Eight queens that do not threaten one another ---------------------------------
Take a chessboard—that is, an 8-by-8 square grid with sixty-four squares. Now take eight queens, the chess pieces that can attack in every direction. Can you arrange the eight queens on the board so that none threatens another? If you try for long enough, you will succeed. One way to look for a suitable configuration is to begin by placing a queen at random on the board. This rules out the row, the column, and the two diagonals threatened by the first piece. You can then place a second queen on one of the squares not ruled out by the first. This blocks another row, another column, and two more diagonals. Next, place a third queen on one of the remaining unthreatened squares, and continue in the same way. If you manage to place all eight queens on the board before running out of available squares, you win! If not, all you can do is start again and try different positions…

An example of a non-attacking configuration (in which no two of the eight queens threaten each other) on a chessboard.

The generalized problem ----------------------