PreprintarXiv licencearXiv
The algebraic geometry of 3-by-3 magic squares of squares
Abstract
The question of whether a 3-by-3 magic square of squares with distinct integer entries exists has been open since the 18th century. We study the geometry of the algebraic surface parameterizing 3-by-3 magic squares of squares, which is a singular complete intersection of six quadrics in projective 8-space. We compute its geometric automorphism group via an argument involving Gale duality. We compute the basic topological invariants and Hodge diamond of its resolution. We provide an explicit rank 518 sublattice of its geometric Picard group, close to the Hodge-theoretic upper bound of 544. Finally, we study the geometry and arithmetic of del Pezzo, K3, and Enriques surfaces that arise as coordinate projections.
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BibTeX
@misc{auel2026algebraic,
title = {{The algebraic geometry of 3-by-3 magic squares of squares}},
author = {Asher Auel and Benjamin Singer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09351},
eprint = {2609.09351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09351},
}
RIS
TY - UNPB
TI - The algebraic geometry of 3-by-3 magic squares of squares
AU - Asher Auel
AU - Benjamin Singer
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09351
UR - https://arxiv.org/abs/2609.09351
AB - The question of whether a 3-by-3 magic square of squares with distinct integer entries exists has been open since the 18th century. We study the geometry of the algebraic surface parameterizing 3-by-3 magic squares of squares, which is a singular complete intersection of six quadrics in projective 8-space. We compute its geometric automorphism group via an argument involving Gale duality. We compute the basic topological invariants and Hodge diamond of its resolution. We provide an explicit rank 518 sublattice of its geometric Picard group, close to the Hodge-theoretic upper bound of 544. Finally, we study the geometry and arithmetic of del Pezzo, K3, and Enriques surfaces that arise as coordinate projections.
ER -