Eight Queens Puzzle

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Eight Queens Puzzle

Last Updated Aug 20, 2008 07:53 AM

The eight queens puzzle is the problem of putting eight chess queens on an 8×8 chessboard such that none of them is able to capture any other using the standard chess queen's moves. (Piece colour is ignored, and any piece is assumed to be able to attack any other). That is to say, no two queens should share the same row, column, or diagonal. This is an example of an n queens puzzle for placing n queens on an n×n chessboard.

Trivial solution method

There is a simple method for getting a solution to any n queens puzzle, except for two and three queens :

Divide n by six. Remember the remainder (it's two for the eight queens puzzle).
Start in the upper-left corner square.
If the remainder you got wasn't two or three, move right one square.
Place a queen on the current square.
Move two squares right and one square down, like a knight moves, and place a queen.
Repeat step six until you run off the edge of the board.
Move one square down from the last queen you placed.
Move all the way to the left.
If the remainder you got was three, you must place the remaining queens either one square down or one square up from where the instructions tell you. The first queen is one square down and it alternates.
If there is a queen in the upper-left corner square, move right one square.
Place a queen on the current square.
Repeat step six until you run out of queens.

History
Over the years, many mathematicians, including Gauss have worked on this puzzle, which is a special case of the generalized problem of placing n "independent" queens on an n by n chessboard, posed as early as 1850 by Franz Nauck. In 1874, S. Gunther proposed a method of finding solutions by using determinants, and J.W.L. Glaisher refined this approach.

This puzzle was used in the popular early 1990s computer game, The 7th Guest.

Solution counts
The eight queens problem has 92 distinct solutions, or 12 distinct solutions if symmetry operations such as rotations and reflections of the board are taken into consideration (via Burnside's lemma.)

The number of possible solutions of placing N queens on a N by N board, considering the symmetries as distinct solutions, is known for N up to 24.

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