Editing Magic square (section)

===For squares of order ''m × n'' where ''m'', ''n'' > 2===
This is a method reminiscent of the [[Kronecker product]] of two matrices, that builds an ''nm'' × ''nm'' magic square from an ''n'' × ''n'' magic square and an ''m'' × ''m'' magic square.<ref name="hartleym">Hartley, M. [http://www.dr-mikes-math-games-for-kids.com/making-big-magic-squares.html "Making Big Magic Squares"].</ref> The "product" of two magic squares creates a magic square of higher order than the two multiplicands. Let the two magic squares be of orders ''m'' and ''n''. The final square will be of order ''m × n''. Divide the square of order ''m × n'' into ''m × m'' sub-squares, such that there are a total of ''n''<sup>2</sup> such sub-squares. In the square of order ''n'', reduce by 1 the value of all the numbers. Multiply these reduced values by ''m''<sup>2</sup>, and place the results in the corresponding sub-squares of the ''m × n'' whole square. The squares of order ''m'' are added ''n''<sup>2</sup> times to the sub-squares of the final square. The peculiarity of this construction method is that each magic subsquare will have different magic sums. The square made of such magic sums from each magic subsquare will again be a magic square. The smallest composite magic square of order 9, composed of two order 3 squares is given below.

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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
|-
|+ Order 3
|-
| 8 || 1 || 6
|-
| 3 || 5 || 7
|-
| 4 || 9 || 2
|}

{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:18em;table-layout:fixed;"
|-
|+ Order 3×3
| style="background-color: silver;"|63 || style="background-color: silver;"|63 || style="background-color: silver;"|63 || 0 || 0 || 0 || style="background-color: silver;"|45 || style="background-color: silver;"|45 || style="background-color: silver;"|45
|-
| style="background-color: silver;"|63 || style="background-color: silver;"|63 || style="background-color: silver;"|63 || 0 || 0 || 0 || style="background-color: silver;"|45 || style="background-color: silver;"|45 || style="background-color: silver;"|45
|-
| style="background-color: silver;"|63 || style="background-color: silver;"|63 || style="background-color: silver;"|63 || 0 || 0 || 0 || style="background-color: silver;"|45 || style="background-color: silver;"|45 || style="background-color: silver;"|45
|-
| 18 || 18 || 18 || style="background-color: silver;"|36 || style="background-color: silver;"|36 || style="background-color: silver;"|36 || 54 || 54 || 54
|-
| 18 || 18 || 18 || style="background-color: silver;"|36 || style="background-color: silver;"|36 || style="background-color: silver;"|36 || 54 || 54 || 54
|-
| 18 || 18 || 18 || style="background-color: silver;"|36 || style="background-color: silver;"|36 || style="background-color: silver;"|36 || 54 || 54 || 54
|-
| style="background-color: silver;"|27 || style="background-color: silver;"|27 || style="background-color: silver;"|27 || 72 || 72 || 72 || style="background-color: silver;"|9 || style="background-color: silver;"|9 || style="background-color: silver;"|9
|-
| style="background-color: silver;"|27 || style="background-color: silver;"|27 || style="background-color: silver;"|27 || 72 || 72 || 72 || style="background-color: silver;"|9 || style="background-color: silver;"|9 || style="background-color: silver;"|9
|-
| style="background-color: silver;"|27 || style="background-color: silver;"|27 || style="background-color: silver;"|27 || 72 || 72 || 72 || style="background-color: silver;"|9 || style="background-color: silver;"|9 || style="background-color: silver;"|9
|}

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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:18em;table-layout:fixed;"
|-
|+ Order 3×3 
| style="background-color: silver;"|71 || style="background-color: silver;"|64 || style="background-color: silver;"|69 || 8 || 1 || 6 || style="background-color: silver;"|53 || style="background-color: silver;"|46 || style="background-color: silver;"|51
|-
| style="background-color: silver;"|66 || style="background-color: silver;"|68 || style="background-color: silver;"|70 || 3 || 5 || 7 || style="background-color: silver;"|48 || style="background-color: silver;"|50 || style="background-color: silver;"|52
|-
| style="background-color: silver;"|67 || style="background-color: silver;"|72 || style="background-color: silver;"|65 || 4 || 9 || 2 || style="background-color: silver;"|49 || style="background-color: silver;"|54 || style="background-color: silver;"|47
|-
| 26 || 19 || 24 || style="background-color: silver;"|44 || style="background-color: silver;"|37 || style="background-color: silver;"|42 || 62 || 55 || 60
|-
| 21 || 23 || 25 || style="background-color: silver;"|39 || style="background-color: silver;"|41 || style="background-color: silver;"|43 || 57 || 59 || 61
|-
| 22 || 27 || 20 || style="background-color: silver;"|40 || style="background-color: silver;"|45 || style="background-color: silver;"|38 || 58 || 63 || 56
|-
| style="background-color: silver;"|35 || style="background-color: silver;"|28 || style="background-color: silver;"|33 || 80 || 73 || 78 || style="background-color: silver;"|17 || style="background-color: silver;"|10 || style="background-color: silver;"|15
|-
| style="background-color: silver;"|30 || style="background-color: silver;"|32 || style="background-color: silver;"|34 || 75 || 77 || 79 || style="background-color: silver;"|12 || style="background-color: silver;"|14 || style="background-color: silver;"|16
|-
| style="background-color: silver;"|31 || style="background-color: silver;"|36 || style="background-color: silver;"|29 || 76 || 81 || 74 || style="background-color: silver;"|13 || style="background-color: silver;"|18 || style="background-color: silver;"|11
|}
{{col-end}}

Since each of the 3×3 sub-squares can be independently rotated and reflected into 8 different squares, from this single 9×9 composite square we can derive 8<sup>9</sup> = 134,217,728 essentially different 9×9 composite squares. Plenty more composite magic squares can also be derived if we select non-consecutive numbers in the magic sub-squares, like in Yang Hui's version of the 9×9 composite magic square. The next smallest composite magic squares of order 12, composed of magic squares of order 3 and 4 are given below.

{{col-begin|width=auto;margin:0.5em auto}}
{{col-break|valign=bottom}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
|-
|+ Order 3
|-
| 2 || 9 || 4
|-
| 7 || 5 || 3
|-
| 6 || 1 || 8
|}

{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
|+ Order 4
|-
| 1 || 14 || 11 || 8
|-
| 12 || 7 || 2 || 13
|-
| 6 || 9 || 16 || 3
|-
| 15 || 4 || 5 || 10
|}

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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:30em;height:30em;table-layout:fixed;"
|-
|+ Order 3 × 4 
|-
| style="background-color: silver;"|2 || style="background-color: silver;"|9 || style="background-color: silver;"|4 || 119 || 126 || 121 || style="background-color: silver;"|92 || style="background-color: silver;"|99 || style="background-color: silver;"|94 || 65 || 72 || 67
|-
| style="background-color: silver;"|7 || style="background-color: silver;"|5 || style="background-color: silver;"|3 || 124 || 122 || 120 || style="background-color: silver;"|97 || style="background-color: silver;"|95 || style="background-color: silver;"|93 || 70 || 68 || 66
|-
| style="background-color: silver;"|6 || style="background-color: silver;"|1 || style="background-color: silver;"|8 || 123 || 118 || 125 || style="background-color: silver;"|96 || style="background-color: silver;"|91 || style="background-color: silver;"|98 || 69 || 64 || 71
|-
| 101 || 108 || 103 || style="background-color: silver;"|56 || style="background-color: silver;"|63 || style="background-color: silver;"|58 || 11 || 18 || 13 || style="background-color: silver;"|110 || style="background-color: silver;"|117 || style="background-color: silver;"|112
|-
| 106 || 104 || 102 || style="background-color: silver;"|61 || style="background-color: silver;"|59 || style="background-color: silver;"|57 || 16 || 14 || 12 || style="background-color: silver;"|115 || style="background-color: silver;"|113 || style="background-color: silver;"|111
|-
| 105 || 100 || 107 || style="background-color: silver;"|60 || style="background-color: silver;"|55 || style="background-color: silver;"|62 || 15 || 10 || 17 || style="background-color: silver;"|114 || style="background-color: silver;"|109 || style="background-color: silver;"|116 
|-
| style="background-color: silver;"|47 || style="background-color: silver;"|54 || style="background-color: silver;"|49 || 74 || 81 || 76 || style="background-color: silver;"|137 || style="background-color: silver;"|144 || style="background-color: silver;"|139 || 20 || 27 || 22
|-
| style="background-color: silver;"|52 || style="background-color: silver;"|50 || style="background-color: silver;"|48 || 79 || 77 || 75 || style="background-color: silver;"|142 || style="background-color: silver;"|140 || style="background-color: silver;"|138 || 25 || 23 || 21
|-
| style="background-color: silver;"|51 || style="background-color: silver;"|46 || style="background-color: silver;"|53 || 78 || 73 || 80 || style="background-color: silver;"|141 || style="background-color: silver;"|136 || style="background-color: silver;"|143 || 24 || 19 || 26
|-
| 128 || 135 || 130 || style="background-color: silver;"|29 || style="background-color: silver;"|36 || style="background-color: silver;"|31 || 38 || 45 || 40 || style="background-color: silver;"|83 || style="background-color: silver;"|90 || style="background-color: silver;"|85 
|-
| 133 || 131 || 129 || style="background-color: silver;"|34 || style="background-color: silver;"|32 || style="background-color: silver;"|30 || 43 || 41 || 39 || style="background-color: silver;"|88 || style="background-color: silver;"|86 || style="background-color: silver;"|84
|- 
| 132 || 127 || 134 || style="background-color: silver;"|33 || style="background-color: silver;"|28 || style="background-color: silver;"|35 || 42 || 37 || 44 || style="background-color: silver;"|87 || style="background-color: silver;"|82 || style="background-color: silver;"|89
|}

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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:30em;height:30em;table-layout:fixed;"
|-
|+ Order 4 × 3 
|-
| style="background-color: silver;"|17 || style="background-color: silver;"|30 || style="background-color: silver;"|27 || style="background-color: silver;"|24 || 129 || 142 || 139 || 136 || style="background-color: silver;"|49 || style="background-color: silver;"|62 || style="background-color: silver;"|59 || style="background-color: silver;"|56 
|-
| style="background-color: silver;"|28 || style="background-color: silver;"|23 || style="background-color: silver;"|18 || style="background-color: silver;"|29 || 140 || 135 || 130 || 141 || style="background-color: silver;"|60 || style="background-color: silver;"|55 || style="background-color: silver;"|50 || style="background-color: silver;"|61
|-
| style="background-color: silver;"|22 || style="background-color: silver;"|25 || style="background-color: silver;"|32 || style="background-color: silver;"|19 || 134 || 137 || 144 || 131 || style="background-color: silver;"|54 || style="background-color: silver;"|57 || style="background-color: silver;"|64 || style="background-color: silver;"|51
|- 
| style="background-color: silver;"|31 || style="background-color: silver;"|20 || style="background-color: silver;"|21 || style="background-color: silver;"|26 || 143 || 132 || 133 || 138 || style="background-color: silver;"|63 || style="background-color: silver;"|52 || style="background-color: silver;"|53 || style="background-color: silver;"|58
|-
| 97 || 110 || 107 || 104 || style="background-color: silver;"|65 || style="background-color: silver;"|78 || style="background-color: silver;"|75 || style="background-color: silver;"|72 || 33 || 46 || 43 || 40
|-
| 108 || 103 || 98 || 109 || style="background-color: silver;"|76 || style="background-color: silver;"|71 || style="background-color: silver;"|66 || style="background-color: silver;"|77 || 44 || 39 || 34 || 45 
|-
| 102 || 105 || 112 || 99 || style="background-color: silver;"|70 || style="background-color: silver;"|73 || style="background-color: silver;"|80 || style="background-color: silver;"|67 || 38 || 41 || 48 || 35
|-
| 111 || 100 || 101 || 106 || style="background-color: silver;"|79 || style="background-color: silver;"|68 || style="background-color: silver;"|69 || style="background-color: silver;"|74 || 47 || 36 || 37 || 42 
|-
| style="background-color: silver;"|81 || style="background-color: silver;"|94 || style="background-color: silver;"|91 || style="background-color: silver;"|88 || 1 || 14 || 11 || 8 || style="background-color: silver;"|113 || style="background-color: silver;"|126 || style="background-color: silver;"|123 || style="background-color: silver;"|120
|-
| style="background-color: silver;"|92 || style="background-color: silver;"|87 || style="background-color: silver;"|82 || style="background-color: silver;"|93 || 12 || 7 || 2 || 13 || style="background-color: silver;"|124 || style="background-color: silver;"|119 || style="background-color: silver;"|114 || style="background-color: silver;"|125 
|-
| style="background-color: silver;"|86 || style="background-color: silver;"|89 || style="background-color: silver;"|96 || style="background-color: silver;"|83 || 6 || 9 || 16 || 3 || style="background-color: silver;"|118 || style="background-color: silver;"|121 || style="background-color: silver;"|128 || style="background-color: silver;"|115 
|-
| style="background-color: silver;"|95 || style="background-color: silver;"|84 || style="background-color: silver;"|85 || style="background-color: silver;"|90 || 15 || 4 || 5 || 10 || style="background-color: silver;"|127 || style="background-color: silver;"|116 || style="background-color: silver;"|117 || style="background-color: silver;"|122
|}
{{col-end}}

For the base squares, there is only one essentially different 3rd order square, while there 880 essentially different 4th-order squares that we can choose from. Each pairing can produce two different composite squares. Since each magic sub-squares in each composite square can be expressed in 8 different forms due to rotations and reflections, there can be 1×880×8<sup>9</sup> + 880×1×8<sup>16</sup> ≈ 2.476×10<sup>17</sup> essentially different 12×12 composite magic squares created this way, with consecutive numbers in each sub-square. In general, if there are ''c''<sub>m</sub> and ''c''<sub>n</sub> essentially different magic squares of order ''m'' and ''n'', then we can form ''c''<sub>m</sub> × ''c''<sub>n</sub> × ( 8<sup>''m''<sup>2</sup></sup> + 8<sup>''n''<sup>2</sup></sup>) composite squares of order ''mn'', provided ''m'' ≠ ''n''. If ''m'' = ''n'', then we can form (''c''<sub>m</sub>)<sup>2</sup> × 8<sup>''m''<sup>2</sup></sup> composite squares of order ''m''<sup>2</sup>.