Editing Magic square (section)

===For associative magic squares===
* An associative magic square remains associative when two rows or columns equidistant from the center are interchanged.<ref name="White-Assoc-Magic">{{cite web | url = https://budshaw.ca/Associative.html | title = Associative Magic Squares | website = budshaw.ca | first1 = S. Harry | last1 = White}}</ref><ref name="Hawley2011">{{cite web | url = https://nrich.maths.org/1338 | title = Magic Squares II | website = nrich.maths.org | first = Del | last = Hawley | publisher = University of Cambridge | date = 2011 }}</ref> For an even square, there are ''n''/2 pairs of rows or columns that can be interchanged; thus 2<sup>''n''/2</sup> × 2<sup>''n''/2</sup> = 2<sup>''n''</sup> equivalent magic squares by combining such interchanges can be obtained. For odd square, there are (''n'' &minus; 1)/2 pairs of rows or columns that can be interchanged; and 2<sup>''n''&minus;1</sup> equivalent magic squares obtained by combining such interchanges. Interchanging all the rows flips the square vertically (i.e. reflected along the horizontal axis), while interchanging all the columns flips the square horizontally (i.e. reflected along the vertical axis). In the example below, a 4×4 associative magic square on the left is transformed into a square on the right by interchanging the second and third row, yielding the famous Durer's magic square.
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
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| 16 || 3 || 2 || 13 
|-
| style="background-color: silver;"|9 || style="background-color: silver;"|6 || style="background-color: silver;"|7 || style="background-color: silver;"|12 
|-
| style="background-color: silver;"|5 || style="background-color: silver;"|10 || style="background-color: silver;"|11 || style="background-color: silver;"|8
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| 4 || 15 || 14 || 1
|}
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
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| 16 || 3 || 2 || 13 
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| 5 || 10 || 11 || 8 
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| 9 || 6 || 7 || 12
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| 4 || 15 || 14 || 1
|}
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* An associative magic square remains associative when two same sided rows (or columns) are interchanged along with corresponding other sided rows (or columns).<ref name="White-Assoc-Magic"/><ref name="Hawley2011"/> For an even square, since there are ''n''/2 same sided rows (or columns), there are ''n''(''n'' &minus; 2)/8 pairs of such rows (or columns) that can be interchanged. Thus, 2<sup>''n''(''n'' &minus; 2)/8</sup> × 2<sup>''n''(''n'' &minus; 2)/8</sup> = 2<sup>''n''(''n'' &minus; 2)/4</sup> equivalent magic squares can be obtained by combining such interchanges. For odd square, since there are (''n'' &minus; 1)/2 same sided rows or columns, there are (''n'' &minus; 1)(''n'' &minus; 3)/8 pairs of such rows or columns that can be interchanged. Thus, there are 2<sup>(''n'' &minus; 1)(''n'' &minus; 3)/8</sup> × 2<sup>(''n'' &minus; 1)(''n'' &minus; 3)/8</sup> = 2<sup>(''n'' &minus; 1)(''n'' &minus; 3)/4</sup> equivalent magic squares obtained by combining such interchanges. Interchanging all the same sided rows flips each quadrants of the square vertically, while interchanging all the same sided columns flips each quadrant of the square horizontally. In the example below, the original square is on the left, whose rows 1 and 2 are interchanged with each other, along with rows 3 and 4, to obtain the transformed square on the right.
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
| 1 || 15 || 14 || 4
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| 12 || 6 || 7 || 9 
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| 8 || 10 || 11 || 5 
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| 13 || 3 || 2 || 16
|}
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
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| 12 || 6 || 7 || 9 
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| 1 || 15 || 14 || 4 
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| 13 || 3 || 2 || 16
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| 8 || 10 || 11 || 5
|}
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* An associative magic square remains associative when its entries are replaced with corresponding numbers from a set of ''s'' arithmetic progressions with the same common difference among ''r'' terms, such that ''r'' × ''s'' = ''n''<sup>2</sup>, and whose initial terms are also in arithmetic progression, to obtain a non-normal magic square. Here either ''s'' or ''r'' should be a multiple of ''n''. Let us have ''s'' arithmetic progressions given by
::<math>
\begin{array}{lllll}
a & a + c & a + 2c & \cdots & a + (r-1)c \\
a + d & a + c + d & a + 2c + d & \cdots & a + (r-1)c + d \\
a + 2d & a + c + 2d & a + 2c + 2d & \cdots & a + (r-1)c + 2d \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
a + (s-1)d & a + c + (s-1)d & a + 2c + (s-1)d & \cdots & a + (r-1)c + (s-1)d \\
\end{array}
</math>
:where ''a'' is the initial term, ''c'' is the common difference of the arithmetic progressions, and ''d'' is the common difference among the initial terms of each progression. The new magic constant will be 
::<math> M = na + \frac{n}{2} \big[ (r-1)c+ (s-1)d \big]. </math>
:If ''s'' = ''r'' = ''n'', then follows the simplification
::<math> M = na + \frac{n}{2}(n-1)(c+d). </math>
:With ''a'' = ''c'' = 1 and ''d'' = ''n'', the usual ''M'' = ''n''(''n''<sup>2</sup>+1)/2 is obtained. For given ''M'' the required ''a'', ''c'', and ''d'' can be found by solving the [[linear Diophantine equation]]. In the examples below, there are order 4 normal magic squares on the left most side. The second square is a corresponding non-normal magic square with ''r'' = 8, ''s'' = 2, ''a'' = 1, ''c'' = 1, and ''d'' = 10 such that the new magic constant is ''M'' = 38. The third square is an order 5 normal magic square, which is a 90 degree clockwise rotated version of the square generated by De la Loubere method. On the right most side is a corresponding non-normal magic square with ''a'' = 4, ''c'' = 1, and ''d'' = 6 such that the new magic constant is ''M'' = 90. 
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
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| 1 || 15 || 14 || 4
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| 12 || 6 || 7 || 9 
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| 8 || 10 || 11 || 5 
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| 13 || 3 || 2 || 16
|}
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
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| 1 || 17 || 16 || 4
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| 14 || 6 || 7 || 11 
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| 8 || 12 || 13 || 5 
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| 15 || 3 || 2 || 18
|}
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
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| 11 || 10 || 4 || 23 || 17
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| 18 || 12 || 6 || 5 || 24
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| 25 || 19 || 13 || 7 || 1
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| 2 || 21 || 20 || 14 || 8
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| 9 || 3 || 22 || 16 || 15
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
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| 16 || 14 || 7 || 30 || 23
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| 24 || 17 || 10 || 8 || 31
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| 32 || 25 || 18 || 11 || 4
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| 5 || 28 || 26 || 19 || 12
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| 13 || 6 || 29 || 22 || 20
|}
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