On a General Framework for Large-Scale Constraint-Based Optimization Re: "Explicit Solutions to the N - Q u e e n s Problem for all N" Jun Gu D e p a r t m e n t of Electrical Engineering University of Calgary Calgary, C a n a d a T2N 1N4 gu@enel.ucalgary.ca The explicit solution for the n-queens problem, mentioned in a letter from Bo Bernhardsson [2], is basically Pauls's solution analyzed by Ahrens (See reference [1] of our previous article in SIGART October issue 1990). The result was in public domain long before 1918 (not 1969). We also mentioned its weakness, namely: The class of solutions provided by analytical methods is very restricted, as Ahrens pointed out in [1]. They can only provide one solution for the n-queens problem and can not provide any solution (much better explicit solutions for the n-queens problem exist). This is not the case for search methods which can find, in principle, any solution. This distinction is crucial for practical applications of the n-queens problem. The general local search framework and general gradient-based conflict-minimization heuristic, described in our previous article [4], can be applied to various similar large-scale, constraintbased optimization problems in our daily computing practice. By applying the same ideas and heuristic, some exciting progress has been made by researchers in the AI community [3]. This is not the case with an entirely problem-specific solution. We believe that we appropriately dealt with analytical solutions in our original paper. Bo Bernhardsson's correspondence overlooked the reference to Ahrens and our comments in [4]. The n-queens problem has become a standard benchmark for testing search and backtracking algorithms in the AI community (note that there is no need to "benchmark" an analytical solution). The complete solution for the 8-queens problem was introduced in September 1850 by Nauck, according to Ahrens [1]: In the "Illustrated Paper" of June 1st, 1850 one can find under the category of chess "One in a Series of Mathematical Works by Dr. Nauck of Schleusingen." The abstract of the paper reads as follows: "One can take 8 chess pieces, each having the rank of queen, and arrange them in such a manner that no queen can be killed by any other." In the September 21st issue of the same year Dr. Nauck published "The 92 Solutions" after being asked by 60 people for the solution. Although the problem was of great interest to the readers, only one person, a man born blind, was able to supply the complete set of solutions. The famous mathematician Gauss also read the problem in the journal. In his correspondence with schoolmate, friend, and avid chess player, astronomer H. C. Schulmacher, there are many references to this problem, and apparently both scholars had a great interest therein. Gauss quickly found 72 solutions, but could not say for sure if this was the maximum number of solutions. The complete set of solutions did not come from Gauss, as is widely believed, but from Nauck as was stated earlier. On his own, Gauss realized that "with a couple of hours and a methodical program, one can work out with certainty that the solutions given by Nauck are complete."

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