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Enumeration of $ 4 \times 4$ magic squares


Authors: Matthias Beck and Andrew van Herick
Journal: Math. Comp. 80 (2011), 617-621
MSC (2010): Primary 05A15, 05C78, 52B20, 52C35, 68R05
DOI: https://doi.org/10.1090/S0025-5718-10-02347-1
Published electronically: March 29, 2010
MathSciNet review: 2728997
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Abstract: A 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.


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Additional Information

Matthias Beck
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132
Email: beck@math.sfsu.edu

Andrew van Herick
Affiliation: 531 Beloit Avenue, Kensington, California 94708
Email: avanherick@gmail.com

DOI: https://doi.org/10.1090/S0025-5718-10-02347-1
Keywords: Magic square, lattice-point counting, rational inside-out convex polytope, arrangement of hyperplanes, Ehrhart theory, rational generating function computation
Received by editor(s): August 11, 2009
Received by editor(s) in revised form: August 30, 2009
Published electronically: March 29, 2010
Additional Notes: We thank a referee and an associate editor for helpful comments on an earlier version of this paper. We are grateful to San Francisco State University’s Center for Computing and Life Sciences for graciously offering the use of their resources. This research was partially supported by the NSF (research grant DMS-0810105).
Article copyright: © Copyright 2010 American Mathematical Society