Editing Eight queens puzzle (section)

== Constructing and counting solutions when ''n'' = 8 ==
The problem of finding all solutions to the 8-queens problem can be quite computationally expensive, as there are 4,426,165,368 possible arrangements of eight queens on an 8×8 board,{{efn|The number of [[combination]]s of 8 squares from 64 is the [[binomial coefficient]] <sub>64</sub>C<sub>8</sub>.}} but only 92 solutions. It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids [[Brute-force search|brute-force computational techniques]]. For example, by applying a simple rule that chooses one queen from each column, it is possible to reduce the number of possibilities to 16,777,216 (that is, 8<sup>8</sup>) possible combinations. Generating [[permutation]]s further reduces the possibilities to just 40,320 (that is, [[factorial|8!]]), which can then be checked for diagonal attacks.

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.

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, one of the 12 fundamental solutions (solution 12 below) is identical to its own 180° rotation, so has only four variants (itself and its reflection, its 90° rotation and the reflection of that).{{efn|Other symmetries are possible for other values of ''n''.  For example, there is a placement of five nonattacking queens on a 5×5 board that is identical to its own 90° rotation. Such solutions have only two variants (itself and its reflection). If ''n'' > 1, it is not possible for a solution to be equal to its own reflection because that would require two queens to be facing each other.}}  Thus, the total number of distinct solutions is 11×8 + 1×4 = 92.

All fundamental solutions are presented below:

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|Solution 1
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|Solution 2
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|Solution 3
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|Solution 4
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|Solution 5
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|Solution 6
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|Solution 7
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|Solution 8
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|Solution 9
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|Solution 10
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|Solution 11
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|Solution 12
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Solution 10 has the additional property that [[No-three-in-line problem|no three queens are in a straight line]].