<P> The problem can be quite computationally expensive, as there are 4,426,165,368 (i.e., C) possible arrangements of eight queens on an 8 × 8 board, but only 92 solutions . It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids brute - force computational techniques . For example, by applying a simple rule that constrains each queen to a single column (or row), though still considered brute force, it is possible to reduce the number of possibilities to 16,777,216 (that is, 8) possible combinations . Generating permutations further reduces the possibilities to just 40,320 (that is, 8!), which are then checked for diagonal attacks . </P> <P> Martin Richards published a program to count solutions to the n - queens problem using bitwise operations. . However, this solution has already been published by Zongyan Qiu . </P> <P> The eight queens puzzle has 92 distinct solutions . If solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions . These are called fundamental solutions; representatives of each are shown below . </P> <P> A fundamental solution usually has eight variants (including its original form) obtained by rotating 90, 180, or 270 ° and then reflecting each of the four rotational variants in a mirror in a fixed position . However, should a solution be equivalent to its own 90 ° rotation (as happens to one solution with five queens on a 5 × 5 board), that fundamental solution will have only two variants (itself and its reflection). Should a solution be equivalent to its own 180 ° rotation (but not to its 90 ° rotation), it will have four variants (itself and its reflection, its 90 ° rotation and the reflection of that). If n> 1, it is not possible for a solution to be equivalent to its own reflection because that would require two queens to be facing each other . Of the 12 fundamental solutions to the problem with eight queens on an 8 × 8 board, exactly one (solution 12 below) is equal to its own 180 ° rotation, and none is equal to its 90 ° rotation; thus, the number of distinct solutions is 11 × 8 + 1 × 4 = 92 (where the 8 is derived from four 90 ° rotational positions and their reflections, and the 4 is derived from two 180 ° rotational positions and their reflections). </P>

Can you place 8 queens on a chessboard