imported>SilkPyjamas: Adding short description: "Mathematical process for constructing magic squares"

2025-03-07T05:08:51Z

Adding short description: "Mathematical process for constructing magic squares"

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{{Short description|Mathematical process for constructing magic squares}}
[[File:SiameseMethod.gif|thumb|307px|A simple example of the Siamese method. Starting with "1", boxes are filled diagonally up and right (↗). When a move would leave the square, it is wrapped around to the last row or first column, respectively. If a filled box is encountered, one moves vertically down one box (↓) instead, then continuing as before.]]
The '''Siamese method''', or '''De la Loubère method''', is a simple method to construct any size of ''n''-odd [[magic squares]] (i.e. number squares in which the sums of all rows, columns and diagonals are identical). The method was brought to [[France]] in 1688 by the French [[mathematician]] and [[diplomat]] [[Simon de la Loubère]],<ref>{{cite book |title=Number Story: From Counting to Cryptography |url=https://archive.org/details/numberstoryfromc00higg_612 |url-access=registration |last=Higgins |first=Peter |year=2008 |publisher=Copernicus |location=New York |isbn=978-1-84800-000-1 |page=[https://archive.org/details/numberstoryfromc00higg_612/page/n63 54] }} footnote 8</ref> as he was returning from his 1687 embassy to the kingdom of [[Siam]].<ref>''Mathematical Circles Squared'' By Phillip E. Johnson, Howard Whitley Eves, p.22</ref><ref>''CRC Concise Encyclopedia of Mathematics'' By Eric W. Weisstein, Page 1839 [https://books.google.com/books?id=uzIekOtnD2gC&pg=PA1839&dq=Loubere+Siam+magic+square&sig=0hwmj7fcQqclTG9uVgwY5fL5bgQ]</ref><ref>''The Zen of Magic Squares, Circles, and Stars'' By Clifford A. Pickover Page 38 [https://books.google.com/books?id=slpObYbvV4gC&pg=PA38&dq=Loubere+Siam+magic+square&sig=to5a7r-pEhNFV9iYe85lSSqtJq4]</ref> The Siamese method makes the creation of [[magic square]]s straightforward.

==Publication==
[[File:Siamese Square.jpg|thumb|A description of the Siamese method in [[Simon de la Loubère]]'s 1693 ''A new historical relation of the kingdom of Siam''.]]
De la Loubère published his findings in his book ''A new historical relation of the kingdom of Siam'' (''Du Royaume de Siam'', 1693), under the chapter entitled ''The problem of the magical square according to the Indians''.<ref name=p228>[http://dlxs.library.cornell.edu/cgi/t/text/pageviewer-idx?c=sea;idno=sea130;view=image;seq=258 ''A new historical relation of the kingdom of Siam'' p.228]</ref>
Although the method is generally qualified as "Siamese", which refers to de la Loubère's travel to the country of Siam, de la Loubère himself learnt it from a Frenchman named M.Vincent (a doctor, who had first travelled to [[Persia]] and then to [[Siam]], and was returning to France with the de la Loubère embassy), who himself had learnt it in the city of [[Surat]] in [[India]]:<ref name=p228/>

{{blockquote|"Mr. Vincent, whom I have so often mentioned in my ''Relations'', seeing me one day in the ship, during our return, studiously to range the Magical Squares after the method of ''[[Claude Gaspard Bachet de Méziriac|Bachet]]'', informed me that the ''Indians'' of ''[[Surat]]te'' ranged them with much more facility, and taught me their method for the unequal squares only, having, he said, forgot that of the equal"|Simon de la Loubère, ''A new historical relation of the kingdom of Siam''.<ref name=p228/>}}

==The method==
The method was surprising in its effectiveness and simplicity:

{{blockquote|"I hope that it will not be unacceptable that I give the rules and the demonstration of this method, which is surprising for its extreme facility to execute a thing, which has appeared difficult to our Mathematicians"|Simon de la Loubère, ''A new historical relation of the kingdom of Siam''.<ref name=p228 />}}

First, an [[arithmetic progression]] has to be chosen (such as the simple progression 1,2,3,4,5,6,7,8,9 for a square with three rows and columns (the [[Lo Shu square]])).

Then, starting from the central box of the first row with the number 1 (or the first number of any arithmetic progression), the fundamental movement for filling the boxes is diagonally '''up and right''' ('''↗'''), one step at a time. When a move would leave the square, it is wrapped around to the last row or first column, respectively.

If a filled box is encountered, one moves vertically '''down one box''' ('''↓''') instead, then continuing as before.

===Order-3 magic squares===

{{col-begin|width=auto; margin:0.5em auto}}
{{col-break|valign=bottom}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
! colspan="3" | step 1
|-
| || 1 ||
|-
| ||. ||
|-
| ||. ||
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
! colspan="3" | step 2
|-
| || 1 ||
|-
| || . ||
|-
| || || 2
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
! colspan="3" | step 3
|-
| || 1 ||
|-
| 3 || ||
|-
| || || 2
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
! colspan="3" | step 4
|-
| || 1 ||
|-
| 3 || ||
|-
| 4 || || 2
|}
{{col-end}}

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

===Order-5 magic squares===

{{col-begin|width=auto; margin:0.5em auto}}
{{col-break|valign=bottom}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Step 1
|-
| || || 1 || ||
|-
| || || . || ||
|-
| || || . || ||
|-
| || || . || ||
|-
| || || . || ||
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Step 2
|-
| || || 1 || ||
|-
| || || . || ||
|-
| || || . || ||
|-
| || || . || || 3
|-
| || || . || 2 ||
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Step 3
|-
| || || 1 || ||
|-
| || 5 || || ||
|-
| 4 || || . || ||
|-
| || || || || 3
|-
| || || || 2 ||
|}
{{col-end}}

{{col-begin|width=auto; margin:0.5em auto}}
{{col-break|valign=bottom}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Step 4
|-
| || || 1 || 8 ||
|-
| || 5 || 7 || ||
|-
| 4 || 6 || . || ||
|-
| || || || || 3
|-
| || || || 2 ||
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Step 5
|-
| || || 1 || 8 || 15
|-
| || 5 || 7 || 14 ||
|-
| 4 || 6 || 13 || ||
|-
| 10 || 12 || || || 3
|-
| 11 || || || 2 || 9
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Step 6
|-
| 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-end}}

===Other sizes===
Any ''n''-odd square ("[[Parity (mathematics)|odd]]-order square") can be thus built into a magic square. The Siamese method does not work however for n-even squares ("[[Parity (mathematics)|even]]-order squares", such as 2 rows/ 2 columns, 4 rows/ 4 columns etc...).

{{col-begin|width=auto; margin:0.5em auto}}
{{col-break|valign=bottom}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
! colspan="3" | Order 3
|-
| 8 || 1 || 6
|-
| 3 || 5 || 7
|-
| 4 || 9 || 2
|}
{{col-break|valign=bottom|gap=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | 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=2em}}
{| class="wikitable" style="margin:0.5em auto;text-align:center;width:18em;height:18em;table-layout:fixed;"
! colspan="9" | 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}}

===Other values===
Any sequence of numbers can be used, provided they form an [[arithmetic progression]] (i.e. the difference of any two successive members of the sequence is a constant). Also, any starting number is possible. For example the following sequence can be used to form an order 3 magic square according to the Siamese method (9 boxes): 5, 10, 15, 20, 25, 30, 35, 40, 45 (the magic sum gives 75, for all rows, columns and diagonals). The magic sum in these cases will be the sum of the arithmetic progression used divided by the order of the magic square.

{| class="wikitable" style="margin:0.5em auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
! colspan="3" | Order 3
|-
| 40 || 5 || 30
|-
| 15 || 25 || 35
|-
| 20 || 45 || 10
|}

===Other starting points===
It is possible not to start the arithmetic progression from the middle of the top row, 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 not be a true magic square:

{| class="wikitable" style="margin:0.5em auto;text-align:center;width:9em;height:9em;table-layout:fixed;"
! colspan="3" | Order 3
|-
| 500 || 700 || 300
|-
| 900 || 200 || 400
|-
| '''100''' || 600 || 800
|}

===Rotations and reflections===
Numerous other magic squares can be deduced from the above by simple [[rotations]] and [[Reflection (mathematics)|reflections]].

==Variations==
A slightly more complicated variation of this method exists in which the first number is placed in the box just above the center box. The fundamental movement for filling the boxes remains '''up and right''' ('''↗'''), one step at a time. However, if a filled box is encountered, one moves vertically '''up two boxes''' instead, then continuing as before.

{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Order 5
|-
| 23 || 6 || 19 || 2 || 15
|-
| 10 || 18 || '''1''' || 14 || 22
|-
| 17 || 5 || 13 || 21 || 9
|-
| 4 || 12 || 25 || 8 || 16
|-
| 11 || 24 || 7 || 20 || 3
|}

Numerous variants can be obtained by simple rotations and reflections. The next square is equivalent to the above (a simple reflexion): the first number is placed in the box just below the center box. The fundamental movement for filling the boxes then becomes diagonally '''down and right''' ('''↘'''), one step at a time. If a filled box is encountered, one moves vertically '''down two boxes''' instead, then continuing as before.<ref name=p229>[http://dlxs.library.cornell.edu/cgi/t/text/pageviewer-idx?c=sea;idno=sea130;view=image;seq=259 ''A new historical relation of the kingdom of Siam'' p229]</ref>

{| class="wikitable" style="margin:0.5em auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | Order 5
|-
| 11 || 24 || 7 || 20 || 3
|-
| 4 || 12 || 25 || 8 || 16
|-
| 17 || 5 || 13 || 21 || 9
|-
| 10 || 18 || '''1''' || 14 || 22
|-
| 23 || 6 || 19 || 2 || 15
|}

These variations, although not quite as simple as the basic Siamese method, are equivalent to the methods developed by earlier Arab and European scholars, such as [[Manuel Moschopoulos]] (1315), [[Johann Faulhaber]] (1580–1635) and [[Claude Gaspard Bachet de Méziriac]] (1581–1638), and allowed to create magic squares similar to theirs.<ref name=p229/><ref>''The Zen of Magic Squares, Circles, and Stars'' by Clifford A. Pickover, 2002 p.37 [https://books.google.com/books?id=QxMdE_HlrQsC&pg=PA45&dq=Faulhaber+magic+squares&sig=ACfU3U16MpAfumGuHO8FEC39p4WkN_z3wQ#PPA37,M1]</ref>

==See also==
*[[Conway's LUX method for magic squares]]
*[[Strachey method for magic squares]]

==Notes and references==
{{reflist}}

{{DEFAULTSORT:Siamese Method}}
[[Category:Magic squares]]
[[Category:Search algorithms]]

Siamese method - Revision history

imported>SilkPyjamas: Adding short description: "Mathematical process for constructing magic squares"