Editing Magic square (section)

===India===
[[File:Hindi Manuscript 317, folio 2b Wellcome L0024035.jpg|thumb|right|220px|The 3×3 magic square in different orientations forming a non-normal 6×6 magic square, from an unidentified 19th century Indian manuscript.]]
The 3×3 magic square first appears in India in ''Gargasamhita'' by Garga, who recommends its use to pacify the nine planets (''navagraha''). The oldest version of this text dates from 100 CE, but the passage on planets could not have been written earlier than 400 CE. The first dateable instance of 3×3 magic square in India occur in a medical text ''Siddhayog'' ({{Circa|900 CE}}) by Vrnda, which was prescribed to women in labor in order to have easy delivery.<ref name="Hayashi"/>

The oldest dateable fourth order magic square in the world is found in an encyclopaedic work written by [[Varāhamihira|Varahamihira]] around 587 CE called ''Brhat Samhita''. The magic square is constructed for the purpose of making perfumes using 4 substances selected from 16 different substances. Each cell of the square represents a particular ingredient, while the number in the cell represents the proportion of the associated ingredient, such that the mixture of any four combination of ingredients along the columns, rows, diagonals, and so on, gives the total volume of the mixture to be 18. Although the book is mostly about divination, the magic square is given as a matter of combinatorial design, and no magical properties are attributed to it. The special features of this magic square were commented on by Bhattotpala ({{Circa|966 CE}})<ref name="Datta"/><ref name="Hayashi">{{cite book |last=Hayashi |first=Takao | date=2008 | edition=2 | pages=1252–1259| publisher=Springer |doi=10.1007/978-1-4020-4425-0_9778 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=978-1-4020-4559-2 |chapter=Magic Squares in Indian Mathematics }}</ref>

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

The square of Varahamihira as given above has sum of 18. Here the numbers 1 to 8 appear twice in the square. It is a [[pandiagonal magic square|pan-diagonal magic square]]. Four different magic squares can be obtained by adding 8 to one of the two sets of 1 to 8 sequence. The sequence is selected such that the number 8 is added exactly twice in each row, each column and each of the main diagonals. One of the possible magic squares shown in the right side. This magic square is remarkable in that it is a 90 degree rotation of a magic square that appears in the 13th century Islamic world as one of the most popular magic squares.<ref name="Hayashi1">{{cite journal |last=Hayashi |first=Takao |title=Varahamihira's Pandiagonal Magic Square of the Order Four | journal=Historia Mathematica | date=1987 | volume=14 | 
 issue=2 | pages=159–166| url=https://core.ac.uk/download/pdf/82500954.pdf |doi=10.1016/0315-0860(87)90019-X |doi-access=free }}</ref>

The construction of 4th-order magic square is detailed in a work titled ''Kaksaputa'', composed by the alchemist [[Nagarjuna (metallurgist)|Nagarjuna]] around 10th century CE. All of the squares given by Nagarjuna are 4×4 magic squares, and one of them is called ''Nagarjuniya'' after him. Nagarjuna gave a method of constructing 4×4 magic square using a primary skeleton square, given an odd or even magic sum.<ref name="Datta">{{cite journal |last1=Datta |first1=Bibhutibhusan |last2=Singh |first2=Awadhesh Narayan |title=Magic Squares in India |journal=Indian Journal of History of Science |date=1992 |volume=27 |issue=1 |pages=51–120 |url=http://124.108.19.235:12000/jspui/bitstream/123456789/11663/1/Vol27_1_5_BDatta.pdf |access-date=2018-01-16 |archive-url=https://web.archive.org/web/20180117131236/http://124.108.19.235:12000/jspui/bitstream/123456789/11663/1/Vol27_1_5_BDatta.pdf |archive-date=2018-01-17 |url-status=dead }}</ref> The Nagarjuniya square is given below, and has the sum total of 100.

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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
| 30 || 16 || 18 || 36
|-
| 10 || 44 || 22 || 24
|-
| 32 || 14 || 20 || 34
|-
| 28 || 26 || 40 || 6
|}
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{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
| '''7''' || 1 || '''4''' || 6
|-
| '''2''' || 8 || '''5''' || 3
|-
| 5 || '''3''' || 2 || '''8'''
|-
| 4 || '''6''' || 7 || '''1'''
|}
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The Nagarjuniya square is a [[pandiagonal magic square|pan-diagonal magic square]]. The Nagarjuniya square is made up of two arithmetic progressions starting from 6 and 16 with eight terms each, with a common difference between successive terms as 4. When these two progressions are reduced to the normal progression of 1 to 8, the adjacent square is obtained.

Around 12th-century, a 4×4 magic square was inscribed on the wall of [[Parshvanatha temple, Khajuraho|Parshvanath]] temple in [[Khajuraho]], India. Several Jain hymns teach how to make magic squares, although they are undateable.<ref name="Hayashi"/>

As far as is known, the first systematic study of magic squares in India was conducted by [[Thakkar Pheru]], a Jain scholar, in his ''Ganitasara Kaumudi'' (c. 1315). This work contains a small section on magic squares which consists of nine verses. Here he gives a square of order four, and alludes to its rearrangement; classifies magic squares into three (odd, evenly even, and oddly even) according to its order; gives a square of order six; and prescribes one method each for constructing even and odd squares. For the even squares, Pheru divides the square into component squares of order four, and puts the numbers into cells according to the pattern of a standard square of order four. For odd squares, Pheru gives the method using [[Knight (chess)|horse move or knight's move]]. Although algorithmically different, it gives the same square as the De la Loubere's method.<ref name="Hayashi"/>

The next comprehensive work on magic squares was taken up by [[Narayana Pandit]], who in the fourteenth chapter of his ''Ganita Kaumudi'' (1356) gives general methods for their construction, along with the principles governing such constructions. It consists of 55 verses for rules and 17 verses for examples. Narayana gives a method to construct all the pan-magic squares of fourth order using knight's move; enumerates the number of pan-diagonal magic squares of order four, 384, including every variation made by rotation and reflection; three general methods for squares having any order and constant sum when a standard square of the same order is known; two methods each for constructing evenly even, oddly even, and of squares when the sum is given. While Narayana describes one older method for each species of square, he claims the method of [[Superposition principle|superposition]] for evenly even and odd squares and a method of interchange for oddly even squares to be his own invention. The superposition method was later re-discovered by [[De la Hire]] in Europe. In the last section, he conceives of other figures, such as circles, rectangles, and hexagons, in which the numbers may be arranged to possess properties similar to those of magic squares.<ref name="Datta"/><ref name="Hayashi"/> Below are some of the magic squares constructed by Narayana:<ref name="Datta"/>

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

The order 8 square is interesting in itself since it is an instance of the most-perfect magic square. Incidentally, Narayana states that the purpose of studying magic squares is to construct ''yantra'', to destroy the ego of bad mathematicians, and for the pleasure of good mathematicians. The subject of magic squares is referred to as ''bhadraganita'' and Narayana states that it was first taught to men by god [[Shiva]].<ref name="Hayashi"/>