Editing Magic square (section)

===A method for constructing a magic square of odd order===
{{See also|Siamese method}}
[[File:Yanghui magic square.GIF|thumb|right|300px|[[Yang Hui]]'s construction method]]
A method for constructing magic squares of odd order was published by the French diplomat de la Loubère in his book, ''A new historical relation of the kingdom of Siam'' (Du Royaume de Siam, 1693), in the chapter entitled ''The problem of the magical square according to the Indians''.<ref>''Mathematical Circles Squared'' By Phillip E. Johnson, Howard Whitley Eves, p. 22</ref> The method operates as follows:

The method prescribes starting in the central column of the first row with the number 1. After that, the fundamental movement for filling the squares is diagonally up and right, one step at a time. If a square is filled with a multiple of the order ''n'', one moves vertically down one square instead, then continues as before. When an "up and to the right" move would leave the square, it is wrapped around to the last row or first column, respectively.

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

Starting from other squares rather than the central column of the first row is possible, but then only the row and column sums will be identical and result in a magic sum, whereas the diagonal sums will differ. The result will thus be a semimagic square and not a true magic square. Moving in directions other than north east can also result in magic squares.

{{col-begin|width=auto;margin:0.5em auto}}
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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: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:18em;height:18em;table-layout:fixed;"
|-
|+ Order 9
|-
| 47 || 58 || 69 || 80 || 1 || 12 || 23 || 34 || 45
|-
| 57 || 68 || 79 || 9 || 11 || 22 || 33 || 44 || 46
|-
| 67 || 78 || 8 || 10 || 21 || 32 || 43 || 54 || 56
|-
| 77 || 7 || 18 || 20 || 31 || 42 || 53 || 55 || 66
|-
| 6 || 17 || 19 || 30 || 41 || 52 || 63 || 65 || 76
|-
| 16 || 27 || 29 || 40 || 51 || 62 || 64 || 75 || 5
|-
| 26 || 28 || 39 || 50 || 61 || 72 || 74 || 4 || 15
|-
| 36 || 38 || 49 || 60 || 71 || 73 || 3 || 14 || 25
|-
| 37 || 48 || 59 || 70 || 81 || 2 || 13 || 24 || 35
|}
{{col-end}}