{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/new-explicit-solution-to-the-n-queens-problem","title":"New explicit solution to the N-Queens Problem and its relation to the Millennium Problem","arxiv_id":"1805.07329","date":"2018-05-18","proceeding":null,"authors":["Dmitrii Mikhailovskii"],"abstract":"Using modular arithmetic of the ring $\\mathbb{Z}_{n+1}$ we obtain a new short solution to the problem of existence of at least one solution to the $N$-Queens problem on an $N \\times N$ 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.","url_abs":"https://arxiv.org/abs/1805.07329v1","url_pdf":"https://arxiv.org/pdf/1805.07329v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"new-explicit-solution-to-the-n-queens-problem","repo_url":"https://github.com/TheInventorMan/N-Queens-GPU","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}