Editing Magic square (section)

===Chinese===
[[File:Suanfatongzong-790-790.jpg|thumb|right|220px|A page displaying 9×9 magic square from Cheng Dawei's ''Suanfa tongzong'' (1593).]]

While ancient references to the pattern of even and odd numbers in the 3×3 magic square appear in the ''[[I Ching]]'', the first unequivocal instance of this magic square appears in the chapter called ''Mingtang'' (Bright Hall) of a 1st-century book ''Da Dai Liji'' (Record of Rites by the Elder Dai), which purported to describe ancient Chinese rites of the Zhou dynasty.<ref name="Yoke">{{cite book |last=Yoke |first=Ho Peng | series=Encyopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures | date=2008 | edition=2 | pages=1252–1259| publisher=Springer |doi=10.1007/978-1-4020-4425-0_9350 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=978-1-4020-4559-2 |chapter=Magic Squares in China }}</ref> 
<ref name="Andrews122">{{cite book |last=Andrews |first=William Symes |title=Magic Squares and Cubes |publisher=Open Court Publishing Company| date=1917| edition=2nd| page=122| url=https://archive.org/details/MagicSquaresAndCubes_754}}</ref><ref name="Cammann">{{cite journal| last = Cammann | first= Schuyler| title=The Evolution of Magic Squares in China | journal=Journal of the American Oriental Society | volume = 80 | issue = 2 | pages= 116–124| date=April 1960 | url= http://www.chinesehsc.org/downloads/cammann/camman_the_evolution_of_magic_squares_in_china.pdf| doi= 10.2307/595587| jstor= 595587}}</ref><ref name="Swetz2008"/> These numbers also occur in a possibly earlier mathematical text called ''Shushu jiyi'' (Memoir on Some Traditions of Mathematical Art), said to be written in 190 BCE. This is the earliest appearance of a magic square on record; and it was mainly used for divination and astrology.<ref name="Yoke"/> The 3×3 magic square was referred to as the "Nine Halls" by earlier Chinese mathematicians.<ref name="Cammann"/> The identification of the 3×3 magic square to the legendary Luoshu chart was only made in the 12th century, after which it was referred to as the Luoshu square.<ref name="Yoke"/><ref name="Cammann"/> The oldest surviving Chinese treatise that displays magic squares of order larger than 3 is [[Yang Hui]]'s ''Xugu zheqi suanfa'' (Continuation of Ancient Mathematical Methods for Elucidating the Strange) written in 1275.<ref name="Yoke"/><ref name="Cammann"/> The contents of Yang Hui's treatise were collected from older works, both native and foreign; and he only explains the construction of third and fourth-order magic squares, while merely passing on the finished diagrams of larger squares.<ref name="Cammann"/> He gives a magic square of order 3, two squares for each order of 4 to 8, one of order nine, and one semi-magic square of order 10. He also gives six magic circles of varying complexity.<ref name="Connor">{{cite web |url= http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Yang_Hui.html|title= Yang Hui|last1= O'Connor|first1= J.J. | last2 = Robertson| first2 = E.F. |website=MacTutor History of Mathematics Archive |access-date= 15 March 2018}}</ref>

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

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

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

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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;"
|-
| style="background-color: silver;"|13 || style="background-color: silver;"|22 || 18 || 27 || style="background-color: silver;"|11 || style="background-color: silver;"|20
|-
| style="background-color: silver;"|31 || style="background-color: silver;"|'''4''' || 36 || '''9''' || style="background-color: silver;"|29 || style="background-color: silver;"|'''2'''
|-
| 12 || 21 || style="background-color: silver;"|14 || style="background-color: silver;"|23 || 16 || 25 
|-
| 30 || '''3''' || style="background-color: silver;"|'''5''' || style="background-color: silver;"|32 || 34 || '''7'''
|-
| style="background-color: silver;"|17 || style="background-color: silver;"|26 || 10 || 19 || style="background-color: silver;"|15 || style="background-color: silver;"|24 
|-
| style="background-color: silver;"|'''8''' || style="background-color: silver;"|35 || 28 || '''1''' || style="background-color: silver;"|'''6''' || style="background-color: silver;"|33
|}
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:14em;height:14em;table-layout:fixed;"
|-
| style="background-color: silver;"|46 || style="background-color: silver;"|8 || style="background-color: silver;"|16 || style="background-color: silver;"|20 || style="background-color: silver;"|29 || style="background-color: silver;"|7 || style="background-color: silver;"|49
|-
| style="background-color: silver;"|3 || 40 || 35 || 36 || 18 || 41 || style="background-color: silver;"|2
|-
| style="background-color: silver;"|44 || 12 || style="background-color: silver;"|33 || style="background-color: silver;"|23 || style="background-color: silver;"|19 || 38 || style="background-color: silver;"|6 
|-
| style="background-color: silver;"|28 || 26 || style="background-color: silver;"|11 || style="background-color: silver;"|25 || style="background-color: silver;"|39 || 24 || style="background-color: silver;"|22
|-
| style="background-color: silver;"|5 || 37 || style="background-color: silver;"|31 || style="background-color: silver;"|27 || style="background-color: silver;"|17 || 13 || style="background-color: silver;"|45 
|-
| style="background-color: silver;"|48 || 9 || 15 || 14 || 32 || 10 || style="background-color: silver;"|47 
|-
| style="background-color: silver;"|1 || style="background-color: silver;"|43 || style="background-color: silver;"|34 || style="background-color: silver;"|30 || style="background-color: silver;"|21 || style="background-color: silver;"|42 || style="background-color: silver;"|4
|}

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

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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:18em;table-layout:fixed;"
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| style="background-color: silver;"|31 || style="background-color: silver;"|76 || style="background-color: silver;"|13 || 36 || 81 || 18 || style="background-color: silver;"|29 || style="background-color: silver;"|74 || style="background-color: silver;"|11
|-
| style="background-color: silver;"|22 || style="background-color: silver;"|40 || style="background-color: silver;"|58 || 27 || 45 || 63 || style="background-color: silver;"|20 || style="background-color: silver;"|38 || style="background-color: silver;"|56
|-
| style="background-color: silver;"|67 || style="background-color: silver;"|'''4''' || style="background-color: silver;"|49 || 72 || '''9''' || 54 || style="background-color: silver;"|65 || style="background-color: silver;"|'''2''' || style="background-color: silver;"|47
|-
| 30 || 75 || 12 || style="background-color: silver;"|32 || style="background-color: silver;"|77 || style="background-color: silver;"|14 || 34 || 79 || 16
|-
| 21 || 39 || 57 || style="background-color: silver;"|23 || style="background-color: silver;"|41 || style="background-color: silver;"|59 || 25 || 43 || 61
|-
| 66 || '''3''' || 48 || style="background-color: silver;"|68 || style="background-color: silver;"|'''5''' || style="background-color: silver;"|50 || 70 || '''7''' || 52
|-
| style="background-color: silver;"|35 || style="background-color: silver;"|80 || style="background-color: silver;"|17 || 28 || 73 || 10 || style="background-color: silver;"|33 || style="background-color: silver;"|78 || style="background-color: silver;"|15
|-
| style="background-color: silver;"|26 || style="background-color: silver;"|44 || style="background-color: silver;"|62 || 19 || 37 || 55 || style="background-color: silver;"|24 || style="background-color: silver;"|42 || style="background-color: silver;"|60
|-
| style="background-color: silver;"|71 || style="background-color: silver;"|'''8''' || style="background-color: silver;"|53 || 64 || '''1''' || 46 || style="background-color: silver;"|69 || style="background-color: silver;"|'''6''' || style="background-color: silver;"|51
|}
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The above magic squares of orders 3 to 9 are taken from Yang Hui's treatise, in which the Luo Shu principle is clearly evident.<ref name="Cammann"/><ref name="Swetz2008"/> The order 5 square is a bordered magic square, with central 3×3 square formed according to Luo Shu principle. The order 9 square is a composite magic square, in which the nine 3×3 sub squares are also magic.<ref name="Cammann"/> After Yang Hui, magic squares frequently occur in Chinese mathematics such as in Ding Yidong's ''Dayan suoyin'' ({{circa|1300}}), [[Cheng Dawei]]'s ''[[Suanfa tongzong]]'' (1593), Fang Zhongtong's ''Shuduyan'' (1661) which contains magic circles, cubes and spheres, Zhang Chao's ''Xinzhai zazu'' ({{circa|1650}}), who published China's first magic square of order ten, and lastly Bao Qishou's ''Binaishanfang ji'' ({{circa|1880}}), who gave various three dimensional magic configurations.<ref name="Yoke"/><ref name="Swetz2008"/> However, despite being the first to discover the magic squares and getting a head start by several centuries, the Chinese development of the magic squares are much inferior compared to the Indian, Middle Eastern, or European developments. The high point of Chinese mathematics that deals with the magic squares seems to be contained in the work of Yang Hui; but even as a collection of older methods, this work is much more primitive, lacking general methods for constructing magic squares of any order, compared to a similar collection written around the same time by the Byzantine scholar [[Manuel Moschopoulos]].<ref name="Cammann"/> This is possibly because of the Chinese scholars' enthralment with the Lo Shu principle, which they tried to adapt to solve higher squares; and after Yang Hui and the fall of [[Yuan dynasty]], their systematic purging of the foreign influences in Chinese mathematics.<ref name="Cammann"/>