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The First Solutions
Chessboard size 43×43

1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 24, 26, 37, 39, 34, 38, 35, 33, 42, 36, 41, 43, 40, 20, 14, 17, 10, 12, 28, 18, 16, 27, 25, 21, 32, 30, 23, 31, 29

Diagram of the first possible solution to the 43 Queens on a 43x43 chessboard problem