Editing Magic square (section)

==Method of composition==
===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
|}

{{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;"|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.

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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
|-
| 2 || 9 || 4
|-
| 7 || 5 || 3
|-
| 6 || 1 || 8
|}

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{| 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
|}

{{col-break|valign=bottom|gap=1em}}
{| 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>.

===For squares of doubly even order===
When the squares are of doubly even order, we can construct a composite magic square in a manner more elegant than the above process, in the sense that every magic subsquare will have the same magic constant. Let ''n'' be the order of the main square and ''m'' the order of the equal subsquares. The subsquares are filled one by one, in any order, with a continuous sequence of ''m''<sup>2</sup>/2 smaller numbers (i.e. numbers less than or equal to ''n''<sup>2</sup>/2) together with their complements to ''n''<sup>2</sup> + 1. Each subsquare as a whole will yield the same magic sum. The advantage of this type of composite square is that each subsquare is filled in the same way and their arrangement is arbitrary. Thus, the knowledge of a single construction of even order will suffice to fill the whole square. Furthermore, if the subsquares are filled in the natural sequence, then the resulting square will be pandiagonal. The magic sum of the subsquares is related to the magic sum of the whole square by <math>M_m = \frac{M_n}{k}</math> where ''n'' = ''km''.<ref name="Sesiano2007"/>

In the examples below, we have divided the order 12 square into nine subsquares of order 4 filled each with eight smaller numbers and, in the corresponding bishop's cells (two cells diagonally across, including wrap arounds, in the 4×4 subsquare), their complements to ''n''<sup>2</sup> + 1 = 145. Each subsquare is pandiagonal with magic constant 290; while the whole square on the left is also pandiagonal with magic constant 870.

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

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

In another example below, we have divided the order 12 square into four order 6 squares. Each of the order 6 squares are filled with eighteen small numbers and their complements using bordering technique given by al-Antaki. If we remove the shaded borders of the order 6 subsquares and form an order 8 square, then this order 8 square is again a magic square. In its full generality, we can take any ''m''<sup>2</sup>/2 smaller numbers together with their complements to ''n''<sup>2</sup> + 1 to fill the subsquares, not necessarily in continuous sequence.

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

===Medjig-method for squares of even order 2''n'', where ''n'' > 2===
In this method a magic square is "multiplied" with a medjig square to create a larger magic square. The namesake of this method derives from mathematical game called medjig created by Willem Barink in 2006, although the method itself is much older.{{citation needed|date=September 2022}} An early instance of a magic square constructed using this method occurs in Yang Hui's text for order 6 magic square.{{citation needed|date=September 2022}} The [[LUX method]] to construct singly even magic squares is a special case of the medjig method, where only 3 out of 24 patterns are used to construct the medjig square.{{citation needed|date=September 2022}}

The pieces of the medjig puzzle are 2×2 squares on which the numbers 0, 1, 2 and 3 are placed. There are three basic patterns by which the numbers 0, 1, 2 and 3 can be placed in a 2×2 square, where 0 is at the top left corner:

{{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:4em;height:4em;table-layout:fixed;"
|-
| 0 || 1 
|-
| 2 || 3 
|}
{{col-break|valign=bottom|gap=3em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:4em;height:4em;table-layout:fixed;"
|-
| 0 || 1 
|-
| 3 || 2 
|}
{{col-break|valign=bottom|gap=3em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:4em;height:4em;table-layout:fixed;"
|-
| 0 || 2 
|-
| 3 || 1 
|}
{{col-end}}

Each pattern can be reflected and rotated to obtain 8 equivalent patterns, giving us a total of 3×8 = 24 patterns. The aim of the puzzle is to take ''n''<sup>2</sup> medjig pieces and arrange them in an ''n'' × ''n'' ''medjig square'' in such a way that each row, column, along with the two long diagonals, formed by the medjig square sums to 3''n'', the magic constant of the medjig square. An ''n'' × ''n'' medjig square can create a 2''n'' × 2''n'' magic square where ''n'' > 2.

Given an ''n''×''n'' medjig square and an ''n''×''n'' magic square base, a magic square of order 2''n''×2''n'' can be constructed as follows:
* Each cell of an ''n''×''n'' magic square is associated with a corresponding 2×2 subsquare of the medjig square
* Fill each 2×2 subsquares of the medjig square with the four numbers from 1 to 4''n''<sup>2</sup> that equal the original number modulo ''n''<sup>2</sup>, i.e. ''x''+''n''<sup>2</sup>''y'' where ''x'' is the corresponding number from the magic square and ''y'' is a number from 0 to 3 in the 2×2 subsquares.

Assuming that we have an initial magic square base, the challenge lies in constructing a medjig square. For reference, the sums of each medjig piece along the rows, columns and diagonals, denoted in italics, are:

{{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:8em;height:6em;table-layout:fixed;"
|-
| style="border-right:solid"| ''1'' || style="border-top:solid"| 0 || style="border-right:solid;border-top:solid"| 1 
|-
| style="border-right:solid"| ''5'' || 2 || style="border-right:solid"| 3 
|-
| ''3'' || style="border-top:solid"| ''2'' || style="border-top:solid"|''4'' || ''3'' 
|}
{{col-break|valign=bottom|gap=3em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:6em;table-layout:fixed;"
|-
| style="border-right:solid"| ''1'' || style="border-top:solid"| 0 || style="border-right:solid;border-top:solid"| 1 
|-
| style="border-right:solid"| ''5'' ||3 || style="border-right:solid"| 2 
|-
| ''4'' || style="border-top:solid"| ''3'' || style="border-top:solid"| ''3'' || ''2''
|}
{{col-break|valign=bottom|gap=3em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:6em;table-layout:fixed;"
|-
| style="border-right:solid"| ''2'' || style="border-top:solid"| 0 || style="border-right:solid;border-top:solid"| 2 
|-
| style="border-right:solid"| ''4'' || 3 || style="border-right:solid"| 1 
|-
| ''5'' || style="border-top:solid"| ''3'' || style="border-top:solid"| ''3'' || ''1''
|}
{{col-end}}

'''Doubly even squares''': The smallest even ordered medjig square is of order 2 with magic constant 6. While it is possible to construct a 2×2 medjig square, we cannot construct a 4×4 magic square from it since 2×2 magic squares required to "multiply" it does not exist. Nevertheless, it is worth constructing these 2×2 medjig squares. The magic constant 6 can be partitioned into two parts in three ways as 6 = 5 + 1 = 4 + 2 = 3 + 3. There exist 96 such 2×2 medjig squares.{{citation needed|date=September 2022}} In the examples below, each 2×2 medjig square is made by combining different orientations of a single medjig piece.

{{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:8em;height:8em;table-layout:fixed;"
|-
|+ Medjig 2×2
|-
| 0 || style="border-right:solid"|1 || 3 || 2
|-
| 2 || style="border-right:solid"|3 || 1 || 0
|-
| style="border-top:solid"|3 || style="border-top:solid;border-right:solid"|2 || style="border-top:solid"|0 || style="border-top:solid"|1 
|- 
| 1 || style="border-right:solid"|0 || 2 || 3
|}
{{col-break|valign=bottom|gap=3em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
|+ Medjig 2×2
|-
| 0 || style="border-right:solid"|1 || 2 || 3
|-
| 3 || style="border-right:solid"|2 || 1 || 0
|-
| style="border-top:solid"|1 || style="border-top:solid;border-right:solid"|0 || style="border-top:solid"|3 || style="border-top:solid"|2
|-
| 2 || style="border-right:solid"|3 || 0 || 1
|}
{{col-break|valign=bottom|gap=3em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
|+ Medjig 2×2
|-
| 0 || style="border-right:solid"|2 || 3 || 1 
|-
| 3 || style="border-right:solid"|1 || 0 || 2
|-
| style="border-top:solid"|0 || style="border-top:solid;border-right:solid"|2 || style="border-top:solid"|3 || style="border-top:solid"|1
|-
| 3 || style="border-right:solid"|1 || 0 || 2
|}
{{col-end}}

We can use the 2×2 medjig squares to construct larger even ordered medjig squares. One possible approach is to simply combine the 2×2 medjig squares together. Another possibility is to wrap a smaller medjig square core with a medjig border. The pieces of a 2×2 medjig square can form the corner pieces of the border. Yet another possibility is to append a row and a column to an odd ordered medjig square. An example of an 8×8 magic square is constructed below by combining four copies of the left most 2×2 medjig square given above:

{{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:8em;height:8em;table-layout:fixed;"
|-
|+ Order 4
|-
| 1 || 14 || 4 || 15
|-
| 8 || 11 || 5 || 10
|-
| 13 || 2 || 16 || 3
|-
| 12 || 7 || 9 || 6
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;"
|-
|+ Medjig 4 × 4
|-
| 0 || style="border-right:solid"| 1 || 3 || style="border-right:solid"|2 || 0 || style="border-right:solid"|1 || 3 || 2
|-
| 2 || style="border-right:solid"|3 || 1 || style="border-right:solid"|0 || 2 || style="border-right:solid"|3 || 1 || 0
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|3 || style="border-top:solid; border-right:solid"|2 || style="border-top:solid"|0 || style="border-top:solid"|1
|-
| 1 || style="border-right:solid"|0 || 2 || style="border-right:solid"|3 || 1 || style="border-right:solid"|0 ||2 || 3
|-
| style="border-top:solid"|0 || style="border-right:solid; border-top:solid "|1 || style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|0 || style="border-top:solid; border-right:solid"|1 || style="border-top:solid"|3 || style="border-top:solid"|2 
|-
| 2 || style="border-right:solid"|3 || 1 || style="border-right:solid"|0 || 2 || style="border-right:solid"|3 || 1 || 0
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid "|2 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|3 || style="border-top:solid; border-right:solid"|2 || style="border-top:solid"|0 || style="border-top:solid"|1
|-
| 1 || style="border-right:solid"|0 || 2 || style="border-right:solid"|3 || 1 || style="border-right:solid"|0 || 2 || 3
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;"
|-
|+ Order 8
|-
| '''1''' || style="border-right:solid"| 17 || 62 || style="border-right:solid"| 46 || '''4''' || style="border-right:solid"|20 || 63 || 47
|-
| 33 || style="border-right:solid"| 49 || 30 || style="border-right:solid"| '''14''' || 36 || style="border-right:solid"|52 || 31 || '''15'''
|-
| style="border-top:solid"| 56 || style="border-right:solid; border-top:solid"| 40 || style="border-top:solid"| '''11''' || style="border-right:solid; border-top:solid"| 27 || style="border-top:solid"| 53 || style="border-top:solid; border-right:solid"| 37 || style="border-top:solid"|'''10''' || style="border-top:solid"|26
|-
| 24 || style="border-right:solid"| '''8''' || 43 || style="border-right:solid"| 59 || 21 || style="border-right:solid"|'''5''' || 42 || 58
|-
| style="border-top:solid"| '''13''' || style="border-right:solid; border-top:solid "| 29 || style="border-top:solid"| 50 || style="border-right:solid; border-top:solid"| 34 || style="border-top:solid"| '''16''' || style="border-top:solid; border-right:solid"| 32 || style="border-top:solid"|51 || style="border-top:solid"|35
|-
| 45 || style="border-right:solid"| 61 || 18 || style="border-right:solid"| '''2''' || 48 || style="border-right:solid"|64 || 19 || '''3'''
|-
| style="border-top:solid"| 60 || style="border-right:solid; border-top:solid "| 44 || style="border-top:solid"| '''7''' || style="border-right:solid; border-top:solid"| 23 || style="border-top:solid"| 57 || style="border-top:solid; border-right:solid"| 41 || style="border-top:solid"| '''6''' || style="border-top:solid"|22
|-
| 28 || style="border-right:solid"| '''12''' || 39 || style="border-right:solid"| 55 || 25 || style="border-right:solid"|'''9''' || 38 || 54
|}
{{col-end}}

The next example is constructed by bordering a 2×2 medjig square core.

{{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:8em;height:8em;table-layout:fixed;"
|-
|+ Order 4
|-
| 1 || 14 || 4 || 15
|-
| 8 || 11 || 5 || 10
|-
| 13 || 2 || 16 || 3
|-
| 12 || 7 || 9 || 6
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;"
|-
|+ Medjig 4 × 4
|-
| 0 || style="border-right:solid"| 1 || 0 || style="border-right:solid"|1 || 2 || style="border-right:solid"|3 || 3 || 2
|-
| 2 || style="border-right:solid"|3 || 3 || style="border-right:solid"|2 || 1 || style="border-right:solid"|0 || 1 || 0
|-
| style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|3 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|3 || style="border-top:solid; border-right:solid"|1 || style="border-top:solid"|0 || style="border-top:solid"|3
|-
| 1 || style="border-right:solid"|2 || 3 || style="border-right:solid"|1 || 0 || style="border-right:solid"|2 ||1 || 2
|-
| style="border-top:solid"|2 || style="border-right:solid; border-top:solid "|1 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|3 || style="border-top:solid; border-right:solid"|1 || style="border-top:solid"|2 || style="border-top:solid"|1 
|-
| 3 || style="border-right:solid"|0 || 3 || style="border-right:solid"|1 || 0 || style="border-right:solid"|2 || 3 || 0
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid "|2 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|2 || style="border-top:solid; border-right:solid"|3 || style="border-top:solid"|0 || style="border-top:solid"|1
|-
| 1 || style="border-right:solid"|0 || 3 || style="border-right:solid"|2 || 1 || style="border-right:solid"|0 || 2 || 3
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;"
|-
|+ Order 8
|-
| '''1''' || style="border-right:solid"| 17 || '''14''' || style="border-right:solid"| 30 || 36 || style="border-right:solid"|52 || 63 || 47
|-
| 33 || style="border-right:solid"| 49 || 62 || style="border-right:solid"| 46 || 20 || style="border-right:solid"|'''4''' || 31 || '''15'''
|-
| style="border-top:solid"| '''8''' || style="border-right:solid; border-top:solid"| 56 || style="border-top:solid"| '''11''' || style="border-right:solid; border-top:solid"| 43 || style="border-top:solid"| 53 || style="border-top:solid; border-right:solid"| 21 || style="border-top:solid"|'''10''' || style="border-top:solid"| 58
|-
| 24 || style="border-right:solid"| 40 || 59 || style="border-right:solid"| 27 || '''5''' || style="border-right:solid"|37 || 26 || 42
|-
| style="border-top:solid"| 45 || style="border-right:solid; border-top:solid "| 29 || style="border-top:solid"| '''2''' || style="border-right:solid; border-top:solid"| 34 || style="border-top:solid"| 64 || style="border-top:solid; border-right:solid"| 32 || style="border-top:solid"| 35 || style="border-top:solid"|19
|-
| 61 || style="border-right:solid"| '''13''' || 50 || style="border-right:solid"| 18 || '''16''' || style="border-right:solid"|48 || 51 || '''3'''
|-
| style="border-top:solid"| 60 || style="border-right:solid; border-top:solid "| 44 || style="border-top:solid"| '''7''' || style="border-right:solid; border-top:solid"| 23 || style="border-top:solid"| 41 || style="border-top:solid; border-right:solid"| 57 || style="border-top:solid"| '''6''' || style="border-top:solid"|22
|-
| 28 || style="border-right:solid"| '''12''' || 55 || style="border-right:solid"| 39 || 25 || style="border-right:solid"|'''9''' || 38 || 54
|}
{{col-end}}

'''Singly even squares''': Medjig square of order 1 does not exist. As such, the smallest odd ordered medjig square is of order 3, with magic constant 9. There are only 7 ways of partitioning the integer 9, our magic constant, into three parts.{{citation needed|date=September 2022}} If these three parts correspond to three of the medjig pieces in a row, column or diagonal, then the relevant partitions for us are:

: 9 = 1 + 3 + 5 = 1 + 4 + 4 = 2 + 3 + 4 = 2 + 2 + 5 = 3 + 3 + 3.

A 3×3 medjig square can be constructed with some trial-and-error, as in the left most square below. Another approach is to add a row and a column to a 2×2 medjig square. In the middle square below, a left column and bottom row has been added, creating an L-shaped medjig border, to a 2×2 medjig square given previously. The right most square below is essentially same as the middle square, except that the row and column has been added in the middle to form a cross while the pieces of 2×2 medjig square are placed at the corners.

{{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:12em;height:12em;table-layout:fixed;"
|-
|+ Medjig 3 × 3
|-
| 2 || style="border-right:solid"| 3 || 0 || style="border-right:solid"|2 || 0 || 2
|-
| 1 || style="border-right:solid"|0 || 3 || style="border-right:solid"|1 || 3 || 1
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|1 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|2 || style="border-top:solid"|0
|-
| 0 || style="border-right:solid"|2 || 0 || style="border-right:solid"|3 || 3 || 1
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid "|2 || style="border-top:solid"|2 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|0 || style="border-top:solid"|2
|-
| 0 || style="border-right:solid"|1 || 3 || style="border-right:solid"|1 || 1 || 3
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;"
|-
|+ Medjig 3 × 3
|-
| 0 || style="border-right:solid"| 3 || 0 || style="border-right:solid"|1 || 3 || 2
|-
| 2 || style="border-right:solid"|1 || 2 || style="border-right:solid"|3 || 1 || 0
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|0 || style="border-top:solid"|1
|-
| 2 || style="border-right:solid"|1 || 1 || style="border-right:solid"|0 || 2 || 3
|-
| style="border-top:solid"|0 || style="border-right:solid; border-top:solid "|1 || style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|3 || style="border-top:solid"|1
|-
| 2 || style="border-right:solid"|3 || 0 || style="border-right:solid"|2 || 0 || 2
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;"
|-
|+ Medjig 3 × 3
|-
| 0 || style="border-right:solid"| 1 || 0 || style="border-right:solid"|3 || 3 || 2
|-
| 2 || style="border-right:solid"|3 || 2 || style="border-right:solid"|1 || 1 || 0
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|3 || style="border-top:solid"|1
|-
| 0 || style="border-right:solid"|2 || 2 || style="border-right:solid"|3 || 0 || 2
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid "|2 || style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|0 || style="border-top:solid"|1
|-
| 1 || style="border-right:solid"|0 || 2 || style="border-right:solid"|1 || 2 || 3
|}
{{col-end}}

Once a 3×3 medjig square has been constructed, it can be converted into a 6×6 magic square. For example, using the left most 3×3 medjig square given above:

{{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
|-
| 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:12em;height:12em;table-layout:fixed;"
|-
|+ Medjig 3 × 3
|-
| 2 || style="border-right:solid"| 3 || 0 || style="border-right:solid"|2 || 0 || 2
|-
| 1 || style="border-right:solid"|0 || 3 || style="border-right:solid"|1 || 3 || 1
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|1 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|2 || style="border-top:solid"|0
|-
| 0 || style="border-right:solid"|2 || 0 || style="border-right:solid"|3 || 3 || 1
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid "|2 || style="border-top:solid"|2 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|0 || style="border-top:solid"|2
|-
| 0 || style="border-right:solid"|1 || 3 || style="border-right:solid"|1 || 1 || 3
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;"
|-
|+ Order 6
|-
| 26 || style="border-right:solid"| 35 || '''1''' || style="border-right:solid"| 19 || '''6''' || 24
|-
| 17 || style="border-right:solid"| '''8''' || 28 || style="border-right:solid"| 10 || 33 || 15
|-
| style="border-top:solid"| 30 || style="border-right:solid; border-top:solid"| 12 || style="border-top:solid"| 14 || style="border-right:solid; border-top:solid"| 23 || style="border-top:solid"| 25 || style="border-top:solid"| '''7'''
|-
| '''3''' || style="border-right:solid"| 21 || '''5''' || style="border-right:solid"| 32 || 34 || 16
|-
| style="border-top:solid"| 31 || style="border-right:solid; border-top:solid "| 22 || style="border-top:solid"| 27 || style="border-right:solid; border-top:solid"| '''9''' || style="border-top:solid"| '''2''' || style="border-top:solid"| 20
|-
| '''4''' || style="border-right:solid"| 13 || 36 || style="border-right:solid"| 18 || 11 || 29
|}
{{col-end}}

There are 1,740,800 such 3×3 medjig squares.<ref>http://budshaw.ca/2xNComposite.html, 2N Composite Squares, S. Harry White, 2009</ref> An easy approach to construct higher order odd medjig square is by wrapping a smaller odd ordered medjig square with a medjig border, just as with even ordered medjig squares. Another approach is to append a row and a column to an even ordered medjig square. Approaches such as the LUX method can also be used. In the example below, a 5×5 medjig square is created by wrapping a medjig border around a 3×3 medjig square given previously:

{{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:10em;height:10em;table-layout:fixed;"
|-
|+ Order 5
|-
| 17 || 24 || 1 || 8 || 15
|-
| 23 || 5 || 7 || 14 || 16
|-
| 4 || 6 || 13 || 20 || 22
|-
| 10 || 12 || 19 || 21 || 3
|-
| 11 || 18 || 25 || 2 || 9 
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:20em;height:20em;table-layout:fixed;"
|-
|+ Medjig 5 × 5
|-
| 0 || style="border-right:solid"| 1 || 3 || style="border-right:solid"|1 || 0 || style="border-right:solid"|1 || 3 || style="border-right:solid"|1 || 3 || 2
|-
| 2 || style="border-right:solid"|3 || 0 || style="border-right:solid"|2 || 2 || style="border-right:solid"|3 || 0 || style="border-right:solid"|2 || 1 || 0
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|2 || style="border-right:solid; border-top:solid"|3 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|1 || style="border-top:solid"|2
|-
| 1 || style="border-right:solid"|2 || 1 || style="border-right:solid"|0 || 3 || style="border-right:solid"|1 || 3 || style="border-right:solid"|1 || 3 || 0
|-
| style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|1 || style="border-top:solid"|1 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|2 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|1 || style="border-top:solid"|3
|-
| 1 || style="border-right:solid"|3 || 0 || style="border-right:solid"|2 || 0 || style="border-right:solid"|3 || 3 || style="border-right:solid"|1 || 0 || 2 
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|2 || style="border-right:solid; border-top:solid"|0 || style="border-top:solid"|0 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|1 || style="border-top:solid"|2
|-
| 1 || style="border-right:solid"|2 || 0 || style="border-right:solid"|1 || 3 || style="border-right:solid"|1 || 1 || style="border-right:solid"|3 || 3 || 0
|-
| style="border-top:solid"|3 || style="border-right:solid; border-top:solid"|2 || style="border-top:solid"|1 || style="border-right:solid; border-top:solid"|3 || style="border-top:solid"|1 || style="border-right:solid; border-top:solid "|0 || style="border-top:solid"|1 || style="border-right:solid; border-top:solid"|3 || style="border-top:solid"|0 || style="border-top:solid"|1
|-
| 1 || style="border-right:solid"|0 || 2 || style="border-right:solid"|0 || 3 || style="border-right:solid"|2 || 2 || style="border-right:solid"|0 || 2 || 3 
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:20em;height:20em;table-layout:fixed;"
|-
|+ Order 10
|-
| '''17''' || style="border-right:solid"| 42 || 99 || style="border-right:solid"| 49 || '''1''' || style="border-right:solid"|26 || 83 || style="border-right:solid"|33 || 90 || 65
|-
| 67 || style="border-right:solid"| 92 || '''24''' || style="border-right:solid"| 74 || 51 || style="border-right:solid"|76 || '''8''' || style="border-right:solid"|58 || 40 || '''15'''
|-
| style="border-top:solid"| 98 || style="border-right:solid; border-top:solid"| '''23''' || style="border-top:solid"| 55 || style="border-right:solid; border-top:solid"| 80 || style="border-top:solid"| '''7''' || style="border-right:solid; border-top:solid"| 57 || style="border-top:solid"| '''14''' || style="border-right:solid; border-top:solid"| 64 || style="border-top:solid"| 41 || style="border-top:solid"| 66
|-
| 48 || style="border-right:solid"| 73 || 30 || style="border-right:solid"| '''5''' || 82 || style="border-right:solid"| 32 || 89 || style="border-right:solid"| 39 || 91 || '''16'''
|-
| style="border-top:solid"| '''4''' || style="border-right:solid; border-top:solid "| 54 || style="border-top:solid"| 81 || style="border-right:solid; border-top:solid"| 31 || style="border-top:solid"| 38 || style="border-right:solid; border-top:solid"| 63 || style="border-top:solid"| 70 || style="border-right:solid; border-top:solid"| '''20''' || style="border-top:solid"| 47 || style="border-top:solid"| 97
|-
| 29 || style="border-right:solid"| 79 || '''6''' || style="border-right:solid"| 56 || '''13''' || style="border-right:solid"| 88 || 95 || style="border-right:solid"| 45 || '''22''' || 72
|-
| style="border-top:solid"| 85 || style="border-right:solid; border-top:solid"| '''10''' || style="border-top:solid"| 87 || style="border-right:solid; border-top:solid"| 62 || style="border-top:solid"| 69 || style="border-right:solid; border-top:solid"| '''19''' || style="border-top:solid"| '''21''' || style="border-right:solid; border-top:solid"| 71 || style="border-top:solid"| 28 || style="border-top:solid"| 53
|-
| 35 || style="border-right:solid"| 60 || '''12''' || style="border-right:solid"| 37 || 94 || style="border-right:solid"| 44 || 46 || style="border-right:solid"| 96 || 78 || '''3'''
|-
| style="border-top:solid"| 86 || style="border-right:solid; border-top:solid "| 61 || style="border-top:solid"| 43 || style="border-right:solid; border-top:solid"| 93 || style="border-top:solid"| 100 || style="border-right:solid; border-top:solid"| '''25''' || style="border-top:solid"| 27 || style="border-right:solid; border-top:solid"| 77 || style="border-top:solid"| '''9''' || style="border-top:solid"| 34
|-
| 36 || style="border-right:solid"| '''11''' || 68 || style="border-right:solid"| '''18''' || 95 || style="border-right:solid"| 75 || 52|| style="border-right:solid"| '''2''' || 59 || 84
|}
{{col-end}}