{"paper":{"title":"An Almost-Optimal Upper Bound on the Push Number of the Torus Puzzle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The torus puzzle can be solved using O(mn log max{m,n}) unit row and column rotations.","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Matteo Caporrella, Stefano Leucci","submitted_at":"2026-01-13T21:28:37Z","abstract_excerpt":"We study the Torus Puzzle, a solitaire game in which the elements of an input $m \\times n$ matrix need to be rearranged into a target configuration via a sequence of unit rotations (i.e., circular shifts) of rows and/or columns. Amano et al. proposed a more permissive variant of the above puzzle, where each row and column rotation can shift the involved elements by any amount of positions. The number of rotations needed to solve the original and the permissive variants of the puzzle are respectively known as the \\emph{push number} and the \\emph{drag number}, where the latter is always smaller "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We provide an algorithm that solves the Torus Puzzle using O(mn ⋅ log max{m, n}) unit rotations in a model that is more restricted than that of the original puzzle. This implies a corresponding upper bound on the push number and reduces the gap between the known upper and lower bounds from Θ(max{m,n}) to Θ(log max{m, n}).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The algorithm constructed for the restricted unit-rotation model can be shown to solve every possible input configuration without hidden exponential blow-up or additional assumptions on the target permutation that would not hold in the original puzzle.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An algorithm solves the torus puzzle with O(mn log max(m,n)) unit rotations in a restricted model, improving the push-number upper bound and shrinking the gap to the Omega(mn) lower bound to a logarithmic factor.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The torus puzzle can be solved using O(mn log max{m,n}) unit row and column rotations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3522e4f4baf4fe30cfcf17100a7b5bc3a341cf6c781cb542d22bee76620e3dfe"},"source":{"id":"2601.08989","kind":"arxiv","version":2},"verdict":{"id":"9e2c9d2b-e75c-418a-9b0d-804341b7c964","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T14:14:53.565963Z","strongest_claim":"We provide an algorithm that solves the Torus Puzzle using O(mn ⋅ log max{m, n}) unit rotations in a model that is more restricted than that of the original puzzle. This implies a corresponding upper bound on the push number and reduces the gap between the known upper and lower bounds from Θ(max{m,n}) to Θ(log max{m, n}).","one_line_summary":"An algorithm solves the torus puzzle with O(mn log max(m,n)) unit rotations in a restricted model, improving the push-number upper bound and shrinking the gap to the Omega(mn) lower bound to a logarithmic factor.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The algorithm constructed for the restricted unit-rotation model can be shown to solve every possible input configuration without hidden exponential blow-up or additional assumptions on the target permutation that would not hold in the original puzzle.","pith_extraction_headline":"The torus puzzle can be solved using O(mn log max{m,n}) unit row and column rotations."},"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"}