<P> Since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized n - queens version . In 1874, S. Gunther proposed a method using determinants to find solutions . J.W.L. Glaisher refined Gunther's approach . </P> <P> In 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming . He published a highly detailed description of a depth - first backtracking algorithm . </P> <P> The problem of finding all solutions to the 8 - queens 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>

Total number of solutions for 8 queens problem
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