{"paper":{"title":"Enumeration of $4 \\times 4$ Magic Squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Van Herick, Matthias Beck","submitted_at":"2009-07-18T07:54:03Z","abstract_excerpt":"A \\emph{magic square} is an $n \\times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the $n=3$ case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane arrangements and Ehrhart's theory of lattice-point enumeration."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3188","kind":"arxiv","version":2},"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"}