was always flooding and that weiqi may have been originally conceived as chart of controlling flow (qi/liberties as they are).
I know nothing about the history, but from the pronunciation alone it is unlikely unless historical literature show otherwise. The 渭 in 渭河 is fourth tone while the 围 in 围棋 is second tone. 围 means to surround, and from what I can find, 渭 is used as a proper noun. So their meaning are different.