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Using modular arithmetic of the ring $\mathbb{Z}_{n+1}$ we obtain a new shortsolution to the problem of existence of at least one solution to the $N$-Queensproblem on an $N \times N$ chessboard. It was proved, that these solutions canbe represented as the Queen function with the width fewer or equal to 3. It isshown, that this estimate could not be reduced. A necessary and sufficientcondition of being a composition of solutions a solution is found. Based on theobtained results we formulate a conjecture about the width of therepresentation of arbitrary solution. If this conjecture is valid, it entailssolvability of the $N$-Queens completion in polynomial time. The connectionbetween the $N$-Queens completion and the Millennium $P$ vs $NP$ Problem isfound by the group of mathematicians from Scotland in August 2017.

New explicit solution to the N-Queens Problem and its relation to the Millennium Problem