Associative magic square

An associative magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an n × n square, filled with the numbers from 1 to n2, this common sum must equal n2 + 1. These squares are also called associated magic squares, regular magic squares, regmagic squares, or symmetric magic squares.[1][2][3]

In the Lo Shu Square, pairs of opposite numbers sum to 10
Detail from Melencolia I showing a 4 × 4 associative square

Examples

For instance, the Lo Shu Square – the unique 3 × 3 magic square – is associative, because each pair of opposite points form a line of the square together with the center point, so the sum of the two opposite points equals the sum of a line minus the value of the center point regardless of which two opposite points are chosen.[4] The 4 × 4 magic square from Albrecht Dürer's 1514 engraving Melencolia I – also found in a 1765 letter of Benjamin Franklin – is also associative, with each pair of opposite numbers summing to 17.[5]

Existence and enumeration

The numbers of possible associative n × n magic squares for n = 3,4,5,..., counting two squares as the same whenever they differ only by a rotation or reflection, are:

1, 48, 48544, 0, 1125154039419854784, ... (sequence A081262 in the OEIS)

The number zero for n = 6 is an example of a more general phenomenon: associative magic squares do not exist for values of n that are singly even (equal to 2 modulo 4).[3] Every associative magic square of even order forms a singular matrix, but associative magic squares of odd order can be singular or nonsingular.[4]

References
  1. Frierson, L. S. (1917), "Notes on pandiagonal and associated magic squares", in Andrews, W. S. (ed.), Magic Squares and Cubes (2nd ed.), Open Court, pp. 229–244
  2. Bell, Jordan; Stevens, Brett (2007), "Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular -queens solutions", Journal of Combinatorial Designs, 15 (3): 221–234, doi:10.1002/jcd.20143, MR 2311190, S2CID 121149492
  3. Nordgren, Ronald P. (2012), "On properties of special magic square matrices", Linear Algebra and Its Applications, 437 (8): 2009–2025, doi:10.1016/j.laa.2012.05.031, MR 2950468
  4. Lee, Michael Z.; Love, Elizabeth; Narayan, Sivaram K.; Wascher, Elizabeth; Webster, Jordan D. (2012), "On nonsingular regular magic squares of odd order", Linear Algebra and Its Applications, 437 (6): 1346–1355, doi:10.1016/j.laa.2012.04.004, MR 2942355
  5. Pasles, Paul C. (2001), "The lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle", American Mathematical Monthly, 108 (6): 489–511, doi:10.1080/00029890.2001.11919777, JSTOR 2695704, MR 1840656, S2CID 341378
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