8 Queens on the Chess Board

The pages linked here show solutions to the problem of placing eight queens on a chess board so that no two of them protect each other.

The solution algorithm was as follows:   Place a queen at a1.   Then place a queen on the first available square on the second rank, which initially is c2.   Then place a queen on the first available square on the third rank, which will be e3 initially.   Continue this process until no more queens can be placed without protecting an already placed queen.   Then advance the queen on the highest rank to the next available square on its rank, if possible.   If no such advance is possible, remove the queen and similarly try to advance the queen which then remains on the highest rank.   Once a queen is successfully advanced along its rank, continue the process until a solution is obtained.   The first solution that this process produces is the one at the link above.

The algorithm produces 4 solutions beginning with the queen at a1.   I write this as 4_@_a1.   All   92   solutions that I found are indicated by the links in the order they were produced.   Using the notation just described, the distribution of solutions indicated by the links is

4_@_a1,    8_@_b1,    16_@_c1,    18_@_d1,    18_@_e1,    16_@_f1,    8_@_g1,    4_@_h1.

The numbers added below the links indicate what I call the knight move groupings of the queens.   At the bottom of each page is the solution which is a duplicate of that page's solution by 180 degree rotation.   Duplicates by 90 degree rotation and reflection through one of the board's center lines are not applicable to reduce the number of distinct solutions, because these operations leave a dark square in the player's right hand corner.   The algorithm produces four solutions which are their own 180 degree duplicates.   Consequently, in my interpretation distinctness, the number of distinct solutions is

[ ( 92 - 4 ) / 2 ] + 4 = 44 + 4 = 48 .

In what follows I will need to refer to the pages described above. Therefore, I will use the designations which are seen in the links - such as   a1-01   which applies to the first page.   As can be seen at the bottom of that page, the   180   degree duplicate for solution   a1-01   is page   e1-14.   Because of the   180   duplications, the   92   solutions can be illustrated on   48   chessboards.   This is done in my first 'expo' pages.

©  Barry Davies  2000  2007  2011