*Introduction*

The n-Queens problem is well known and solved on any *regular chessboard*. When the problem has extended to the *toroidal (modular) board* it was discussed rarely in the math literature (1) and, consequently, some uncertainty occurred regarding the number of possible solutions ( e.g. *oeis.A051906**) *on such a board. I hope, the previous enthusiastic article (2) gave some new facts regarding the ‘translated solutions’ that are kind of ostensibly solutions (see Chapter 4 in the mentioned article).

In this article, I would like to give a more concise presentation of the sQuaricon pattern matrices and go one step ahead…

I am a long-time enthusiast in math and would like to keep this status in future. So, all my claims exposed here should be taken with reasonable caution at least until the full experts’ verification. This article is a preprint version in work…

*this is the 3rd Revised version, as of July 21, 2019*

I will try to show and prove (with a little bit outdated math knowledge and notification) ** 4 Claims** that make

**Claim-1:*** the n-Queens-**Completion problem** on modular board is **solvable in the polynomial time** for any n=p, (proof in…*

very soon here will be published full article of simple but efficient way how to solve *n-queens completion problem*…

ARTICLE PUBLISHED — LINK

It was exciting surprise for me when on Facebook page of my younger son I have seen a picture of 8-queens-arrangement on the chessboard that is very similar to my video-game sQuaricon, and later explored what is going about it.

I’ve gone to the web page of* Journal of Artificial Intelligence Research* and carefully red article of professors Ian P. Gent, Christopher Jefferson and Peter Nightingale “Complexity of n-Queens Completion”. *link to the work*

Founder of sQuaricon - Prague, a game company, develops unique logic-puzzle-brainteaser, number theory enthusiast with aspiration (!) to synchronize the robots