Papers › New explicit solution to the N-Queens Problem and its relation to the Millennium Problem
New explicit solution to the N-Queens Problem and its relation to the Millennium Problem
Dmitrii Mikhailovskii
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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.
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