Eight Queens

Place eight queens on a chessboard so that no two attack each other: no two in the same row, the same column, or the same diagonal.

Solution

Take the rows in turn, from the bottom, and put the queen in columns $$1, \ 5, \ 8, \ 6, \ 3, \ 7, \ 2, \ 4.$$ No column is repeated, so no two queens share a file. For the diagonals, note that two queens lie on a common diagonal exactly when their row-plus-column totals agree, or their row-minus-column differences agree; running through the eight placements, all eight sums are different and all eight differences are different, so no diagonal holds two queens. Drawing the board with row $8$ at the top: $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \cdot & \cdot & \cdot & Q & \cdot & \cdot & \cdot & \cdot\\\hline \cdot & Q & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & Q & \cdot\\\hline \cdot & \cdot & Q & \cdot & \cdot & \cdot & \cdot & \cdot\\\hline \cdot & \cdot & \cdot & \cdot & \cdot & Q & \cdot & \cdot\\\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & Q\\\hline \cdot & \cdot & \cdot & \cdot & Q & \cdot & \cdot & \cdot\\\hline Q & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\\hline \end{array}$$ There are $92$ ways in all to place the eight, which reduce to just $12$ essentially different arrangements once rotations and mirror images of the board are counted as one.