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 $n\times n$ square, filled with the numbers from $1$ to $n^{2}$ , this common sum must equal $n^{2}+1$ . These squares are also called associated magic squares, regular magic squares, regmagic squares, or symmetric magic squares.^[1]^[2]^[3]

Examples

For instance, the Lo Shu Square, the unique $3\times 3$ 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 $4\times 4$ 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 $n\times n$ magic squares for $n=3,4,5,\dots$ , 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 in the position for $6\times 6$ associative magic squares is an example of a more general phenomenon: these squares do not exist for values of $n$ that are singly even (that is, 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

^ 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
^ Bell, Jordan; Stevens, Brett (2007), "Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular $n$ -queens solutions", Journal of Combinatorial Designs, 15 (3): 221–234, doi:10.1002/jcd.20143, MR 2311190
^ ^a ^b 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
^ ^a ^b 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
^ 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

External links

Weisstein, Eric W., "Associative Magic Square", MathWorld

[andrews-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

[belste-2] Bell, Jordan; Stevens, Brett (2007), "Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular $n$ -queens solutions", Journal of Combinatorial Designs, 15 (3): 221–234, doi:10.1002/jcd.20143, MR 2311190

[nordgren-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

[llnww-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

[pasles-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

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