{"paper":{"title":"The sliding tile puzzle, roots to polynomials, and $\\textbf{P}$ vs. $\\textbf{NP}$ complexity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Roy Burson","submitted_at":"2025-01-17T00:11:48Z","abstract_excerpt":"This work explores the relationship between solution space and time complexity in the context of the $\\textbf{P}$ vs. $\\textbf{NP}$ problem, particularly through the lens of the sliding tile puzzle and root finding algorithms. We focus on the trade-off between finding a solution and verifying it, highlighting how understanding the structure of the solution space can inform the complexity of these problems. By examining the relationship between the number of possible configurations and the time complexity required to traverse this space we demonstrate that the minimal time to verify a solution "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.10480","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.10480/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}