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97:– 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. The 4 × 4 magic square from
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Lee, Michael Z.; Love, Elizabeth; Narayan, Sivaram K.; Wascher, Elizabeth; Webster, Jordan D. (2012), "On nonsingular regular magic squares of odd order",
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Pasles, Paul C. (2001), "The lost squares of Dr. Franklin: Ben
Franklin's missing squares and the secret of the magic circle",
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Bell, Jordan; Stevens, Brett (2007), "Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular
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136:= 3,4,5,..., counting two squares as the same whenever they differ only by a rotation or reflection, are:
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158:= 6 is an example of a more general phenomenon: associative magic squares do not exist for values of
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Nordgren, Ronald P. (2012), "On properties of special magic square matrices",
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Mathematical concept of arrangement of numbers in a square
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201:"Notes on pandiagonal and associated magic squares"
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207:(2nd ed.), Open Court, pp. 229–244
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27:, pairs of opposite numbers sum to 10
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170:4). Every associative magic square of
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124:The numbers of possible associative
324:Linear Algebra and Its Applications
285:Linear Algebra and Its Applications
178:, but associative magic squares of
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112:– also found in a 1765 letter of
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240:Journal of Combinatorial Designs
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551:Prime reciprocal magic square
360:American Mathematical Monthly
65:, this common sum must equal
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418:"Associative Magic Square"
565:Higher dimensional shapes
556:Most-perfect magic square
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337:10.1016/j.laa.2012.04.004
298:10.1016/j.laa.2012.05.031
120:Existence and enumeration
610:Pandiagonal magic square
605:Associative magic square
546:Pandiagonal magic square
199:Frierson, L. S. (1917),
71:associated magic squares
47:associative magic square
205:Magic Squares and Cubes
83:symmetric magic squares
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238:-queens solutions",
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132:magic squares for
93:For instance, the
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231:{\displaystyle n}
114:Benjamin Franklin
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38:Melencolia I
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35:Detail from
656:Magic graph
636:Word square
164:singly even
572:Magic cube
496:Magic star
186:References
423:MathWorld
268:121149492
162:that are
676:Category
89:Examples
577:classes
388:1840656
380:2695704
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23:In the
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376:JSTOR
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55:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
