Editing Magic square (section)

==Some famous magic squares==
[[File:Magic square Lo Shu.png|thumb|right|220px|Lo Shu from "The Astronomical Phenomena" (''Tien Yuan Fa Wei''). Compiled by Bao Yunlong in 13th century, published during the [[Ming dynasty]], 1457–1463.]]

===Luo Shu magic square===
{{main|Lo Shu Square}}
Legends dating from as early as 650 BCE tell the story of the [[Lo Shu]] (洛書) or "scroll of the river Lo".<ref name="Swetz2008">{{cite book| first = Frank J. | last = Swetz| title= The Legacy of the Luoshu | publisher = A.K. Peters/CRC Press | date=2008 | edition= 2nd}}</ref> According to the legend, there was at one time in [[History of China#Ancient China|ancient China]] a huge flood. While the [[Yu the Great|great king Yu]] was trying to channel the water out to sea, a [[turtle]] emerged from it with a curious pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15. According to the legend, thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods{{Citation needed|date=September 2024}}. The [[Lo Shu Square]], as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.

===Magic square in Parshavnath temple===
[[File:Magic square at the Parshvanatha temple, Khajuraho.png|thumb|right|upright|220px|Magic Square at the [[Parshvanatha temple, Khajuraho|Parshvanatha temple]], in [[Khajuraho]], [[India]]]]
There is a well-known 12th-century 4×4 normal magic square inscribed on the wall of the [[Parshvanatha temple, Khajuraho|Parshvanath]] temple in [[Khajuraho]], India.<ref name="Datta"/><ref name="Hayashi"/><ref name="Andrews">{{cite book |last=Andrews |first=William Symes |title=Magic Squares and Cubes |publisher=Open Court Publishing Company| date=1917| edition=2nd| pages=124–126| url=https://archive.org/details/MagicSquaresAndCubes_754}}</ref>
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
| 7 || 12 || 1 ||14
|-
| 2 || 13 || 8 || 11
|-
| 16 || 3 || 10 || 5
|-
| 9 || 6 || 15 || 4
|}
This is known as the ''Chautisa Yantra'' (''Chautisa'', 34; ''[[wikt:hi:यन्त्र|Yantra]]'', lit. "device"), since its magic sum is 34. It is one of the three 4×4 [[pandiagonal magic square]]s and is also an instance of the [[most-perfect magic square]]. The study of this square led to the appreciation of pandiagonal squares by European mathematicians in the late 19th century. Pandiagonal squares were referred to as Nasik squares or Jain squares in older English literature.

===Albrecht Dürer's magic square===
[[Image:Albrecht Dürer - Melencolia I (detail).jpg|thumb|Detail of ''Melencolia I'']]
The order four normal magic square [[Albrecht Dürer]] immortalized in his 1514 engraving ''[[Melencolia I]]'', referred to above, is believed to be the first seen in European art. The square associated with Jupiter appears as a talisman used to drive away melancholy. It is very similar to [[Yang Hui]]'s square, which was created in China about 250 years before Dürer's time. As with every order 4 normal magic square, the magic sum is 34. But in the Durer square this sum is also found 
in each of the quadrants, in the center four squares, and in the corner squares (of the 4×4 as well as the four contained 3×3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four [[Queen (chess)|queens]] in the two solutions of the [[eight queens puzzle|4 queens puzzle]]<ref>{{cite web |url=http://www.muljadi.org/MagicSquares.htm |title=Virtual Home of Paul Muljadi |access-date=2005-03-18 |url-status=dead |archive-url=https://web.archive.org/web/20051109234521/http://www.muljadi.org/MagicSquares.htm |archive-date=2005-11-09 }}</ref>), the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14), the sum of the middle two entries of the two outer columns and rows (5+9+8+12 and 3+2+15+14), and in four kite or cross shaped quartets (3+5+11+15, 2+10+8+14, 3+9+7+15, and 2+6+12+14). The two numbers in the middle of the bottom row give the date of the engraving: 1514. It has been speculated that the numbers 4,1 bordering the publication date correspond to Durer's initials D,A.  But if that had been his intention, he could have inverted the order of columns 1 and 4  to achieve "A1514D" without compromising the square's properties. 
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
| 16 || 3 || 2 || 13
|-
| 5 || 10 || 11 || 8
|-
| 9 || 6 || 7 || 12
|-
| 4 || 15 || 14 || 1
|}
Dürer's magic square can also be extended to a magic cube.<ref>"[http://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-duerer-s-square Magic cube with Dürer's square]" Ali Skalli's magic squares and magic cubes</ref>

===Sagrada Família magic square===
[[File:Barcelona Sagrada Familia passion facade magic square.jpg|right|thumb|220px|A magic square on the Sagrada Família church façade]]

The Passion façade of the [[Sagrada Família]] church in [[Barcelona]], conceptualized by [[Antoni Gaudí]] and designed by sculptor [[Josep Maria Subirachs|Josep Subirachs]], features a trivial order 4 magic square: The magic constant of the square is 33, the age of [[Jesus]] at the time of the [[Passion (Christianity)|Passion]].<ref>{{cite web| title= The magic square on the Passion façade: keys to understanding it | date = 7 February 2018| url=https://blog.sagradafamilia.org/en/divulgation/the-magic-square-the-passion-facade-keys-to-understanding-it/}}</ref> Structurally, it is very similar to the [[Melencolia I|Melancholia magic square]], but it has had the numbers in four of the cells reduced by 1.
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
| 1 || 14 || 14 || 4
|-
| 11 || 7 || 6 || 9
|-
| 8 || 10 || 10 || 5
|-
| 13 || 2 || 3 || 15
|}

Trivial squares such as this one are not generally mathematically interesting and only have historical significance. Lee Sallows has pointed out that, due to Subirachs's ignorance of magic square theory, the renowned sculptor made a needless blunder, and supports this assertion by giving several examples of non-trivial 4{{resx}}4 magic squares showing the desired magic constant of 33.<ref>Letters: The Mathematical Intelligencer; 2003; 25; 4: pp. 6–7.</ref>

Similarly to Dürer's magic square, the Sagrada Familia's magic square can also be extended to a magic cube.<ref>{{Cite web |last=Skalli |first=Ali |date=14 October 2009 |title=Magic cube with Gaudi's square |url=https://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-gaudi-s-square |archive-url=https://web.archive.org/web/20211215212634/https://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-gaudi-s-square |archive-date=15 December 2021}}</ref>

===Parker square===
The '''Parker square''', named after recreational mathematician and maths YouTuber [[Matt Parker]],<ref>{{cite arXiv|eprint=1908.03236|quote=Some ’near misses’ have been found such as the Parker Square [2] |last1=Cain |first1=Onno |title=Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares |date=2019 |class=math.RA }}</ref> is an attempt to create a 3{{resx}}3 magic square of [[Square number|squares]] — a prized unsolved problem since [[Leonhard Euler|Euler]].<ref>{{cite web|last1=Boyer|first1=Christian|title=Latest research on the "3x3 magic square of squares" problem|url=http://www.multimagie.com/English/SquaresOfSquaresSearch.htm|website=multimagie|access-date=16 June 2019|quote=The two corresponding prizes are still to be won!}}</ref> Discovered in 2016, the Parker square is a trivial semimagic square, since it uses some numbers more than once, and the diagonal {{math|23{{sup|2}} + 37{{sup|2}} + 47{{sup|2}}}} sums to {{val|4107}}, not {{val|3051}} as for all the other rows and columns, and the other diagonal. A true 3{{resx}}3 magic square of square numbers has to date not been discovered (despite computer searches). In a ''[[Numberphile]]'' video from June 2023, Mathematician Tony Várilly-Alvarado used mathematics to speculate that the existence of such a square is "probably impossible".<ref>{{Cite AV media |url=https://www.youtube.com/watch?v=Kdsj84UdeYg |title=Magic Squares of Squares (are PROBABLY impossible) - Numberphile |date=2023-06-13 |last=Numberphile |access-date=2025-03-05 |via=YouTube}}</ref> In February 2025, Parker upped his years-old bounty of US$1,000 to $10,000 to find a fully magic 3{{resx}}3 square using square numbers.<ref>{{Cite AV media |url=https://www.youtube.com/watch?v=stpiBy6gWOA |title=A Magic Square Breakthrough - Numberphile |date=2025-02-09 |last=Numberphile |access-date=2025-03-05 |via=YouTube}}</ref>

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:8em;height:8em;"
|-
| 29{{sup|2}}|| 1{{sup|2}}|| 47{{sup|2}}
|-
| 41{{sup|2}}|| 37{{sup|2}}|| 1{{sup|2}}
|-
| 23{{sup|2}}|| 41{{sup|2}}|| 29{{sup|2}}
|}

===Gardner square===
The Gardner square, named after recreational mathematician [[Martin Gardner]], similar to the Parker square,
is given as a problem to determine a, b, c and d.{{Citation needed|date=January 2024}}

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:8em;height:8em;"
|-
| 127{{sup|2}}|| 46{{sup|2}}|| 58{{sup|2}}
|-
| 2{{sup|2}}|| b{{sup|2}}|| c{{sup|2}}
|-
| a{{sup|2}}|| 82{{sup|2}}|| d{{sup|2}}
|}

This solution for a = 74, b = 113, c = 94 and d = 97 gives a semimagic square; the diagonal {{math|127{{sup|2}} + b{{sup|2}} + d{{sup|2}}}} sums to {{val|38307}}, not {{val|21609}} as for all the other rows and columns, and the other diagonal.<ref>{{cite journal |last1=Gardner |first1=Martin |title=The magic of 3x3 |journal=Quantum |date=January 1996 |volume=6 |issue=3 |pages=24–26 |url=https://static.nsta.org/pdfs/QuantumV6N3.pdf |access-date=6 January 2024 |issn=1048-8820}}</ref><ref>{{cite journal |last1=Gardner |first1=Martin |title=The latest magic |journal=Quantum |date=March 1996 |volume=6 |issue=4 |page=60 |url=https://static.nsta.org/pdfs/QuantumV6N4.pdf |access-date=6 January 2024 |issn=1048-8820}}</ref><ref>{{cite journal |last1=Boyer |first1=Christian |title=Some Notes on the Magic Squares of Squares Problem |journal=The Mathematical Intelligencer |date=12 November 2008 |volume=27 |issue=2 |pages=52–64 |doi=10.1007/BF02985794}}</ref>
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:19em;height:8em;border-style:none;"
|-
| style="background-color: white; border-style: none;" | || 127{{sup|2}}||  46{{sup|2}}|| 58{{sup|2}} || 21609
|-
| style="background-color: white; border-style: none;" | || 2{{sup|2}}  || 113{{sup|2}}|| 94{{sup|2}} || 21609
|-
| style="background-color: white; border-style: none;" | || 74{{sup|2}} ||  82{{sup|2}}|| 97{{sup|2}} || 21609
|-
|           21609                    ||    21609    ||    21609    ||    21609    || 38307 
|}