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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.13393","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T11:51:05Z","cross_cats_sorted":[],"title_canon_sha256":"1a6c2bb849a18c64563c1f7f94b538e066692eedacac3694f18db8410f874283","abstract_canon_sha256":"8fcca7da7d8f3969e08e34d3c4646daec9ed7f2c3f3573a8b241a82289efd7c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:47.691877Z","signature_b64":"+p4oJ8PJ7194ytbn6J2iT1tcUJqQkjXW9W2S4BfqGBNmbi6gDy3+mlv4eUQyNmuAQToRSRYd2Ejh7PH3h4xdBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0bbb65d4636b76cd6819a9340425d4da7973ae26d38e9ffe1b83c839ecd0bdb5","last_reissued_at":"2026-05-18T02:44:47.691454Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:47.691454Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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