{"paper":{"title":"Note on a magic rectangle set on dihedral group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Magic rectangle sets exist for every dihedral group of order mnk when m and n are even.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sylwia Cichacz","submitted_at":"2026-05-13T11:51:05Z","abstract_excerpt":"Let $\\Gamma$ be a group of order $mnk$ and $MRS_{\\Gamma}(m,n;k)=(a_{i,j}^s)_{m\\times n}$ be a collection of $k$ arrays $m\\times n$ whose entries are all distinct elements of $\\Gamma$. If there exist elements $\\rho,\\sigma\\in\\Gamma$ such that for every row $i$, there exists an ordering of elements such that\n  $$\n  a_{i,j_1}^s a_{i,j_2}^s \\dots a_{i,j_{n-1}}^s a_{i,j_n}^s= \\rho\n  $$\n  and for every column $j$ there exists an ordering of elements such that\n  $$\n  a_{i_1,j}^s a_{i_2,j}^s \\dots a_{i_{m-1},j}^s a_{i_m,j}^s = \\sigma,\n  $$\n  then $MRS_{\\Gamma}(m,n;k)$ is called a \\emph{$\\Gamma$-magic r"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that MRS_Γ(m,n;k) exists for every dihedral group Γ of order mnk, provided that m and n are even. As a consequence, we obtain broad existence results for magic rectangles and magic squares over dihedral groups.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The construction requires m and n even so that elements can be paired and ordered to produce constant products despite the non-commutative relations in the dihedral group.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Magic rectangle sets exist over dihedral groups of order mnk whenever m and n are even.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Magic rectangle sets exist for every dihedral group of order mnk when m and n are even.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0423f9742731d591ef84be00acaac4ec4ea138bf3b2447a49caa64cbf8fe74aa"},"source":{"id":"2605.13393","kind":"arxiv","version":1},"verdict":{"id":"1bfe141c-52c4-498a-b322-30aaf89c946d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:18:01.541852Z","strongest_claim":"We prove that MRS_Γ(m,n;k) exists for every dihedral group Γ of order mnk, provided that m and n are even. As a consequence, we obtain broad existence results for magic rectangles and magic squares over dihedral groups.","one_line_summary":"Magic rectangle sets exist over dihedral groups of order mnk whenever m and n are even.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The construction requires m and n even so that elements can be paired and ordered to produce constant products despite the non-commutative relations in the dihedral group.","pith_extraction_headline":"Magic rectangle sets exist for every dihedral group of order mnk when m and n are even."},"references":{"count":11,"sample":[{"doi":"","year":2025,"title":"Cichacz, Partition of Abelian groups into zero-sum sets by complete mappings and its application to the existence of a magic rectangle set,J","work_id":"d7a8cdf1-3b63-4d32-bf3b-1d3fc7bb9c8a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"S. Cichacz, D. Froncek, Magic squares on Abelian groups,Discrete Math. 349(7)(2026), 115033","work_id":"a84a69d4-527f-4436-b15e-3117767405a6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"S. Cichacz, D. Froncek, Semi-magic dihedral squares, Preprint arXiv:2602.20774 [math.CO] (2026)","work_id":"38fe732b-30ca-4621-86b7-3c2014033926","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"S. Cichacz, T. Hinc, A magic rectangle set on Abelian groups and its application,Discrete Appl. Math.288(2021), 201–210. 10","work_id":"28e07f94-5437-4615-bcd2-b754ac8da8dc","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"C. J. Colbourn, J. H. Dinitz, eds.,Handbook of combinatorial designs, second edn.,Discrete Mathematics and its Applications (Boca Raton), Chapman & Hill/CRC Press, Boca Raton, FL 2007","work_id":"6fa7eae8-42d7-44a6-9b65-b2809a468a1e","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"a4a7d44c48d3ebe059acba9d6fb8372a7d47f5f24fcd10b85b8e8726bf659d9e","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"}