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
*n*x*n* 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 *n*x*n* -size table, which has
also remain unsolved!

Mathematical formula is known only for up to *k* = *6*
queens, and in this work I present the step-by-step
solution for ** k = 3 queens** ! In contrast
of

*-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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** 1. K3Queens_en_rev2.pdf **w/o Appendix I-II** ........... FREE!**

** 2. K3Queens_en_rev2.F.pdf **Full with Appendix I-II** ...... $24.99**

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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!