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| In [[recreational mathematics]], a '''magic square''' is an arrangement of numbers (usually [[integer]]s) in a [[square (geometry)|square]] grid, where the numbers in each row, and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number. A magic square has the same number of rows as it has columns, and in conventional math notation, "''n''" stands for the number of rows (and columns) it has. Thus, a magic square always contains ''n''<sup>2</sup> numbers, and its size (the number of rows [and columns] it has) is described as being "of order ''n''".<ref>"[http://demonstrations.wolfram.com/MagicSquare/ Magic Square]" by Onkar Singh, [[Wolfram Demonstrations Project]].</ref> A magic square that contains the integers from 1 to ''n''<sup>2</sup> is called a ''normal'' magic square. (The term "magic square" is also sometimes used to refer to any of various types of [[word square]]s.)
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| It is possible to construct a normal magic square of any size except 2 × 2 (that is, where ''n'' = 2), although the solution to a magic square where ''n'' = 1 is trivial, since it consists simply of a single cell containing the number 1. The smallest nontrivial case, shown below, is a 3 × 3 grid (that is, a magic square of order 3).
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| <div class="center">[[Image:Magicsquareexample.svg]]</div>
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| The constant that is the sum of every row, column and diagonal is called the [[magic constant]] or magic sum, ''M''. Every normal magic square has a unique constant determined solely by the value of ''n'', which can be calculated using this formula:
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| :<math>M = \frac{n(n^2+1)}{2}.</math>
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| For example, if ''n'' = 3, the formula says ''M'' = [3 (3<sup>2</sup> + 1)]/2, which simplifies to 15. | |
| For normal magic squares of order ''n'' = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260. (See sequence [[OEIS:A006003|A006003]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]])
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| ==History==
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| [[File:Yuan dynasty iron magic square.jpg|thumb|right|220px|Iron plate with an order 6 magic square in [[Arabic numbers]] from China, dating to the [[Yuan Dynasty]] (1271–1368).]]
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| Magic squares were known to Chinese mathematicians as early as 650 BCE,<ref name="Swaney">Swaney, Mark. [http://web.archive.org/web/20040807015853/http://www.netmastersinc.com/secrets/magic_squares.htm].</ref> and to [[Islamic mathematics|Arab mathematicians]] possibly as early as the 7th century CE, when the [[Arab]]s conquered northwestern parts of the [[Indian subcontinent]] and learned [[Indian mathematics]] and [[Indian astronomy|astronomy]], including other aspects of [[combinatorial mathematics]].{{Citation needed|date=March 2009}} The first magic squares of order 5 and 6 appear in an encyclopedia from [[Baghdad]] ''circa'' 983 CE, the ''[[Encyclopedia of the Brethren of Purity]]'' (''Rasa'il Ihkwan al-Safa''); simpler magic squares were known to several earlier Arab mathematicians.<ref name="Swaney" /> Some of these squares were later used in conjunction with magic letters, as in ([[Shams Al-ma'arif]]), to assist Arab [[illusionists]] and magicians.<ref>The most famous Arabic book on magic, named "Shams Al-ma'arif ({{lang-ar|كتاب شمس المعارف}}), for [[Ahmed bin Ali Al-boni]], who died about 1225 (622 AH). Reprinted in [[Beirut]] in 1985</ref>
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| ===Lo Shu square (3×3 magic square)===
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| {{main|Lo Shu Square}}
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| [[Chinese literature]] dating from as early as 650 BCE tells the legend of [[Lo Shu]] or "scroll of the river Lo".<ref name="Swaney"/> According to the legend, there was at one time in [[History of China#Ancient China|ancient China]] a huge flood. While the great king Yu ({{lang|zh|{{linktext|禹}}}}) was trying to channel the water out to sea, a [[turtle]] emerged from it with a curious figure / 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, which is also the number of days in each of the 24 [[Solar term|cycles]] of the [[Chinese year|Chinese solar year]]. 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.
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
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| | 4 || 9 || 2
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| | 3 || 5 || 7
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| | 8 || 1 || 6
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| |}
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| 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.
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| The Square of Lo Shu is also referred to as the Magic Square of [[Saturn]].
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| ===Persia===
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| [[Image:Magic square 4x4 Shams Al-maarif2 (Arabic magics).jpg|thumb|widthpx|Original script from the ''[[Shams al-Ma'arif]]''.]]
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| [[Image:Magic square 4x4 Shams Al-maarif3 (Arabic magics).jpg|thumb|widthpx|Printed version of the previous manuscript. [[Eastern Arabic numerals]] were used.]]
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| Although the early history of magic squares in Persia is not known, it has been suggested that they were known in pre-Islamic times.<ref>J. P. Hogendijk, A. I. Sabra, ''The Enterprise of Science in Islam: New Perspectives'', Published by MIT Press, 2003, ISBN 0-262-19482-1, p. xv.</ref> It is clear, however, that the study of magic squares was common in [[Islamic Golden Age|medieval Islam]] in Persia, and it was thought to have begun after the introduction of [[chess]] into the region.<ref>[[Helaine Selin]], [[Ubiratan D'Ambrosio]], ''Mathematics Across Cultures: The History of Non-western Mathematics'', Published by Springer, 2001, ISBN 140200260, p. 160.</ref> The 10th-century Persian mathematician [[Buzjani]], for example, left a manuscript that on page 33 contains a series of magic squares, filled by numbers in arithmetic progression, in such a way that the sums of each row, column and diagonal are equal.<ref>Sesiano, J., ''Abūal-Wafā\rasp's treatise on magic squares'' (French), Z. Gesch. Arab.-Islam. Wiss. 12 (1998), 121–244.</ref>
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| ===Arabia===
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| Magic squares were known to [[Islamic mathematicians]] in Arabia as early as the 7th century CE. They may have learned about them when the [[Arab]]s came into contact with Indian culture and learned Indian astronomy and mathematics – including other aspects of [[combinatorial mathematics]]. Alternatively, the idea may have come to them from China. The first magic squares of order 5 and 6 known to have been devised by Arab mathematicians appear in an encyclopedia from [[Baghdad]] ''circa'' 983 AD, the [[Rasa'il Ikhwan al-Safa]] (the [[Encyclopedia of the Brethren of Purity]]); simpler magic squares were known to several earlier Arab mathematicians.<ref name="Swaney"/>
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| The Arab mathematician [[Ahmad al-Buni]], who worked on magic squares around 1250 CE, attributed mystical properties to them, although no details of these supposed properties are known. There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.<ref name="Swaney"/>
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| ===India===
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| The 3×3 magic square has been a part of rituals in India since Vedic times, and still is today. The Ganesh yantra is a 3×3 magic square. There is a well-known 10th-century 4×4 magic square on display in the [[Parshvanath]] [[Jain]] temple in [[Khajuraho]], India.<ref>Magic Squares and Cubes By William Symes Andrews, 1908, Open court publish company</ref>
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
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| |-
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| | 7 || 12 || 1 ||14
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| |-
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| | 2 || 13 || 8 || 11
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| |-
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| | 16 || 3 || 10 || 5
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| |-
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| | 9 || 6 || 15 || 4
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| |}
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| This is known as the Chautisa Yantra. Each row, column, and diagonal, as well as each 2×2 sub-square, the corners of each 3×3 and 4×4 square, the two sets of four symmetrical numbers (1+11+16+6 and 2+12+15+5), and the sum of the middle two entries of the two outer columns and rows (12+1+6+15 and 2+16+11+5), sums to 34.
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| In this square, every second diagonal number adds to 17. In addition to squares, there are eight trapeziums – two in one direction, and the others at a rotation of 90 degrees, such as (12, 1, 16, 5) and (13, 8, 9, 4). And in addition to trapeziums, four triangles are also present, where three numbers connect to a corner – for example, the numbers 2, 3, 15 connect to 14 form a triangle. This triangle can also be rotated 90 degrees.
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| '''''Alternate representation of Chautisa Yantra'''''
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
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| |-
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| | 1 || 15 || 6 ||12
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| |-
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| | 14 || 4 || 9 || 7
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| |-
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| | 11 || 5 || 16 || 2
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| |-
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| | 8 || 10 || 3 || 13
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| |}
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| The [[Kuberakolam|Kubera-Kolam]], a magic square of order three, is commonly painted on floors in India. It is essentially the same as the Lo Shu Square, but with 19 added to each number, giving a magic constant of 72.
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
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| |-
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| | 23 || 28 || 21
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| |-
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| | 22 || 24 || 26
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| |-
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| | 27 || 20 || 25
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| |}
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| ===Europe===
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| [[File:Sigillum Iovis.jpg|thumb|215px|right|This page from [[Athanasius Kircher|Athanasius Kircher's]] ''Oedipus Aegyptiacus'' (1653) belongs to a treatise on magic squares and shows the ''Sigillum Iovis'' associated with Jupiter]]
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| In 1300, building on the work of the Arab [[Al-Buni]], Greek Byzantine scholar [[Manuel Moschopoulos]] wrote a mathematical treatise on the subject of magic squares, leaving out the mysticism of his predecessors.<ref>[http://mtcs.truman.edu/~thammond/history/ManuelMoschopoulos.html Manuel Moschopoulos – Mathematics and the Liberal Arts]</ref> Moschopoulos was essentially unknown to the Latin west. He was not, either, the first Westerner to have written on magic squares. They appear in a Spanish manuscript written in the 1280s, presently in the Biblioteca Vaticana (cod. Reg. Lat. 1283a) due to [[Alfonso X]] of Castille.<ref>See ''Alfonso X el Sabio, Astromagia (Ms. Reg. lat. 1283a)'', a cura di A.D'Agostino, Napoli, Liguori, 1992</ref> In that text, each magic square is assigned to the respective planet, as in the Islamic literature.<ref>Mars magic square appears in figure 1 of "Saturn and Melancholy: Studies in the History of Natural Philosophy, Religion, and Art" by [[Raymond Klibansky]], [[Erwin Panofsky]] and [[Fritz Saxl]], Basic Books (1964)</ref> Magic squares surface again in Italy in the 14th century, and specifically in Florence. In fact, a 6×6 and a 9×9 square are exhibited in a manuscript of the ''Trattato d'Abbaco'' (Treatise of the Abacus) by [[Paolo dell'Abbaco]], aka Paolo Dagomari, a mathematician, astronomer and astrologer who was, among other things, in close contact with [[Jacopo Alighieri]], a son of [[Dante]]. The squares can be seen on folios 20 and 21 of MS. 2433, at the Biblioteca Universitaria of Bologna. They also appear on folio 69rv of Plimpton 167, a manuscript copy of the ''Trattato dell'Abbaco'' from the 15th century in the Library of Columbia University.<ref>In a 1981 article ("Zur Frühgeschichte der magischen Quadrate in Westeuropa" i.e. "Prehistory of Magic Squares in Western Europe", Sudhoffs Archiv Kiel (1981) vol. 65, pp. 313–338) German scholar Menso Folkerts lists several manuscripts in which the "Trattato d'Abbaco" by Dagomari contains the two magic square. Folkerts quotes a 1923 article by Amedeo Agostini in the Bollettino dell'Unione Matematica Italiana: "A. Agostini in der Handschrift Bologna, Biblioteca Universitaria, Ms. 2433, f. 20v-21r; siehe Bollettino della Unione Matematica Italiana 2 (1923), 77f. Agostini bemerkte nicht, dass die Quadrate zur Abhandlung des Paolo dell’Abbaco gehören und auch in anderen Handschriften dieses Werks vorkommen, z. B. New York, Columbia University, Plimpton 167, f. 69rv; Paris, BN, ital. 946, f. 37v-38r; Florenz, Bibl. Naz., II. IX. 57, f. 86r, und Targioni 9, f. 77r; Florenz, Bibl. Riccard., Ms. 1169, f. 94-95."</ref> It is interesting to observe that Paolo Dagomari, like Pacioli after him, refers to the squares as a useful basis for inventing mathematical questions and games, and does not mention any magical use. Incidentally, though, he also refers to them as being respectively the Sun's and the Moon's squares, and mentions that they enter astrological calculations that are not better specified. As said, the same point of view seems to motivate the fellow Florentine [[Luca Pacioli]], who describes 3×3 to 9×9 squares in his work ''De Viribus Quantitatis''.<ref>This manuscript text (circa 1496–1508) is also at the Biblioteca Universitaria in Bologna. It can be seen in full at the address http://www.uriland.it/matematica/DeViribus/Presentazione.html</ref> Pacioli states: ''A lastronomia summamente hanno mostrato li supremi di quella commo Ptolomeo, al bumasar ali, al fragano, Geber et gli altri tutti La forza et virtu de numeri eserli necessaria'' (Masters of astronomy, such as [[Ptolemy]], [[Albumasar]], [[Alfraganus]], [[Jabir ibn Aflah|Jabir]] and all the others, have shown that the force and the virtue of numbers are necessary to that science) and then goes on to describe the seven planetary squares, with no mention of magical applications.
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| Magic squares of order 3 through 9, assigned to the seven planets, and described as means to attract the influence of planets and their angels (or demons) during magical practices, can be found in several manuscripts all around Europe starting at least since the 15th century. Among the best known, the ''Liber de Angelis'', a magical handbook written around 1440, is included in Cambridge Univ. Lib. MS Dd.xi.45.<ref>See Juris Lidaka, ''The Book of Angels, Rings, Characters and Images of the Planets'' in ''Conjuring Spirits'', C. Fangier ed. (Pennsylvania State University Press, 1994)</ref> The text of the ''Liber de Angelis'' is very close to that of ''De septem quadraturis planetarum seu quadrati magici'', another handbook of planetary image magic contained in the Codex 793 of the Biblioteka Jagiellońska (Ms BJ 793).<ref>Benedek Láng, ''Demons in Krakow, and Image Magic in a Magical Handbook'', in ''Christian Demonology and Popular Mythology'', Gábor Klaniczay and Éva Pócs eds. (Central European University Press, 2006)</ref> The magical operations involve engraving the appropriate square on a plate made with the metal assigned to the corresponding planet,<ref>According to the correspondence principle, each of the seven planets is associated to a given metal: lead to Saturn, iron to Mars, gold to the Sun, etc.</ref> as well as performing a variety of rituals. For instance, the 3×3 square, that belongs to Saturn, has to be inscribed on a lead plate. It will, in particular, help women during a difficult childbirth.
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| In 1514 [[Albrecht Dürer]] immortalizes a 4×4 square in his famous engraving "Melancholia I".
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| In about 1510 [[Heinrich Cornelius Agrippa]] wrote ''De Occulta Philosophia'', drawing on the [[Hermeticism|Hermetic]] and [[Magic (paranormal)|magical]] works of [[Marsilio Ficino]] and [[Pico della Mirandola]]. In its 1531 edition, he expounded on the magical virtues of the seven magical squares of orders 3 to 9, each associated with one of the [[astrology|astrological]] planets, much in the same way as the older texts did. This book was very influential throughout Europe until the [[counter-reformation]], and Agrippa's magic squares, sometimes called [[Kamea]]s, continue to be used within modern ceremonial magic in much the same way as he first prescribed.<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 |isbn=1-85327-075-X}}</ref>
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
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| ! colspan="3" | [[Saturn (astrology)|Saturn]]=15
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| | 4 || 9 || 2
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| |-
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| | 3 || 5 || 7
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| |-
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| | 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;"
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| ! colspan="4" | [[Jupiter (astrology)|Jupiter]]=34
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| |-
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| | 4 || 14 || 15 || 1
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| |-
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| | 9 || 7 || 6 || 12
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| |-
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| | 5 || 11 || 10 || 8
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| |-
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| | 16 || 2 || 3 || 13
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| |}
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
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| ! colspan="5" | [[Mars (astrology)|Mars]]=65
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| |-
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| | 11 || 24 || 7 || 20 || 3
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| |-
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| | 4 || 12 || 25 || 8 || 16
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| |-
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| | 17 || 5 || 13 || 21 || 9
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| |-
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| | 10 || 18 || 1 || 14 || 22
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| | 23 || 6 || 19 || 2 || 15
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| |}
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;"
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| ! colspan="6" | [[Sun (astrology)|Sol]]=111
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| | 6 || 32 || 3 || 34 || 35 || 1
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| |-
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| | 7 || 11 || 27 || 28 || 8 || 30
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| |-
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| | 19 || 14 || 16 || 15 || 23 || 24
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| |-
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| | 18 || 20 || 22 || 21 || 17 || 13
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| |-
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| | 25 || 29 || 10 || 9 || 26 || 12
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| |-
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| | 36 || 5 || 33 || 4 || 2 || 31
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| |}
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:14em;height:14em;table-layout:fixed;"
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| ! colspan="7" | [[Venus (astrology)|Venus]]=175
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| |-
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| | 22 || 47 || 16 || 41 || 10 || 35 || 4
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| |-
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| | 5 || 23 || 48 || 17 || 42 || 11 || 29
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| |-
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| | 30 || 6 || 24 || 49 || 18 || 36 || 12
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| |-
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| | 13 || 31 || 7 || 25 || 43 || 19 || 37
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| |-
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| | 38 || 14 || 32 || 1 || 26 || 44 || 20
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| |-
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| | 21 || 39 || 8 || 33 || 2 || 27 || 45
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| |-
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| | 46 || 15 || 40 || 9 || 34 || 3 || 28
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| |}
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;"
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| ! colspan="8" | [[Mercury (astrology)|Mercury]]=260
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| | 8 || 58 || 59 || 5 || 4 || 62 || 63 || 1
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| | 49 || 15 || 14 || 52 || 53 || 11 || 10 || 56
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| |-
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| | 41 || 23 || 22 || 44 || 45 || 19 || 18 || 48
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| |-
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| | 32 || 34 || 35 || 29 || 28 || 38 || 39 || 25
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| |-
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| | 40 || 26 || 27 || 37 || 36 || 30 || 31 || 33
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| |-
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| | 17 || 47 || 46 || 20 || 21 || 43 || 42 || 24
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| |-
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| | 9 || 55 || 54 || 12 || 13 || 51 || 50 || 16
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| |-
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| | 64 || 2 || 3 || 61 || 60 || 6 || 7 || 57
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| |}
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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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| ! colspan="9" | [[Moon (astrology)|Luna]]=369
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| | 37 || 78 || 29 || 70 || 21 || 62 || 13 || 54 || 5
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| | 6 || 38 || 79 || 30 || 71 || 22 || 63 || 14 || 46
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| |-
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| | 47 || 7 || 39 || 80 || 31 || 72 || 23 || 55 || 15
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| |-
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| | 16 || 48 || 8 || 40 || 81 || 32 || 64 || 24 || 56
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| |-
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| | 57 || 17 || 49 || 9 || 41 || 73 || 33 || 65 || 25
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| |-
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| | 26 || 58 || 18 || 50 || 1 || 42 || 74 || 34 || 66
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| |-
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| | 67 || 27 || 59 || 10 || 51 || 2 || 43 || 75 || 35
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| |-
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| | 36 || 68 || 19 || 60 || 11 || 52 || 3 || 44 || 76
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| |-
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| | 77 || 28 || 69 || 20 || 61 || 12 || 53 || 4 || 45
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| |}
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| |}
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| </div>
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| [[Image:Hagiel sigil derivation.svg|thumb|The derivation of the [[Sigil (magic)|sigil]] of Hagiel, the [[planetary intelligence]] of [[Venus (astrology)|Venus]], drawn on the magic square of Venus. Each [[Hebrew]] letter provides a numerical value, giving the vertices of the sigil.]]
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| The most common use for these Kameas is to provide a pattern upon which to construct the sigils of [[spirit]]s, [[angel]]s or [[demon]]s; the letters of the entity's name are converted into numbers, and lines are traced through the pattern that these successive numbers make on the kamea.
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| In a magical context, the term ''magic square'' is also applied to a variety of [[word square]]s or number squares found in magical [[grimoire]]s, including some that do not follow any obvious pattern, and even those with differing numbers of rows and columns. 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 |year=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]]:
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| | S || A || T || O || R
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| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:8em;table-layout:fixed;"
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| | 6 || 66 || 848 || 938
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| | 8 || 11 || 544 || 839
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| |
| | 1 || 11 || 383 || 839
| |
| |-
| |
| | 2 || 73 || 774 || 447
| |
| |}
| |
| |
| |
| | 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 || || || ||
| |
| |}
| |
| |
| |
| | 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
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| | |
| ===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, and 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>http://www.muljadi.org/MagicSquares.htm</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. The numbers 1 and 4 at either side of the date correspond to the letters 'A' and 'D' which are the initials of the artist.
| |
| | |
| {| 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>
| |
| | |
| Dürer's magic square and his ''Melencolia I'' both also played large roles in [[Dan Brown]]'s 2009 novel, [[The Lost Symbol|''The Lost Symbol'']].
| |
| | |
| ===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]], conceptualized by [[Antoni Gaudí]] and designed by sculptor [[Josep Maria Subirachs|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→''n''<sup>2</sup> rule.
| |
| | |
| Similarly to Dürer's magic square, the Sagrada Familia's magic square can also be extended to a magic cube.<ref>"[http://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-gaudi-s-square Magic cube with Gaudi's square ]" Ali Skalli's magic squares and magic cubes</ref>
| |
| | |
| ==Types and construction==
| |
| There are many ways to construct magic squares, but the standard (and most simple) way is to follow certain configurations/formulas which generate regular patterns.
| |
| Magic squares exist for all values of ''n'', with only one exception: it is impossible to construct a magic square of order 2. Magic squares can be classified into three types: odd, doubly even (''n'' divisible by four) and singly even (''n'' even, but not divisible by four). Odd and doubly even magic squares are easy to generate; the construction of singly even magic squares is more difficult but several methods exist, including the [[LUX method for magic squares]] (due to [[John Horton Conway]]) and the [[Strachey method for magic squares]].
| |
| | |
| [[Group theory]] was also used for constructing new magic squares of a given order from one of them.<ref>[http://www.gaspalou.fr/magic-squares/index.htm Structure of Magic and Semi-Magic Squares, Methods and Tools for Enumeration]</ref>
| |
| | |
| {{unsolved|mathematics|How many n×n magic squares for n>5?}}
| |
| The numbers of different ''n''×''n'' magic squares for ''n'' from 1 to 5, not counting rotations and reflections are: 1, 0, 1, 880, 275305224 {{OEIS|id=A006052}}. The number for ''n'' = 6 has been estimated to be {{nowrap|1=(0.17745 ± 0.00016) × 10<sup>20</sup>.}}<ref>Pinn K. and Wieczerkowski C., (1998) [http://arxiv.org/abs/cond-mat/9804109 "Number of Magic Squares From Parallel Tempering Monte Carlo"], ''Int. J. Mod. Phys. C'' '''9''' 541</ref><ref name=Arxiv>[http://arxiv.org/abs/cond-mat/9804109 "Number of Magic Squares From Parallel Tempering Monte Carlo], arxiv.org, April 9, 1998. Retrieved November 2, 2013.</ref>
| |
| | |
| | |
| ===Method for constructing a magic square of order 3===
| |
| In the 19th century, [[Edouard Lucas]] devised the general formula for order 3 magic squares. Consider the following table made up of positive integers a, b and c:
| |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:26em;height:6em;table-layout:fixed;"
| |
| |-
| |
| | c + a || c - a - b || c + b
| |
| |-
| |
| | c - a + b || c || c + a - b
| |
| |-
| |
| | c - b || c + a + b || c - a
| |
| |}
| |
| These 9 numbers will be distinct positive integers forming a magic square so long as 0 < a < b < c - a and b ≠ 2a. Moreover, every 3 x 3 square of distinct positive integers is of this form.
| |
| | |
| ===Method for constructing a magic square of odd order===
| |
| {{See also|Siamese method}}
| |
| [[File:Yanghui magic square.GIF|thumb|right|300px|[[Yang Hui]]'s construction method]]
| |
| A method for constructing magic squares of odd order was published by the French diplomat de la Loubère in his book, ''A new historical relation of the kingdom of Siam'' (Du Royaume de Siam, 1693), in the chapter entitled ''The problem of the magical square according to the Indians''.<ref>''Mathematical Circles Squared'' By Phillip E. Johnson, Howard Whitley Eves, p.22</ref> The method operates as follows:
| |
| | |
| The method prescribes starting in the central column of the first row with the number 1. After that, the fundamental movement for filling the squares is diagonally up and right, one step at a time. If a filled square is encountered, one moves vertically down one square instead, then continues as before. When an "up and to the right" move would leave the square, it is wrapped around to the last row or first column, respectively.
| |
| | |
| <div class="center">
| |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | step 1
| |
| |-
| |
| | ||1 ||
| |
| |-
| |
| | || ||
| |
| |-
| |
| | || ||
| |
| |}
| |
| |
| |
| | 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" | step 2
| |
| |-
| |
| | ||1 ||
| |
| |-
| |
| | || ||
| |
| |-
| |
| | || ||2
| |
| |}
| |
| |
| |
| | 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" | step 3
| |
| |-
| |
| | ||1 ||
| |
| |-
| |
| | 3 || ||
| |
| |-
| |
| | || || 2
| |
| |}
| |
| |
| |
| | 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" | step 4
| |
| |-
| |
| | || 1 ||
| |
| |-
| |
| | 3 || ||
| |
| |-
| |
| | 4 || || 2
| |
| |}
| |
| | |
| |}
| |
| | |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | step 5
| |
| |-
| |
| | ||1 ||
| |
| |-
| |
| | 3 || 5 ||
| |
| |-
| |
| | 4 || || 2
| |
| |}
| |
| |
| |
| | 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" | step 6
| |
| |-
| |
| | || 1 || 6
| |
| |-
| |
| | 3 || 5 ||
| |
| |-
| |
| | 4 || || 2
| |
| |}
| |
| | |
| |
| |
| | 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" | step 7
| |
| |-
| |
| | || 1 || 6
| |
| |-
| |
| | 3 || 5 || 7
| |
| |-
| |
| | 4 || || 2
| |
| |}
| |
| |
| |
| | 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" | step 8
| |
| |-
| |
| | 8 || 1 || 6
| |
| |-
| |
| | 3 || 5 || 7
| |
| |-
| |
| | 4 || || 2
| |
| |}
| |
| |
| |
| | 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" | step 9
| |
| |-
| |
| | 8 || 1 || 6
| |
| |-
| |
| | 3 || 5 || 7
| |
| |-
| |
| | 4 || 9 || 2
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| | |
| Starting from other squares rather than the central column of the first row is possible, 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 be a semimagic square and not a true magic square. Moving in directions other than north east can also result in magic squares.
| |
| | |
| ----
| |
| | |
| <div class="center">
| |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | Order 3
| |
| |-
| |
| | 8 || 1 || 6
| |
| |-
| |
| | 3 || 5 || 7
| |
| |-
| |
| | 4 || 9 || 2
| |
| |}
| |
| |
| |
| | 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" | 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
| |
| |}
| |
| |
| |
| | 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" | 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
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| | |
| The following formulae help construct magic squares of odd order
| |
| | |
| {| class="wikitable" style="margin: 1em auto 1em auto"
| |
| ! colspan="5" | Order <math>n</math>
| |
| |-
| |
| | Squares (''n'') || Last no. || Middle no. ||Sum (''M'') || ''I''<sub>th</sub> row and ''J''<sub>th</sub> column no.
| |
| |-
| |
| | <math> n </math> || <math>{n^2}</math> || <math>\frac{n^2+1}{2}</math> || <math>\left(\frac{n^2+1}{2}\right) n</math> || <math>n((I+J-1+\left\lfloor\frac{n}{2}\right\rfloor)\,\bmod\,n)+((I+2J-2)\,\bmod\,n)+1</math>
| |
| |}
| |
| | |
| Example:
| |
| | |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | style="padding: 0 1em" |
| |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:27em;height:10em;table-layout:fixed;"
| |
| |-
| |
| ! colspan="4" | Order 5
| |
| |-
| |
| | Squares (''n'') || Last no. || Middle no. || Sum (''M'')
| |
| |-
| |
| | 5 || 25 || 13 || 65
| |
| |}
| |
| | |
| |}
| |
| | |
| The "''middle number''" is always in the diagonal bottom left to top right.<br />
| |
| The "''last number''" is always opposite the number '''1''' in an outside column or row.
| |
| | |
| ===A method of constructing a magic square of doubly even order===
| |
| [[Doubly even]] means that ''n'' is an even multiple of an even integer; or 4''p'' (e.g. 4, 8, 12), where ''p'' is an integer.
| |
| | |
| '''Generic pattern'''
| |
| All the numbers are written in order from left to right across each row in turn, starting from the top left hand corner. The resulting square is also known as a [[mystic square]]. Numbers are then either retained in the same place or interchanged with their diametrically opposite numbers in a certain regular pattern. In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with their diametrically opposite numbers.
| |
| | |
| '''A construction of a magic square of order 4'''
| |
| Go left to right through the square counting and filling in on the diagonals only. Then continue by going left to right from the top left of the table and fill in counting down from 16 to 1. As shown below.
| |
| | |
| <div class="center">
| |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | ''M'' = Order 4
| |
| |-
| |
| | 1 || || || 4
| |
| |-
| |
| | || 6 || 7 ||
| |
| |-
| |
| | ||10 || 11 ||
| |
| |-
| |
| | 13 || || || 16
| |
| |}
| |
| |
| |
| | 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" | ''M'' = Order 4
| |
| |-
| |
| | 1 ||15 ||14 || 4
| |
| |-
| |
| |12 || 6 || 7 ||9
| |
| |-
| |
| | 8 ||10 || 11 ||5
| |
| |-
| |
| | 13 ||3 ||2 || 16
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| | |
| '''An extension of the above example for Orders 8 and 12'''
| |
| First generate a "truth" table, where a '1' indicates selecting from the square where the numbers are written in order 1 to n<sup>2</sup> (left-to-right, top-to-bottom), and a '0' indicates selecting from the square where the numbers are written in reverse order ''n''<sup>2</sup> to 1. For ''M'' = 4, the "truth" table is as shown below, (third matrix from left.)
| |
| | |
| <div class="center">
| |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | ''M'' = Order 4
| |
| |-
| |
| | 1 || 2 || 3 || 4
| |
| |-
| |
| | 5 || 6 || 7 || 8
| |
| |-
| |
| | 9 ||10 || 11 || 12
| |
| |-
| |
| | 13 || 14 || 15 || 16
| |
| |}
| |
| |
| |
| | 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" | ''M'' = Order 4
| |
| |-
| |
| | 16 ||15 ||14 || 13
| |
| |-
| |
| |12 || 11 || 10 ||9
| |
| |-
| |
| | 8 || 7 || 6 ||5
| |
| |-
| |
| | 4 || 3 || 2 || 1
| |
| |}
| |
| | |
| |
| |
| | 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" | ''M'' = Order 4
| |
| |-
| |
| | 1 || 0 || 0 || 1
| |
| |-
| |
| | 0 || 1 || 1 || 0
| |
| |-
| |
| | 0 || 1 || 1 || 0
| |
| |-
| |
| | 1 || 0 || 0 || 1
| |
| |}
| |
| |
| |
| | 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" | ''M'' = Order 4
| |
| |-
| |
| | 1 ||15 ||14 || 4
| |
| |-
| |
| |12 || 6 || 7 ||9
| |
| |-
| |
| | 8 ||10 || 11 ||5
| |
| |-
| |
| | 13 ||3 ||2 || 16
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| Note that a) there are equal number of '1's and '0's; b) each row and each column are "palindromic"; c) the left- and right-halves are mirror images; and d) the top- and bottom-halves are mirror images (c & d imply b.) The truth table can be denoted as (9, 6, 6, 9) for simplicity (1-nibble per row, 4 rows.) Similarly, for M=8, two choices for the truth table are (A5, 5A, A5, 5A, 5A, A5, 5A, A5) or (99, 66, 66, 99, 99, 66, 66, 99) (2-nibbles per row, 8 rows.) For M=12, the truth table (E07, E07, E07, 1F8, 1F8, 1F8, 1F8, 1F8, 1F8, E07, E07, E07) yields a magic square (3-nibbles per row, 12 rows.) It is possible to count the number of choices one has based on the truth table, taking rotational symmetries into account.
| |
| | |
| ===Medjig-method of constructing magic squares of even number of rows===
| |
| This method is based on a 2006 published mathematical game called [[medjig]] (author: [[Willem Barink]], editor: Philos-Spiele). The pieces of the medjig puzzle are squares divided in four quadrants on which the numbers 0, 1, 2 and 3 are dotted in all sequences. There are 18 squares, with each sequence occurring 3 times. The aim of the puzzle is to take 9 squares out of the collection and arrange them in a 3 × 3 "medjig-square" in such a way that each row and column formed by the quadrants sums to 9, along with the two long diagonals.
| |
| | |
| The medjig method of constructing a magic square of order 6 is as follows:
| |
| * Construct any 3 × 3 medjig-square (ignoring the original game's limit on the number of times that a given sequence is used).
| |
| * Take the 3 × 3 magic square and divide each of its squares into four quadrants.
| |
| * Fill these quadrants with the four numbers from 1 to 36 that equal the original number modulo 9, i.e. ''x''+9''y'' where ''x'' is the original number and ''y'' is a number from 0 to 3, following the pattern of the medjig-square.
| |
| | |
| Example:
| |
| | |
| <div class="center">
| |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | Order 3
| |
| |-
| |
| | 8 || 1 || 6
| |
| |-
| |
| | 3 || 5 || 7
| |
| |-
| |
| | 4 || 9 || 2
| |
| |}
| |
| |
| |
| | 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" | Medjig 3 × 3
| |
| |-
| |
| | 2 || 3 || 0 || 2 || 0 || 2
| |
| |-
| |
| | 1 || 0 || 3 || 1 || 3 || 1
| |
| |-
| |
| | 3 || 1 || 1 || 2 || 2 || 0
| |
| |-
| |
| | 0 || 2 || 0 || 3 || 3 || 1
| |
| |-
| |
| | 3 || 2 || 2 || 0 || 0 || 2
| |
| |-
| |
| | 0 || 1 || 3 || 1 || 1 || 3
| |
| |}
| |
| |
| |
| | 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" | Order 6
| |
| |-
| |
| | 26 || 35 || 1 || 19 || 6 || 24
| |
| |-
| |
| | 17 || 8 || 28 || 10 || 33 || 15
| |
| |-
| |
| | 30 || 12 || 14 || 23 || 25 || 7
| |
| |-
| |
| | 3 || 21 || 5 || 32 || 34 || 16
| |
| |-
| |
| | 31 || 22 || 27 || 9 || 2 || 20
| |
| |-
| |
| | 4 || 13 || 36 || 18 || 11 || 29
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| | |
| Similarly, for any larger integer ''N'', a magic square of order 2''N'' can be constructed from any ''N'' × ''N'' medjig-square with each row, column, and long diagonal summing to 3''N'', and any ''N'' × ''N'' magic square (using the four numbers from 1 to 4''N''<sup>2</sup> that equal the original number modulo ''N''<sup>2</sup>).
| |
| | |
| ===Construction of panmagic squares===
| |
| Any number ''p'' in the order-''n'' square can be uniquely written in the form {{nowrap|1=''p'' = ''an'' + ''r''}}, with ''r'' chosen from {{nowrap|1={1,...,''n''}.}} Note that due to this restriction, ''a'' and ''r'' are ''not'' the usual quotient and remainder of dividing ''p'' by ''n''. Consequently the problem of constructing can be split in two problems easier to solve. So, construct two matching square grids of order ''n'' satisfying panmagic properties, one for the ''a''-numbers {{nowrap|1=(0,..., ''n''−1),}} and one for the ''r''-numbers {{nowrap|1=(1,...,''n'').}} This requires a lot of puzzling, but can be done. When successful, combine them into one [[pandiagonal magic square|panmagic square]].
| |
| Van den Essen and many others supposed this was also the way [[Benjamin Franklin]] (1706–1790) constructed his famous [[Franklin squares]]. Three panmagic squares are shown below. The first two squares have been constructed April 2007 by [[Barink]], the third one is some years older, and comes from [[Donald Morris]], who used, as he supposes, the Franklin way of construction.
| |
| <div class="center">
| |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | Order 8, sum 260
| |
| |-
| |
| | 62 || 4 || 13 || 51 || 46 || 20 || 29 || 35
| |
| |-
| |
| | 5 || 59 || 54 || 12 || 21 || 43 || 38 || 28
| |
| |-
| |
| | 52 || 14 || 3 || 61 || 36 || 30 || 19 || 45
| |
| |-
| |
| | 11 || 53 || 60 || 6 || 27 || 37 || 44 || 22
| |
| |-
| |
| | 64 || 2 || 15 || 49 || 48 || 18 || 31 || 33
| |
| |-
| |
| | 7 || 57 || 56 || 10 || 23 || 41 || 40 || 26
| |
| |-
| |
| | 50 || 16 || 1 || 63 || 34 || 32 || 17 || 47
| |
| |-
| |
| | 9 || 55 || 58 || 8 || 25 || 39 || 42 || 24
| |
| |}
| |
| |- style="horizontal-align:bottom;"
| |
| | style="padding: 0 1em" |
| |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:24em;height:24em;table-layout:fixed;"
| |
| |-
| |
| ! colspan="12" | Order 12, sum 870
| |
| |-
| |
| | 138 || 8 || 17 || 127 || 114 || 32 || 41 || 103 || 90 || 56 || 65 || 79
| |
| |-
| |
| | 19 || 125 || 140 || 6 || 43 || 101 || 116 || 30 || 67 || 77 || 92 || 54
| |
| |-
| |
| | 128 || 18 || 7 || 137 || 104 || 42 || 31 || 113 || 80 || 66 || 55 || 89
| |
| |-
| |
| | 5 || 139 || 126 || 20 || 29 || 115 || 102 || 44 || 53 || 91 || 78 || 68
| |
| |-
| |
| | 136 || 10 || 15 || 129 || 112 || 34 || 39 || 105 || 88 || 58 || 63 || 81
| |
| |-
| |
| | 21 || 123 || 142 || 4 || 45 || 99 || 118 || 28 || 69 || 75 || 94 || 52
| |
| |-
| |
| | 130 || 16 || 9 || 135 || 106 || 40 || 33 || 111 || 82 || 64 || 57 || 87
| |
| |-
| |
| | 3 || 141 || 124 || 22 || 27 || 117 || 100 || 46 || 51 || 93 || 76 || 70
| |
| |-
| |
| | 134 || 12 || 13 || 131 || 110 || 36 || 37 || 107 || 86 || 60 || 61 || 83
| |
| |-
| |
| | 23 || 121 || 144 || 2 || 47 || 97 || 120 || 26 || 71 || 73 || 96 || 50
| |
| |-
| |
| | 132 || 14 || 11 || 133 || 108 || 38 || 35 || 109 || 84 || 62 || 59 || 85
| |
| |-
| |
| | 1 || 143 || 122 || 24 || 25 || 119 || 98 || 48 || 49 || 95 || 74 || 72
| |
| |}
| |
| |- style="vertical-align:bottom;"
| |
| | style="padding: 0 1em" |
| |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:24em;height:24em;table-layout:fixed;"
| |
| |-
| |
| ! colspan="12" | Order 12, sum 870
| |
| |-
| |
| | 1 || 120 || 121 || 48 || 85 || 72 || 73 || 60 || 97 || 24 || 25 || 144
| |
| |-
| |
| | 142 || 27 || 22 || 99 || 58 || 75 || 70 || 87 || 46 || 123 || 118 || 3
| |
| |-
| |
| | 11 || 110 || 131 || 38 || 95 || 62 || 83 || 50 || 107 || 14 || 35 || 134
| |
| |-
| |
| | 136 || 33 || 16 || 105 || 52 || 81 || 64 || 93 || 40 || 129 || 112 || 9
| |
| |-
| |
| | 8 || 113 || 128 || 41 || 92 || 65 || 80 || 53 || 104 || 17 || 32 || 137
| |
| |-
| |
| | 138 || 31 || 18 || 103 || 54 || 79 || 66 || 91 || 42 || 127 || 114 || 7
| |
| |-
| |
| | 5 || 116 || 125 || 44 || 89 || 68 || 77 || 56 || 101 || 20 || 29 || 140
| |
| |-
| |
| | 139 || 30 || 19 || 102 || 55 || 78 || 67 || 90 || 43 || 126 || 115 || 6
| |
| |-
| |
| | 12 || 109 || 132 || 37 || 96 || 61 || 84 || 49 || 108 || 13 || 36 || 133
| |
| |-
| |
| | 135 || 34 || 15 || 106 || 51 || 82 || 63 || 94 || 39 || 130 || 111 || 10
| |
| |-
| |
| | 2 || 119 || 122 || 47 || 86 || 71 || 74 || 59 || 98 || 23 || 26 || 143
| |
| |-
| |
| | 141 || 28 || 21 || 100 || 57 || 76 || 69 || 88 || 45 || 124 || 117 || 4
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| | |
| The order 8 square satisfies all panmagic properties, including the Franklin ones. It consists of 4 perfectly panmagic 4×4 units. Note that both order 12 squares show the property that any row or column can be divided in three parts having a sum of 290 (= 1/3 of the total sum of a row or column). This property compensates the absence of the more standard panmagic Franklin property that any 1/2 row or column shows the sum of 1/2 of the total. For the rest the order 12 squares differ a lot. The Barink 12×12 square is composed of 9 perfectly panmagic 4×4 units, moreover any 4 consecutive numbers starting on any odd place in a row or column show a sum of 290. The Morris 12×12 square lacks these properties, but on the contrary shows constant Franklin diagonals. For a better understanding of the constructing decompose the squares as described above, and see how it was done. And note the difference between the Barink constructions on the one hand, and the Morris/Franklin construction on the other hand.
| |
| | |
| In the book ''Mathematics'' in the Time-Life Science Library Series, magic squares by Euler and Franklin are shown. Franklin designed this one so that any four-square subset (any four contiguous squares that form a larger square, or any four squares equidistant from the center) total 130. In Euler's square, the rows and columns each total 260, and halfway they total 130 – and a [[chess]] knight, making its L-shaped moves on the square, can touch all 64 boxes in consecutive numerical order.
| |
| | |
| ===Construction similar to the Kronecker Product===
| |
| There is a method reminiscent of the [[Kronecker product]] of two matrices, that builds an ''nm'' × ''nm'' magic square from an ''n'' × ''n'' magic square and an ''m'' × ''m'' magic square.<ref name="hartleym">Hartley, M. [http://www.dr-mikes-math-games-for-kids.com/making-big-magic-squares.html "Making Big Magic Squares"].</ref>
| |
| | |
| ===The construction of a magic square using genetic algorithms===
| |
| A magic square can be constructed using [[genetic algorithms]].<ref name="GAMagicSquare">[http://www.dcs.napier.ac.uk/~benp/summerschool/jdemos/herdy/magic_problem2.html Evolving a Magic Square using Genetic Algorithms]</ref> In this process an initial population of squares with random values is generated. The ''fitness'' scores of these individual squares are calculated based on the degree of deviation in the sums of the rows, columns, and diagonals. The population of squares ''reproduce'' by exchanging values, together with some random mutations. Those squares with a higher fitness score are more likely to reproduce. The fitness scores of the next generation squares are calculated, and this process continues until a magic square is found or a time limit is reached.
| |
| | |
| ==Generalizations== | |
| | |
| ===Extra constraints===
| |
| Certain extra restrictions can be imposed on magic squares. If not only the main diagonals but also the [[broken diagonal]]s sum to the magic constant, the result is a [[panmagic square]].
| |
| | |
| If raising each number to the ''n''th power yields another magic square, the result is a bimagic (n = 2), a trimagic (n = 3), or, in general, a [[multimagic square]].
| |
| | |
| A magic square in which the number of letters in the name of each number in the square generates another magic square is called an [[alphamagic square]].
| |
| | |
| ===Different constraints===
| |
| Sometimes the rules for magic squares are relaxed, so that only the rows and columns but not necessarily the diagonals sum to the magic constant (this is usually called a '''semimagic square''').
| |
| | |
| In [[heterosquare]]s and [[antimagic square]]s, the 2''n'' + 2 sums must all be ''different''.
| |
| | |
| ===Multiplicative magic squares===
| |
| Instead of ''adding'' the numbers in each row, column and diagonal, one can apply some other operation. For example, a multiplicative magic square has a constant ''product'' of numbers. A multiplicative magic square can be derived from an additive magic square by raising 2 (or any other integer) to the power of each element. For example, the original Lo-Shu magic square becomes:
| |
| | |
| <div class="center">
| |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:9em;height:9em;table-layout:fixed;"
| |
| |-
| |
| ! colspan="3" | ''M'' = 32768
| |
| |-
| |
| | 16 || 512 || 4
| |
| |-
| |
| | 8 || 32 || 128
| |
| |-
| |
| | 256 || 2 || 64
| |
| |}
| |
| </div>
| |
| | |
| Other examples of multiplicative magic squares include:
| |
| | |
| <div class="center">
| |
| {|
| |
| |- style="vertical-align:bottom;"
| |
| | 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" | ''M'' = 216
| |
| |-
| |
| | 2 || 9 || 12
| |
| |-
| |
| | 36 || 6 || 1
| |
| |-
| |
| | 3 || 4 || 18
| |
| |}
| |
| |
| |
| | 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" | ''M'' = 6720
| |
| |-
| |
| | 1 || 6 || 20 || 56
| |
| |-
| |
| | 40 || 28 || 2 || 3
| |
| |-
| |
| | 14 || 5 || 24 || 4
| |
| |-
| |
| | 12 || 8 || 7 || 10
| |
| |}
| |
| | |
| |}
| |
| </div>
| |
| [[Ali Skalli]]'s non iterative method of construction is also applicable to multiplicative magic squares.
| |
| On the 7×7 example below, the products of each line, each column and each diagonal is 6,227,020,800.
| |
| | |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:12em;table-layout:fixed;"
| |
| |-
| |
| ! colspan="7" | Skalli multiplicative 7 × 7
| |
| |-
| |
| |27 ||50 ||66 ||84 ||13 ||2 ||32
| |
| |-
| |
| |24 ||52 ||3 ||40 ||54 ||70 ||11
| |
| |-
| |
| |56 ||9 ||20 ||44 ||36 ||65 ||6
| |
| |-
| |
| |55 ||72 ||91 ||1 ||16 ||36 ||30
| |
| |-
| |
| |4 ||24 ||45 ||60 ||77 ||12 ||26
| |
| |-
| |
| |10 ||22 ||48 ||39 ||5 ||48 ||63
| |
| |-
| |
| |78 ||7 ||8 ||18 ||40 ||33 ||60
| |
| |}
| |
| | |
| ===Multiplicative magic squares of complex numbers===
| |
| Still using [[Ali Skalli]]'s non iterative method, it is possible to produce an infinity of multiplicative magic squares of [[complex numbers]]<ref>"[http://sites.google.com/site/aliskalligvaen/home-page/-multiplicative-of-complex-numbers-8x8 8x8 multiplicative magic square of complex numbers]" Ali Skalli's magic squares and magic cubes</ref> belonging to <math>\mathbb C</math> set. On the example below, the real and imaginary parts are integer numbers, but they can also belong to the entire set of real numbers <math>\mathbb R</math>.
| |
| The product is: '''−352,507,340,640 − 400,599,719,520 ''i'''''.
| |
| | |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:40em;height:12em;table-layout:fixed;"
| |
| |-
| |
| ! colspan="7" | Skalli multiplicative 7 × 7 of [[complex numbers]]
| |
| |-
| |
| | 21+14i ||−70+30i ||−93−9i ||−105−217i ||16+50i ||4−14i ||14−8i
| |
| |-
| |
| | 63−35i ||28+114i ||−14i ||2+6i ||3−11i ||211+357i ||−123−87i
| |
| |-
| |
| | 31−15i ||13−13i ||−103+69i ||−261−213i ||49−49i ||−46+2i ||−6+2i
| |
| |-
| |
| | 102−84i ||−28−14i ||43+247i ||−10−2i ||5+9i ||31−27i ||−77+91i
| |
| |-
| |
| | −22−6i ||7+7i ||8+14i ||50+20i ||−525−492i ||−28−42i ||−73+17i
| |
| |-
| |
| | 54+68i ||138−165i ||−56−98i ||−63+35i ||4−8i ||2−4i ||70−53i
| |
| |-
| |
| | 24+22i ||−46−16i ||6−4i ||17+20i ||110+160i ||84−189i ||42−14i
| |
| |}
| |
| | |
| ===Other magic shapes===
| |
| Other shapes than squares can be considered. The general case is to consider a design with ''N'' parts to be magic if the ''N'' parts are labeled with the numbers 1 through ''N'' and a number of identical sub-designs give the same sum. Examples include magic [[dodecahedron]]s, [[magic triangle]]s<ref name=Ely>Magic Designs,Robert B. Ely III, Journal of Recreational Mathematics volume 1 number 1, January 1968</ref> [[magic star]]s, and [[magic hexagon]]s. Going up in dimension results in [[magic cube]]s, [[magic tesseract]]s and other [[magic hypercube]]s.
| |
| | |
| Edward Shineman has developed yet another design in the shape of magic diamonds.
| |
| | |
| Possible magic shapes are constrained by the number of equal-sized, equal-sum subsets of the chosen set of labels. For example, if one proposes to form a magic shape labeling the parts with {1, 2, 3, 4}, the sub-designs will have to be labeled with {1,4} and {2,3}.<ref name=Ely />
| |
| | |
| ===Other component elements===
| |
| [[File:Geomagic square - Magic Crystals.jpg|thumb|left| text-bottom |160px|Geomagic square]]
| |
| [[File:Geomagic square - Diamonds.jpg|thumb|right| text-bottom |160px| Geomagic square]]
| |
| Magic squares may be constructed which contain geometric shapes rather than numbers, as in the [[geomagic square]]s introduced by [[Lee Sallows]].<ref>[http://www.guardian.co.uk/science/2011/apr/03/magic-squares-geomagic-lee-sallows Magic squares are given a whole new dimension], [[The Observer]], April 3, 2011</ref>
| |
| | |
| ===Combined extensions===
| |
| One can combine two or more of the above extensions, resulting in such objects as ''multiplicative multimagic hypercubes''. Little seems to be known about this subject.
| |
| | |
| ==Related problems==
| |
| Over the years, many mathematicians, including [[Euler]], [[Arthur Cayley|Cayley]] and [[Benjamin Franklin]] have worked on magic squares, and discovered fascinating relations.
| |
| | |
| ===Magic square of primes===
| |
| [[Rudolf Ondrejka]] (1928–2001) discovered the following 3×3 magic square of [[prime number|primes]], in this case nine [[Chen prime]]s:
| |
| | |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:7em;height:7em;table-layout:fixed;"
| |
| |-
| |
| | 17 || 89 || 71
| |
| |-
| |
| | 113 || 59 || 5
| |
| |-
| |
| | 47 || 29 || 101
| |
| |}
| |
| | |
| The [[Green–Tao theorem]] implies that there are arbitrarily large magic squares consisting of primes.
| |
| | |
| Using [[Ali Skalli]]'s non-iterative method of magic squares construction, it is easy to create magic squares of primes<ref>"[http://sites.google.com/site/aliskalligvaen/home-page/-5x5-square-of-prime-numbers magic square of primes]" Ali Skalli's magic squares and magic cubes</ref> of any dimension. In the example below, many symmetries appear (including all sorts of crosses), as well as the horizontal and vertical translations of all those. The magic constant is 13665.
| |
| | |
| {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:12em;table-layout:fixed;"
| |
| |-
| |
| ! colspan="5" | Skalli Primes 5 × 5
| |
| |-
| |
| | 2087 || 2633 || 2803 || 2753 || 3389
| |
| |-
| |
| | 2843 || 2729 || 3347 || 2099 || 2647
| |
| |-
| |
| | 3359 || 2113 || 2687 || 2819 || 2687
| |
| |-
| |
| | 2663 || 2777 || 2699 || 3373 || 2153
| |
| |-
| |
| | 2713 || 3413 || 2129 || 2621 || 2789
| |
| |}
| |
| | |
| It is believed that an infinite number of Skalli's magic squares of prime exist, but no demonstration exists to date. However, it is possible to easily produce a considerable number of them, not calculable in the absence of demonstration.
| |
| | |
| ===''n''-Queens problem===
| |
| In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into [[Eight queens puzzle|''n''-queens]] solutions, and vice versa.<ref>O. Demirörs, N. Rafraf, and M. M. Tanik. "Obtaining ''n''-queens solutions
| |
| from magic squares and constructing magic squares from ''n''-queens solutions". ''Journal of Recreational Mathematics'', 24:272–280, 1992</ref>
| |
| <!--{{Citation needed|date=September 2009}}--><!-- paper not listed in references --><!-- added reference -->
| |
| | |
| ==See also==
| |
| <div style="-moz-column-count:3; column-count:3;">
| |
| * [[Arithmetic sequence]]
| |
| * [[Combinatorial design]]
| |
| * [[Freudenthal magic square]]
| |
| * [[John R. Hendricks]]
| |
| * [[Hexagonal tortoise problem]]
| |
| * [[Latin square]]
| |
| * [[Magic circle (mathematics)|Magic circle]]
| |
| * [[Magic cube classes]]
| |
| * [[Magic series]]
| |
| * [[Most-perfect magic square]]
| |
| * [[Nasik magic hypercube]]
| |
| * [[Prime reciprocal magic square]]
| |
| * [[Room square]]
| |
| * [[Matrix (mathematics)#Square matrices|Square matrices]]
| |
| * [[Sriramachakra]]
| |
| * [[Sudoku]]
| |
| * [[Unsolved problems in mathematics]]
| |
| * [[Vedic square]]
| |
| </div>
| |
| | |
| ==Notes==
| |
| {{reflist|3}}
| |
| | |
| ==References==
| |
| <div class="references-small">
| |
| * {{MathWorld|urlname=MagicSquare|title=Magic Square}}
| |
| * [http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=528&bodyId=784 Magic Squares] at [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 4×4 using Javascript)
| |
| * [http://www.hbmeyer.de/backtrack/mag4en.htm Magic Squares of Order 4,5,6, and some theory]
| |
| * [http://www.dcs.napier.ac.uk/~benp/summerschool/jdemos/herdy/magic_problem2.html Evolving a Magic Square using Genetic Algorithms]
| |
| * [http://sites.google.com/site/aliskalligvaen/home-page Magic squares and magic cubes]: examples of magic squares and magic cubes built with Ali Skalli's non iterative method
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| | |
| </div>
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| | |
| ==Further reading==
| |
| *{{cite book | last = Block | first = Seymour | title = Before Sudoku: The World of Magic Squares | publisher = Oxford University Press | location = | year = 2009 | isbn = 0195367901 }}
| |
| * {{cite journal
| |
| | author = McCranie, Judson
| |
| |date=November 1988
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| | title = Magic Squares of All Orders
| |
| | journal = Mathematics Teacher
| |
| | volume =
| |
| | issue =
| |
| | pages = 674–78
| |
| | doi =
| |
| | url =}}
| |
| * {{cite journal
| |
| | author = Semanisinova, Ingrid; Trenkler, Marian
| |
| |date=August 2007
| |
| | title = Discovering the Magic of Magic Squares
| |
| | journal = Mathematics Teacher
| |
| | volume = 101
| |
| | issue = 1
| |
| | pages = 32–39
| |
| | doi =
| |
| | url = http://www.nctm.org/publications/article.aspx?id=19305}}
| |
| * {{cite journal
| |
| | author = King, J. R.
| |
| | year = 1963
| |
| | month =
| |
| | title = Magic Square Numbers
| |
| | journal =
| |
| | volume =
| |
| | issue =
| |
| | pages =
| |
| | doi =
| |
| | url =}}
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| | |
| ==External links==
| |
| * {{cite web|last=Eaves|first=Laurence|title=Magic Square|url=http://www.sixtysymbols.com/videos/magicsquare.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|authorlink=Laurence Eaves|year=2009}}
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| * {{dmoz|Science/Math/Recreations/Magic_Square}}
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| | |
| {{DEFAULTSORT:Magic Square}}
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| [[Category:Magic squares| ]]
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| [[Category:Matrices]]
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| [[Category:Unsolved problems in mathematics]]
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| [[Category:Magic symbols]]
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