{"paper":{"title":"Cartesian Magicness of 3-Dimensional Boards","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gee-Choon Lau, Ho-Kuen Ng, Wai-Chee Shiu","submitted_at":"2018-05-13T14:04:00Z","abstract_excerpt":"A $(p,q,r)$-board that has $pq+pr+qr$ squares consists of a $(p,q)$-, a $(p,r)$-, and a $(q,r)$-rectangle. Let $S$ be the set of the squares. Consider a bijection $f : S \\to [1,pq+pr+qr]$. Firstly, for $1 \\le i \\le p$, let $x_i$ be the sum of all the $q+r$ integers in the $i$-th row of the $(p,q+r)$-rectangle. Secondly, for $1 \\le j \\le q$, let $y_j$ be the sum of all the $p+r$ integers in the $j$-th row of the $(q,p+r)$-rectangle. Finally, for $1\\le k\\le r$, let $z_k$ be the the sum of all the $p+q$ integers in the $k$-th row of the $(r,p+q)$-rectangle. Such an assignment is called a $(p,q,r)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.04890","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}