View source for Associative magic square

{{Short description|Mathematical concept of arrangement of numbers in a square}}
[[File:Magic Square Lo Shu.svg|thumb|upright|In the [[Lo Shu Square]], pairs of opposite numbers sum to&nbsp;10]]
[[File:Albrecht Dürer - Melencolia I (detail).jpg|thumb|Detail from ''[[Melencolia I]]'' showing a 4&thinsp;×&thinsp;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''&thinsp;×&thinsp;''n'' square, filled with the numbers from 1 to ''n''<sup>2</sup>, this common sum must equal ''n''<sup>2</sup>&nbsp;+&thinsp;1. These squares are also called '''associated magic squares''', '''regular magic squares''', '''regmagic squares''', or '''symmetric magic squares'''.{{r|andrews|belste|nordgren}}

==Examples==
For instance, the [[Lo Shu Square]] – the unique 3&thinsp;×&thinsp;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.{{r|llnww}} The 4&thinsp;×&thinsp;4 magic square from [[Albrecht Dürer]]{{'s}} 1514 engraving {{nowrap|''[[Melencolia I]]''}} – also found in a 1765 letter of [[Benjamin Franklin]] – is also associative, with each pair of opposite numbers summing to 17.{{r|pasles}}

==Existence and enumeration==
The numbers of possible associative ''n''&thinsp;×&thinsp;''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, ... {{OEIS|A081262}}
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 [[modular arithmetic|modulo]] 4).{{r|nordgren}} Every associative magic square of [[parity (mathematics)|even]] order forms a [[singular matrix]], but associative magic squares of [[parity (mathematics)|odd]] order can be singular or nonsingular.{{r|llnww}}

==References==
{{reflist|refs=

<ref name=andrews>{{citation
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 | contribution = Notes on pandiagonal and associated magic squares
 | contribution-url = https://archive.org/details/MagicSquaresCubesAndrewsEdited/page/n237/mode/2up
 | edition = 2nd
 | pages = 229–244
 | publisher = Open Court
 | title = Magic Squares and Cubes
 | year = 1917}}</ref>

<ref name=belste>{{citation
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 | doi = 10.1002/jcd.20143
 | issue = 3
 | journal = Journal of Combinatorial Designs
 | mr = 2311190
 | pages = 221–234
 | title = Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular <math>n</math>-queens solutions
 | volume = 15
 | year = 2007| s2cid = 121149492
 }}</ref>

<ref name=llnww>{{citation
 | last1 = Lee | first1 = Michael Z.
 | last2 = Love | first2 = Elizabeth
 | last3 = Narayan | first3 = Sivaram K.
 | last4 = Wascher | first4 = Elizabeth
 | last5 = Webster | first5 = Jordan D.
 | doi = 10.1016/j.laa.2012.04.004
 | issue = 6
 | journal = Linear Algebra and Its Applications
 | mr = 2942355
 | pages = 1346–1355
 | title = On nonsingular regular magic squares of odd order
 | volume = 437
 | year = 2012| doi-access = free
 }}</ref>

<ref name=nordgren>{{citation
 | last = Nordgren | first = Ronald P.
 | doi = 10.1016/j.laa.2012.05.031
 | issue = 8
 | journal = Linear Algebra and Its Applications
 | mr = 2950468
 | pages = 2009–2025
 | title = On properties of special magic square matrices
 | volume = 437
 | year = 2012| doi-access = free
 }}</ref>

<ref name=pasles>{{citation
 | last = Pasles | first = Paul C.
 | doi = 10.1080/00029890.2001.11919777
 | issue = 6
 | journal = American Mathematical Monthly
 | jstor = 2695704
 | mr = 1840656
 | pages = 489–511
 | title = The lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle
 | volume = 108
 | year = 2001| s2cid = 341378
 }}</ref>

}}

==External links==
*{{mathworld|urlname=AssociativeMagicSquare|title=Associative Magic Square|mode=cs2}}
{{Magic polygons}}
[[Category:Magic squares]]