Mathematical concept of arrangement of numbers in a square
In the
Lo Shu Square, pairs of opposite numbers sum to 10Detail from Melencolia I showing a 4 × 4 associative 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]
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]
^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,
MR2311190,
S2CID121149492
^
abNordgren, 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,
MR2950468
^
abLee, 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,
MR2942355