The k=3 Queens problem

Numerical Solution of the k=3 Queens problem!

A RECREATIVE COMBINATORIAL ANALYSIS

(based on work by Antal Pinter at 1973)

INTRODUCTION

One of the most popular version of theoretical chess problems in the recreative mathematics is the eight Queens problem, that is, in how many ways can eight Queens of the chessboard arranged in such a way that they do not attack each other. The task is originate still from 1848, given by a German chess player, M. Betzel, and then dealt with it dr. F. Nauck and itself Gauss too. The task was solved, although that is particularly not so difficult, anyone could find all the positions of the various 92 with a small effort, but the problem in his general form on nxn board is still unsolved! We can informed about various records from time to time, recently computer tournaments held annually considering this matter, but according to my knowledge, the maximum table size where solution exists is still only up to n = 26 !

The main interest in this subject is not ends with finding out of each possible position, in fact, it only begins with this. The ultimate goal is the exact mathematical expression of the problem of arbitrary number of k-queen's on the nxn -size table, which has also remain unsolved!

Mathematical formula is known only for up to k6 queens, and in this work I present the step-by-step solution for k = 3 queens !  In contrast of  E.Landau's formula from 1896,  given separately for odd and even tables, here is shown a detailed procedure for obtaining a generalized unique formula!

-The formula makes the sequence: A047659 in the OEIS.

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*** Release notes ***

What's New in the Second edition of  Numerical solution of the k=3 Queens problem ?

involve error corrections, text revisions and minor improvements on some tables

append New Chapter, the Numerical solution of the k= 2 Queens problem , as for easier

understanding and more detailed view of the backtrack process with several illustrations!

append New Chapter, the Closing of the open binomial array , as an heuristic method

of solving the sum of expression from the artical:

, or in closed form

where the number of term is  z = [(2n-5+(-1)n ]/4

-This expression also appears as sequence: A002624 -OEIS.

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P.S.!!!   Vaclav Kotesovec's  impressive and marvelous book, the „Non-attacking chess pieces“  ( http://www.kotesovec.cz/) dispelled any doubt about the final results achieved so far concerning the k-Queens problem, and it became clear the complete chronology of the succesive discoveries made on  the k-queen formulas, so they briefly looks as follows:

k=2 queens: Edouard Lucas, 1891 &  Antal Pinter, 6.1973
k=3 queens: Edmund Landau, 1896 & Antal Pinter, 6-8.1973
k=4 queens: Vaclav Kotešovec, 1992
k=5 queens: Vaclav Kotešovec, 4.4.2010
k=6 queens: Artem M.Karavaev, 10.5.2010 & Vaclav Kotešovec, 6.12.2010
k= 7 queens: ... not yet known!