Exact Enumeration of Seven-Line 3 x 3 Semimagic Squares of Distinct Squares Through Root 30000

Reproducibility package for an exact bounded enumeration of 3 x 3 semimagic arrays of nine distinct positive squares whose three rows, three columns, and one transversal have a common sum. All roots are bounded by 30,000, and arrays are identified under independent row and column permutations and transposition. The catalogue contains 450 canonical classes, including 43 primitive classes. Among the primitive classes, 23 have a common sum that is itself a square. No fully magic square occurs within the stated domain. The release includes the complete machine-readable catalogue, the frozen exact enumeration engine, an independent verifier, validation and optimization records, the research paper and its reproducible PDF builder, citation metadata, licenses, and SHA-256 integrity hashes. The result is exhaustive only for positive, pairwise-distinct roots not exceeding 30,000 and for the seven-line subfamily. It is not an exhaustive search over arbitrary semimagic squares and does not settle the existence of a 3 x 3 magic square of nine distinct squares. The seven-line normal form and the common-difference search strategy are prior art; the contribution of this release is the bounded exhaustive catalogue, symmetry quotient, completeness argument, validation chain, and reproducible publication package.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-07-22
DOI
https://doi.org/10.5281/zenodo.21495702
Primary Topic
Advanced Combinatorial Mathematics
Type
preprint
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preprint

Exact Enumeration of Seven-Line 3 x 3 Semimagic Squares of Distinct Squares Through Root 30000

Mustapha RHERRAD
Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
preprint

Exact Enumeration of Seven-Line 3 x 3 Semimagic Squares of Distinct Squares Through Root 30000

Mustapha RHERRAD
preprint en

Abstract

Reproducibility package for an exact bounded enumeration of 3 x 3 semimagic arrays of nine distinct positive squares whose three rows, three columns, and one transversal have a common sum. All roots are bounded by 30,000, and arrays are identified under independent row and column permutations and transposition. The catalogue contains 450 canonical classes, including 43 primitive classes. Among the primitive classes, 23 have a common sum that is itself a square. No fully magic square occurs within the stated domain. The release includes the complete machine-readable catalogue, the frozen exact enumeration engine, an independent verifier, validation and optimization records, the research paper and its reproducible PDF builder, citation metadata, licenses, and SHA-256 integrity hashes. The result is exhaustive only for positive, pairwise-distinct roots not exceeding 30,000 and for the seven-line subfamily. It is not an exhaustive search over arbitrary semimagic squares and does not settle the existence of a 3 x 3 magic square of nine distinct squares. The seven-line normal form and the common-difference search strategy are prior art; the contribution of this release is the bounded exhaustive catalogue, symmetry quotient, completeness argument, validation chain, and reproducible publication package.

Zenodo (CERN European Organization for Nuclear Research)
Mystic Seaport (US)
Advanced Combinatorial Mathematics
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