Abstract. Using  modular arithmetic of the  ring  Zn+1  we obtain a new short solution to  the  problem of existence of at least  one solution to  the  N-Queens problem on an  N × N  chessboard. It was proved, that these  solutions can be represented as the  Queen function with  the  width fewer  or equal  to  3.  It is shown, that this  estimate could  not be reduced. A necessary and  sufficient condition of being  a composition of solutions a solution is found. Based on the  obtained results we formulate a conjecture about the  width of the  representation of arbitrary solution. If this conjecture is valid,  it entails solvability of the  N-Queens completion in polynomial time.  The  connection between the  N-Queens completion and  the  Millennium P  vs  NP   Problem is found  by  the  group of mathematicians  from Scotland in August 2017.

Original article: https://arxiv.org/pdf/1805.07329