Difference between revisions 2800544 and 3474266 on mswiki

{{proses|BukanTeamBiasa}}
Dalam [[matematik rekreasi]], sebuah '''segi empat sama ajaib''' pada aturan ''n'' adalah suatu urutan bilangan ''n''², biasanya [[integer]] berlainan, dalam sebuah [[segi empat sama]], seperti mana yang bilangan ''n'' dalam semua barisan, semua column, dan kedua jumlah diagonal ke konstan sama.<ref>"[http://demonstrations.wolfram.com/MagicSquare/ Magic Square]" by Onkar Singh, [[The Wolfram Demonstrations Project]].</ref> Sebuah segiempat sama ajaib '''biasa''' mengandungi integer dari 1 ke ''n''². Istilah "segiempat sama ajaib" juga kadang-kadang digunakan untuk merujukkan pada mana-mana jenis pelbagai [[segi empat sama kata]].

Segiempat sama ajaib bermuncul untuk semua aturan ''n'' ≥ 1 kecuali ''n'' = 2, walaupun kesnya ''n'' = 1 adalah trivial&mdash;ia mengandungi suatu sel tunggal yang mengandungi nombor 1. Kes bukan-trivial terkecil, ditunjuk di bawah, adalah aturan 3.
<center>[[Image:Magicsquareexample.svg]]</center>
Jumlah konstant dalam setiap row, column dan diagonal digelar [[konstan ajaib]] atau jumlah ajaib, ''M''. Konstan ajaib pada segiempat sama ajaib terpulang hanya pada ''n'' dan mempunyai nilai
:<math>M(n) = \frac{n^3+n}{2}.</math>

Untuk segiempat sama ajaib biasa pada aturan ''n'' = 3, 4,&nbsp;5, …, konstant ajaibnya adalah:
:15, 34, 65, 111, 175, 260, … (urutan [[OEIS:A006003|A006003]] pada [[On-Line Encyclopedia of Integer Sequences|OEIS]]).

== Sejarah segiempat sama ajaib ==
=== Segiempat sama Lo Shu (3&times;3 segiempat sama) ===
[[Sasatera China]] melatar belakang seawal [[650 SM]] menceritakan lagenda [[Lo Shu]] atau "scroll dari sungai Lo".<ref name="Swaney"/> Di [[China silam]], ada suatu banjir yang besar. Rakyatnya cuba untuk memberikan pengorbanan ke dewa sungai pada salah satu sungai banjir, sungai Lo, untuk menyenangkan kemarahannya. Kemudian, di situ bermuncul dari seekor [[kura-kura]] dengan suatu angka/corak pada kulitnya; ada titik-titik bulat bilangan yang diatur dalam suatu corak petak sembilan tiga seperti mana jumlah bilangan dalam setiap baris, lajur dan diagonal yang sama; 15. Nombor ini juga sama dengan bilangan hari setiap 24 kitaran [[tahun China|tahun matahari China]]. Corak ini, dalam sesetengah cara, telah digunakan oleh orang-orang yang mengawal sungai itu. 

{| 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 
|}

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.

The Square of Lo Shu is also referred to as the Magic Square of [[Saturn]] or [[Cronos]]. Its numerical value is obtained from the workings of the [[I Ching]] when the [[Trigram]]s are placed in an order given in the first river map, the [[Ho Tu]] or [[Yellow River]]. The Ho Tu produces 4 squares of [[Hexagram]]s 8 x 8 in its outer values of 1 to 6, 2 to 7, 3 to 8, and 4 to 9, and these outer squares can then be symmetrically added together to give an inner central square of 5 to 10. The central values of the Ho Tu are those of the Lo Shu (so they work together), since in the total value of 15 x 2 (light and dark) is found the number of years in the cycle of [[equinoctial precession]] (12,960 x 2 = 25,920).
The Ho Tu produces a total of 40 light and 40 dark numbers called the days and nights (the alternations of light and dark), and a total of 8 x 8 x 8 Hexagrams whose opposite symmetrical addition equals 8640, therefore each value of a square is called a season as it equals 2160. 8640 is the number of hours in a 360-day year, and 2160 years equals an [[aeon]] (12 aeons = 25,920 yrs).

To validate the values contained in the 2 river maps (Ho Tu and Lo Shu) the [[I Ching]] provides numbers of Heaven and Earth that are the 'Original Trigrams' (father and mother) from 1 to 10. Heaven or a Trigram with all unbroken lines (light lines - [[yin and yang|yang]]) have odd numbers 1,3,5,7,9, and Earth a Trigram with all broken lines have even numbers 2,4,6,8,10. If each of the Trigram's lines is given a value by multiplying the numbers of Heaven and Earth, then the value of each line in Heaven 1 would be 1 + 2 + 3 = 6, and its partner in the Ho Tu of Earth 6 would be 6 + 12 + 18 = 36, these 2 'Original Trigrams' thereby produce 6 more Trigrams (or children in all their combinations) -- and when the sequences of Trigrams are placed at right angles to each other they produce an 8 x 8 square of Hexagrams (or cubes) that each have 6 lines of values. From this simple point the complex structure of the maths evolves as a hexadecimal progression, and it is the hexagon that is the link to the turtle or tortoise shell. In Chinese texts of the I Ching the moon is symbolic of water (darkness) whose transformations or changes create the light or fire - the dark value 6 creates the light when its number is increased by 1.
This same principle can be found in ancient calendars such as the [[Egyptian calendar|Egyptian]], as the 360 day year of 8640 hrs was divided by 72 to produce the 5 extra days or 120 hours on which the gods were born. It takes 72 years for the heavens to move 1 degree through its Precession.

=== Kepentingan kebudayaan pada segiempat sama ajaib ===
Magic squares have fascinated humanity throughout the ages, and have been around for over 4,000 years. They are found in a number of cultures, including [[Egypt]] and India, engraved on stone or metal and worn as [[amulet|talisman]]s, the belief being that magic squares had [[astrology|astrological]] and divinatory qualities, their usage ensuring longevity and prevention of diseases.

The Kubera-Kolam is a floor painting used in India which is in the form of a magic square of order three. It is essentially the same as the Lo Shu Square, but with 19 added to each number, giving a magic constant of 72.

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
|-
| 23 || 28 || 21 
|-
| 22 || 24 || 26 
|-
| 27 || 20 || 25
 
|}

===Arabia===
Segiempat sama ajaib dikenali pada [[ahli matematik Islam|ahli matematik Arab]], mungkin seawal abad ke-7, apabila orang [[Arab]] berhubung dengan budaya India atau Asia Selatan, dan mempelajari matematik dan astronomi India, termasuk aspek-aspek [[matematik berkombinasi]]. Ia juga telah dicadangkan bahawa gagasan tiba melalui China. Segiempat sama ajaib pertama pada aturan 5 dan 6 bermuncul di sebuah ensiklopedia dari [[Baghdad]] ''sekitar'' 983 M, [[Rasa'il Ihkwan al-Safa]] (Ensiklopedia Brethern of Purity); segiempat sama ajaib yang lebih mudah dikenali pada beberapa ahli matematik awal Arab.<ref name="Swaney">Swaney, Mark. [http://www.arthurmag.com/magpie/?p=449 History of Magic Squares].</ref>

Ahli matematik Arab [[Ahmad al-Buni]], yang bekerja pada segiempat sama ajaib sewaktu 1200 M, menganggap ciri-ciri mistikal pada mereka, walaupun tiada rinci pada ciri-ciri sepatutnya ini dikenali. Ada juga rujukan pada kegunaan segiempat sama ajaib pada pengiraan astrologi, suatu amalan yang kelihatan berasal dari orang Arab.<ref name="Swaney"/>

===India===

Segiempat sama ajaib 3x3 telah digunakan sebagai sebahagian dari upacara di India sejak zaman vedic, dan berlanjut untuk digunakan untuk sampai ia tidak layak digunakan lagi. Suatu segiempat sama ajaib yang terkenal di India dapat dilihat di [[Khajuraho]] di kuil [[Jain]] [[Parshvanath]]. Ia bermula dari abad ke-10 <ref>Magic Squares and Cubes By William Symes Andrews, 1908, Open court publish company</ref>.

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

Ini dirujukkan sebagai Chautisa Yantra, sejak setiap sub-segi empat sama berjumlah ke 34.

===Eropah===
Pada 1300, membangun pada karya Arab [[Al-Buni]], sarjana Greek Byzantine [[Manuel Moschopoulos]] menulis suatu perjanjian matematik pada tajuk segiempat sama ajaib, mengetepikan mistikisme pewaris terdahulunya.<ref>[http://mtcs.truman.edu/~thammond/history/ManuelMoschopoulos.html Manuel Moschopoulos - Mathematics and the Liberal Arts]</ref> Moschopoulos is thought to be the first Westerner to have written on the subject. In the 1450s the Italian [[Luca Pacioli]] studied magic squares and collected a large number of examples.<ref name="Swaney"/>

Pada sekitar 1510 [[Heinrich Cornelius Agrippa]] menulis ''De Occulta Philosophia'', menulis pada [[Hermetikisms|Hermetik]] dan karya [[Ajaib (paranormal)|ajaib]] [[Marsilio Ficino]] dan [[Pico della Mirandola]], dan dalam ia dia expounded pada magical virtues aturan 3 ke 9, setiapnya berkaitan dengan salam satu planet [[astrologi]]. Buku ini mempunyai pengaruh yang sangat besar sepanjang Eropah sehingga [[counter-reformation]], dan segiempat sama ajaib Agrippa, kadang-kadang digelar [[Kamea]], berlanjut digunakan dalam ajaib majlis moden dalam cara yang hampir sama dengan yang diprajelasnya terdahulunya.<ref name="Swaney"/><ref name="DruryDict">{{cite book |last=Drury |first=Nevill |authorlink=Nevill Drury |title=Dictionary of Mysticism and the Esoteric Traditions |year=1992 |location=Bridport, Dorset |publisher=Prism Press |id=ISBN 1-85327-075-X}}</ref>

<center>
<table>
 <tr valign="bottom">
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
! colspan="3" | [[Zuhal (astrologi)|Zuhal]]=15
|-
| 4 || 9 || 2 
|-
| 3 || 5 || 7 
|-
| 8 || 1 || 6 
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
! colspan="4" | [[Musytari (astrologi)|Musytari]]=34
|-
| 4 || 14 || 15 || 1
|-
| 9 || 7 || 6 || 12
|-
| 5 || 11 || 10 || 8
|-
| 16 || 2 || 3 || 13
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
! colspan="5" | [[Marikh (astrologi)|Marikh]]=65
|-
| 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
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;"
! colspan="6" | [[Matahari (astrologi)|Sol]]=111
|-
| 6 || 32 || 3 || 34 || 35 || 1
|-
| 7 || 11 || 27 || 28 || 8 || 30
|-
| 19 || 14 || 16 || 15 || 23 || 24
|-
| 18 || 20 || 22 || 21 || 17 || 13
|-
| 25 || 29 || 10 || 9 || 26 || 12
|-
| 36 || 5 || 33 || 4 || 2 || 31
|}
</td>
 </tr>
</table>

<table>
 <tr valign="bottom">
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:14em;height:14em;table-layout:fixed;"
! colspan="7" | [[Venus (astrology)|Venus]]=175
|-
| 22 || 47 || 16 || 41 || 10 || 35 || 4
|-
| 5 || 23 || 48 || 17 || 42 || 11 || 29
|-
| 30 || 6 || 24 || 49 || 18 || 36 || 12
|-
| 13 || 31 || 7 || 25 || 43 || 19 || 37
|-
| 38 || 14 || 32 || 1 || 26 || 44 || 20
|-
| 21 || 39 || 8 || 33 || 2 || 27 || 45
|-
| 46 || 15 || 40 || 9 || 34 || 3 || 28
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;"
! colspan="8" | [[Utarid (astrologi)|Utarid]]=260
|-
| 8 || 58 || 59 || 5 || 4 || 62 || 63 || 1
|-
| 49 || 15 || 14 || 52 || 53 || 11 || 10 || 56
|-
| 41 || 23 || 22 || 44 || 45 || 19 || 18 || 48
|-
| 32 || 34 || 35 || 29 || 28 || 38 || 39 || 25
|-
| 40 || 26 || 27 || 37 || 36 || 30 || 31 || 33
|-
| 17 || 47 || 46 || 20 || 21 || 43 || 42 || 24
|-
| 9 || 55 || 54 || 12 || 13 || 51 || 50 || 16
|-
| 64 || 2 || 3 || 61 || 60 || 6 || 7 || 57
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:18em;table-layout:fixed;"
! colspan="9" | [[Bulan (astrologi)|Luna]]=369
|-
| 37 || 78 || 29 || 70 || 21 || 62 || 13 || 54 || 5
|-
| 6 || 38 || 79 || 30 || 71 || 22 || 63 || 14 || 46
|-
| 47 || 7 || 39 || 80 || 31 || 72 || 23 || 55 || 15
|-
| 16 || 48 || 8 || 40 || 81 || 32 || 64 || 24 || 56
|-
| 57 || 17 || 49 || 9 || 41 || 73 || 33 || 65 || 25
|-
| 26 || 58 || 18 || 50 || 1 || 42 || 74 || 34 || 66
|-
| 67 || 27 || 59 || 10 || 51 || 2 || 43 || 75 || 35
|-
| 36 || 68 || 19 || 60 || 11 || 52 || 3 || 44 || 76
|-
| 77 || 28 || 69 || 20 || 61 || 12 || 53 || 4 || 45
|}
</td>
 </tr>
</table>
</center>
[[Image:Hagiel sigil derivation.svg|thumb|Asalnya [[Sigil (ajaib)|sigil]] dari Hagiel, [[planetary intelligence]] [[Venus (astrologi)|Venus]], diletakkan pada segiempat sama ajaib Venus. Setiap huruf [[Hebrew]] memberikan suatu nilai angka, memberikan verteks-verteks sigil.]]
Kegunaan yang terumum untuk Kamea-Kamea ini adalah untuk memberikan suatu corak yang mana untuk membinakan [[Sigil (ajaib)|sigil]] pada [[mambang]], [[malaikat]] atau [[jin]]; huruf-huruf nama entitinya ditukarkan ke bilangan, dan barisannya dikesan balik melalui corak yang bilangan berlanjutan ini membuat pada kamea. 
Dalam suatu konteks ajaib, istilah  ''segiempat sama ajaib'' juga digunakan pada pelbagai [[segiempat sama kata]] atau segiempat sama nombor yang didapati pada [[grimoire]]s, termasuk sesetengah yang tidak mengikut mana-mana corak yang terbuka, dan walaupun dengan yang berlainan nombor pada barisan dan column. They are generally intended for use as talismans. For instance the following squares are: The [[Sator Arepo Tenet Opera Rotas|Sator square]], one of the most famous magic squares found in a number of grimoires including the ''[[Key of Solomon]]''; a square "to overcome envy", from ''The Book of Power'';<ref>"The Book of Power: Cabbalistic Secrets of Master Aptolcater, Mage of Adrianople", transl. 1724. In {{cite book |last=Shah |first=Idries |authorlink=Idries Shah |date=1957 |title=The Secret Lore of Magic |location=London |publisher=Frederick Muller Ltd}}</ref> and two squares from ''[[The Book of the Sacred Magic of Abramelin the Mage]]'', the first to cause the illusion of a superb palace to appear, and the second to be worn on the head of a child during an angelic [[invocation]]:

<center>
<table>
 <tr valign="bottom">
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
| S || A || T || O || R
|-
| A || R || E || P || O
|-
| T || E || N || E || T
|-
| O || P || E || R || A
|-
| R || O || T || A || S
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
| 6 || 66 || 848 || 938
|-
| 8 || 11 || 544 || 839
|-
| 1 || 11 || 383 || 839
|-
| 2 || 73 || 774 || 447
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
| H || E || S || E || B
|-
| E || Q || A || L ||
|-
| S ||   ||   ||   ||
|-
| E ||   || G ||   ||
|-
| B ||   ||   ||   ||
|}
</td>
 <td style="padding: 0 1em">
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:10em;table-layout:fixed;"
| A || D || A || M
|-
| D || A || R || A
|-
| A || R || A || D
|-
| M || A || D || A
|-
| H || O || M || O<!-- Please don't remove this row. It is indeed correct. The whole point of including this "square" is that it is an example of the many odd squares in magical literature that don't follow an obvious mathematical pattern. "Homo" is Latin for "man".-->
|}
</td>
 </tr>
</table>
</center>

=== Albrecht Dürer's magic square ===

[[Image:Albrecht Dürer - Melencolia I (detail).jpg|thumb|Detail of ''Melencolia I'']]

The order-4 magic square in [[Albrecht Dürer]]'s engraving ''[[Melencolia I]]'' is believed to be the first seen in European art. It is very similar to [[Yang Hui]]'s square, which was created in China about 250 years before Dürer's time. The sum 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, the corner squares, 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]] [http://www.muljadi.org/MagicSquares.htm]), the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14) and the sum of the middle two entries of the two outer columns and rows (e.g. 5+9+8+12), as well as several kite-shaped quartets, e.g. 3+5+11+15; the two numbers in the middle of the bottom row give the date of the engraving: [[1514]].

{| 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 
|}

=== The Sagrada Família magic square ===
[[Image:Ms sf 2.jpg|right|thumb|280px|A magic square on the Sagrada Família church façade.]]

The Passion façade of the [[Sagrada Família]] church in [[Barcelona]], designed by sculptor [[Josep Subirachs]], features a 4×4 magic square:

The magic constant of the square is 33, the age of [[Jesus]] at the time of the [[Passion (Christianity)|Passion]]. Structurally, it is very similar to the 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 
|}

While having the same pattern of summation, this is not a ''normal'' magic square as above, as two numbers (10 and 14) are duplicated and two (12 and 16) are absent, failing the 1→n² rule.


== Lihat juga ==
<div style="-moz-column-count:3; column-count:3;">
* [[Arithmetic sequence]]
* [[Antimagic square]]
* [[Bimagic square]]
* [[Eight queens puzzle]]
* [[Heterosquare]]
* [[Latin square]]
* [[Multimagic square]] (also known as a '''Satanic square''')
* [[Magic series]]
* [[Magic star]]
* [[Most-perfect magic square]]
* [[Panmagic square]] (also known as a '''Diabolic square''')
* [[Prime reciprocal magic square]]
* [[Sator Arepo Tenet Opera Rotas]]
* [[Trimagic square]]
* [[Unsolved problems in mathematics]]
* [[Sudoku]]
* [[Word square]]
* [[Yang Hui]]
* [[Magic cube]]
* [[Magic cube classes]]
* [[Magic tesseract]]
* [[Magic hypercube]]
* [[Magic hypercubes]]
* [[Matrix (mathematics)]]
* [[Nasik magic hypercube]]
* [[John R. Hendricks]]
* [[Vedic square]]
</div>

== Nota ==
<references/>

== Rujukan ==
<div class="references-small">
*{{MathWorld|urlname=MagicSquare|title=Magic Square}}
*[http://web.archive.org/web/20070625162103/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=528&bodyId=784 Magic Squares] at [http://web.archive.org/web/20060212072618/http://mathdl.maa.org/convergence/1/ Convergence]
{{Wikisource1911Enc|Magic Square}}
* W. S. Andrews, ''Magic Squares and Cubes''. (New York: Dover, 1960), originally printed in 1917
* John Lee Fults, ''Magic Squares''. (La Salle, Illinois: Open Court, 1974).
* [[Cliff Pickover]], ''The Zen of Magic Squares, Circles, and Stars'' (Princeton, New Jersey: Princeton University Press)
* Leonhard Euler, ''On magic squares''  ( [http://arxiv.org/pdf/math/0408230 pdf] )
* Mark Farrar, ''Magic Squares'' ( [http://www.MagicSquaresBook.com/] )
* Asker Ali Abiyev, ''The Natural Code of Numbered Magic Squares (1996)'', ( http://www1.gantep.edu.tr/~abiyev/abiyeving.htm )
* [http://cboyer.club.fr/multimagie/English/BensonDickinson.htm William H. Benson] and [[Oswald Jacoby]], "New Recreations with Magic Squares". (New York: Dover, 1976).
* [http://www.doermann.com/square/index.html A 'perfect' magic square ] presented as a magic trick (Online Generator - Magic Square 4x4 using Javascript)
* [http://www.faust.fr.bw.schule.de/mhb/backtrack/mag4en.htm Magic Squares of Order 4,5,6, and some theory]
* [http://www.jethroma.com/portfolio.php Magic Square Program using genetic algorithms] at [http://www.jethroma.com Jethro Ma - Nanotechnology Engineering]
* [http://www.magic-square-museum.com/ Magic Square Museum]: the first Second Life museum about Magic Square. Vulcano (89,35,25)
</div>

== Bacaan lanjut ==
* Charney, Noah ''The Art Thief'' Atria (2007), a novel with a key plot point involving a magic square.
* {{cite journal
 | author = McCranie, Judson
 | year = 1988
 | month = November
 | title = '''Magic Squares of All Orders'''
 | journal = Mathematics Teacher
 | volume = 
 | issue = 
 | pages = 674–78
 | url = 
 | format = 
 | accessdate = 
 }}
* {{cite journal
 | author = King, J. R.
 | year = 1963
 | month = 
 | title = '''Magic Square Numbers'''
 | journal =
 | volume = 
 | issue = 
 | pages = 
 | url = 
 | format = 
 | accessdate = 
 }}

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