Magic Square
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Magic Square
The smallest (and unique up to rotation and reflection) non-trivial case of a magic square, measuring 3×3

In recreational mathematics, a magic square[1] is a ${\displaystyle n\times n}$ square grid (where ${\displaystyle n}$ is the number of cells on each side) filled with distinct positive integers in the range ${\displaystyle 1,2,...,n^{2}}$ such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal.[2] The sum is called the magic constant or magic sum of the magic square. A square grid with ${\displaystyle n}$ cells on each side is said to have order n.

In regard to magic sum, the problem of magic squares only requires the sum of each row, column and diagonal to be equal, it does not require the sum to be a particular value. Thus, although magic squares may contain negative integers, they are just variations by adding or multiplying a negative number to every positive integer in the original square.[3][4]

Magic squares are also called normal magic squares, in the sense that there are non-normal magic squares[5] which integers are not restricted in ${\displaystyle 1,2,...,n^{2}}$. However, in some places, "magic squares" is used as a general term to cover both the normal and non-normal ones, especially when non-normal ones are under discussion. Moreover, the term "magic squares" is sometimes also used to refer to various types of word squares.

Magic squares have a long history, dating back to at least 650 BC in China. At various times they have acquired magical or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations.

## Properties of magic squares

### Magic constant

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 constant dependent on the order ${\displaystyle n}$, calculated by the formula ${\displaystyle M=n(n^{2}+1)/2}$, since the sum of ${\displaystyle 1,2,...,n^{2}}$ is ${\displaystyle n^{2}(n^{2}+1)/2}$ which when divided by the order ${\displaystyle n}$ is the magic constant. For normal magic squares of orders n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS).

### Magic square of order 1 is trivial

The 1×1 magic square, with only one cell containing the number 1, is called trivial, because it is typically not under consideration when discussing magic squares; but it is indeed a magic square by definition, if we regard a single cell as a square of order one.

### Magic square of order 2 cannot be constructed

Normal magic squares of all sizes can be constructed except 2×2 (that is, where order n = 2).[6]

### Equivalent magic squares

Any magic square can be rotated and reflected to produce 8 trivially distinct squares. In magic square theory, all of these are generally deemed equivalent and the eight such squares are said to make up a single equivalence class.[7][8]

### Number of magic squares of a given order

Excluding rotations and reflections, there is exactly one 3×3 magic square, exactly 880 4×4 magic squares, and exactly 275,305,224 5×5 magic squares. For the 6×6 case, there are estimated to be approximately 1.8 × 1019 squares.[9]

### An invariant

The moment of inertia of a magic square has been defined as the sum over all cells of the number in the cell times the squared distance from the center of the cell to the center of the square; here the unit of measurement is the width of one cell.[9] (Thus for example a corner cell of a 3×3 square has a distance of ${\displaystyle {\sqrt {2}},}$ a non-corner edge cell has a distance of 1, and the center cell has a distance of 0.) Then all magic squares of a given order have the same moment of inertia as each other. For the order-3 case the moment of inertia is always 60, while for the order-4 case the moment of inertia is always 340. In general, for the n×n case the moment of inertia is ${\displaystyle n^{2}(n^{4}-1)/12.}$[9]

## History

Iron plate with an order 6 magic square in Eastern Arabic Numerals from China, dating to the Yuan Dynasty (1271-1368).

Magic squares were known to Chinese mathematicians as early as 650 BC, and explicitly given since 570 AD,[10] and to Islamic mathematicians possibly as early as the seventh century AD. The magic squares of order 3 to 9 appear in an encyclopedia from Baghdad circa 983, the Encyclopedia of the Brethren of Purity (Rasa'il Ihkwan al-Safa); simpler magic squares were known to several earlier Arab mathematicians.[10] Some of these squares were later used in conjunction with magic letters, as in Shams Al-ma'arif, to assist Arab illusionists and magicians.[11]

### China

Legends dating from as early as 650 BC tell the story of the Lo Shu () or "scroll of the river Lo".[12] According to the legend, there was at one time in ancient China a huge flood. While the great king Yu was trying to channel the water out to sea, a turtle emerged from it with a curious pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15. According to the legend, thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods.

 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.

While ancient references to the pattern of even and odd numbers in the 3×3 magic square appears in the I Ching, the first unequivocal instance of this magic square appears in a 1st century book Da Dai Liji (Record of Rites by the Elder Dai).[13][14][15] These numbers also occur in a possibly earlier mathematical text called Shushu jiyi (Memoir on Some Traditions of Mathematical Art), said to be written in 190 BCE. This is the earliest appearance of a magic square on record; and it was mainly used for divination and astrology.[13] The 3×3 magic square was referred to as the "Nine Halls" by earlier Chinese mathematicians.[15] The identification of the 3×3 magic square to the legendary Luoshu chart was only made in the 12th century, after which it was referred to as the Luoshu square.[13][15] The oldest surviving Chinese treatise on the systematic methods for constructing larger magic squares is Yang Hui's Xugu zheqi suanfa (Continuation of Ancient Mathematical Methods for Elucidating the Strange) written in 1275.[13][15] The contents of Yang Hui's treatise were collected from older works, both native and foreign; and he only explains the construction of third and fourth order magic squares, while merely passing on the finished diagrams of larger squares.[15] After Yang Hui, magic squares frequently occur in Chinese mathematics such as in Ding Yidong's Dayan suoyin (circa 1300), Chen Dawei's Suanfa tongzong (1593), Fang Zhongtong's Shuduyan (1661) which contains magic circles, cubes and spheres, Zhang Chao's Xinzhai zazu (circa 1650), and lastly Bao Qishou's Binaishanfang ji written in later half of 19th century.[13] However, despite being the first to discover the magic squares and getting a head start by several centuries, the Chinese development of the magic squares are much inferior compared to the Islamic, the Indian, or the European developments. The high point of Chinese mathematics that deals with the magic squares seems to be contained in the work of Yang Hui; but even as a collection of older methods, this work is much more primitive, lacking general methods for constructing magic squares of any order, compared to a similar collection written around the same time by the Byzantine scholar Manuel Moschopoulos.[15] This is possibly because of the Chinese scholars' enthralment with the Lo Shu principle, which they tried to adapt to solve higher squares; and after Yang Hui and the fall of Yuan dynasty, their systematic purging of the foreign influences in Chinese mathematics.[15]

### Persia

Original script from the Shams al-Ma'arif.
Printed version of the previous manuscript. Eastern Arabic numerals were used.

Although the early history of magic squares in Persia is not known, it has been suggested that they were known in pre-Islamic times.[16] It is clear, however, that the study of magic squares was common in medieval Islam in Persia, and it was thought to have begun after the introduction of chess into the region.[17] 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 arithmetical progression, in such a way that the sums of each row, column and diagonal are equal.[18]

### Arabia

Magic squares were known to Islamic mathematicians in Arabia as early as the seventh century. They may have learned about them when the Arabs 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 magic squares of order 3 to 9 known to have been devised by Arab mathematicians appear in an encyclopedia from Baghdad circa 983, the Rasa'il Ikhwan al-Safa (the Encyclopedia of the Brethren of Purity); simpler magic squares were known to several earlier Arab mathematicians.[10]

The magic square of order three was described as a child-bearing charm[19] since its first literary appearances in the works of J?bir ibn Hayy?n (fl. c. 721- c. 815)[20] and al-Ghaz?l? (1058-1111)[21] and it was preserved in the tradition of the planetary tables, known from H.C.Agrippa's work,[22] too.

The Arab mathematician Ahmad al-Buni, who gave general methods on constructing magic squares around 1250, 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.[10]

### India

Magic Square at the Parshvanatha temple, in Khajuraho India

The 3×3 magic square has been a part of rituals in India since Vedic times, and still is today. For instance, the 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.

 23 28 21 22 24 26 27 20 25

The earliest unequivocal occurrence of magic square is found in a work called Kaksaputa, composed by the alchemist Nagarjuna around 1st century AD. All of the squares given by Nagarjuna are 4×4 magic squares, and one of them is called Nagarjuniya after him. Nagarjuna gave a method of constructing 4×4 magic square using a primary skeleton square, given an odd or even magic sum. Incidentally, the special Nagarjuniya square cannot be constructed from the method he expounds.[23] The Nagarjuniya square is given below, and has the sum total of 100.

The Nagarjuniya square is a pan-diagonal magic square, where the broken diagonals (e.g. 16+22+34+28, 18+24+32+26, etc) sum to 100. It is also an instance of a most perfect magic square, where every 2×2 sub-square, four corners of any 3×3 sub-square, four corners of the 4×4 square, the four corners of any 2×4 or 4×2 sub-rectangle, and the four corners of oblong diagonals (18+24+32+26 and 10+16+34+40) all sum to 100. Furthermore, the corners of eight trapezoids (16+18+32+34, 44+22+28+6, etc) all sum to 100. The Nagarjuniya square is made up of two arithmetic progressions starting from 6 and 16 with eight terms each, with a common difference between successive terms as 4. When these two progressions are reduced to the normal progression of 1 to 8, we obtain the adjacent square.

The oldest datable magic square in India is found in an encyclopaedic work of Varahamihira around 587 AD called Brhat Samhita for the purpose of making perfumes using 4 substances selected from 16 different substances. Each cell of the square represents a particular ingredient, while the number in the cell represents the proportion of the associated ingredient, such that the mixture of any four combination of ingredients along the columns, rows, diagonals, and so on, gives the total volume of the mixture to be 18. Although the book is mostly about divination, the magic square is given as a matter of combinatorial design, and no magical properties are attributed to it.[23][24]

 2 3 5 8 5 8 2 3 4 1 7 6 7 6 4 1

The square of Varahamihira as given above has sum of 18. Here the numbers 1 to 8 appear twice in the square. It is a pan-diagonal magic square. It is also an instance of most perfect magic square. Four different magic squares can be obtained by adding 8 to one of the two sets of 1 to 8 sequence. The sequence is selected such that the number 8 is added exactly twice in each row, each column and each of the main diagonals.[25] One of the possible magic squares is given below:

 10 3 13 8 5 16 2 11 4 9 7 14 15 6 12 1

This magic square is remarkable in that it is a 90 degree rotation of a magic square that appears in the 13th century Islamic world as one of the most popular magic squares.[25] His book also contains a method for constructing a magic square of order four when a constant sum is given. It also contains the Nagarjuniya square.[24]

The first datable instance of 3×3 magic square in India occur in a medical text Siddhayog (ca. 900 AD) by Vrnda, which was prescribed to a woman in labor in order to have easy delivery.[24] Vrnda's square is given below, which sum to 30.

 16 6 8 2 10 18 12 14 4

There is a well-known 12th-century 4×4 magic square on display in the Parshvanath temple in Khajuraho, India. [23][24][26]

 7 12 1 14 2 13 8 11 16 3 10 5 9 6 15 4

This is known as the Chautisa Yantra of which the total is 34. It is one of the three 4×4 pandiagonal magic squares and is also an instance of the most-perfect magic square. Several Jain hyms teach how to make magic squares, although they are undatable.[24]

As far as is known, the first systematic study of magic squares was conducted by Thakkar Pheru, a Jain scholar, in his Ganitasara Kaumudi (ca.1315 AD). This work contains a small section on magic squares which consists of nine verses. Here he gives a square of order four, and alludes to its rearrangement; classifies magic squares into three (odd, evenly even, and evenly odd) according to its order; gives a square of order six; and prescribes one method each for constructing even and odd squares.[24] For the even squares, Pheru divides the square into component squares of order four, and puts the numbers into cells according to the pattern of a standard square of order four.[24] For odd squares, Pheru gives the method using horse move or knight's move. Although algorithmically different, it gives the same square as the De la Loubere's method.[24] Below is Pheru's square of order six.[24]

 1 32 34 33 5 6 30 8 28 27 11 7 24 23 15 16 14 19 13 20 21 22 17 18 12 26 9 10 29 25 31 2 4 3 35 36

The next comprehensive work on magic figures was taken up by Narayana Pandit, who in the fourteenth chapter of his Ganita Kaumudi (1356 AD) gives general methods for the constructions of all sorts of magic squares with the principles governing such constructions. It comprises of 55 verses for rules and 17 verses for examples. Narayana gives the method to make a magic squares of order four using knight's move; enumerates the number of pan-diagonal magic squares of order four, 384, including every variation made by rotation and inversion; three general methods for squares having any order and constant sum when a standard square of the same order is known; two methods each for constructing evenly even, evenly odd, and odd squares when the sum is given. While Narayana recounts some older methods of construction, his folding method seems to be his own invention, which was later re-discovered by De la Hire. In the last section, he conceives of other figures, such as circles, rectangles, and hexagons, in which the numbers may be arranged to possess properties similar to those of magic squares.[23][24] Below is an example of 4×4 magic square constructed by knight's move as given by Narayana. This is also a pan-diagonal as well as a most-perfect magic square.

 1 14 4 15 8 11 5 10 13 2 16 3 12 7 9 6

Incidentally, Narayana states that the purpose of studying magic squares is to construct yantra, to destroy the ego of bad mathematicians, and for the pleasure of good mathematicians.[24]

### Europe

This page from Athanasius Kircher's Oedipus Aegyptiacus (1653) belongs to a treatise on magic squares and shows the Sigillum Iovis associated with Jupiter

Around 1315, 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, where he gave a method for odd square and two methods for evenly even squares.[27] 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.[28] In that text, each magic square is assigned to the respective planet, as in the Islamic literature.[29] 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.[30] 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.[31] 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 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.

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.[32] 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 Jagiello?ska (Ms BJ 793).[33] The magical operations involve engraving the appropriate square on a plate made with the metal assigned to the corresponding planet,[34] 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.

In 1514 Albrecht Dürer immortalized a 4×4 square, of order four, in his famous engraving Melencolia I. It is described in more detail below.

In about 1510 Heinrich Cornelius Agrippa wrote De Occulta Philosophia, drawing on the Hermetic and 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 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 kameas, continue to be used within modern ceremonial magic in much the same way as he first prescribed.[10][35]

The derivation of the sigil of Hagiel, the planetary intelligence of Venus, drawn on the magic square of Venus. Each Hebrew letter provides a numerical value, giving the vertices of the sigil.

The most common use for these kameas is to provide a pattern upon which to construct the sigils of spirits, angels or demons; 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. In a magical context, the term magic square is also applied to a variety of word squares or number squares found in magical grimoires, 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 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;[36] 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:

### Albrecht Dürer's magic square

Detail of Melencolia I

The order-4 magic square Albrecht Dürer immortalized in his 1514 engraving Melencolia I, referred to above, 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 queens in the two solutions of the 4 queens puzzle[37]), 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 respectively to the letters "A" and "D," which are the initials of the artist.

 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.[38]

Dürer's magic square and his Melencolia I both also played large roles in Dan Brown's 2009 novel, The Lost Symbol.

The Passion façade of the Sagrada Família church in Barcelona, conceptualized by Antoni Gaudí and 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. Structurally, it is very similar to the Melancholia magic square, but it has had the numbers in four of the cells reduced by 1.

 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.

Similarly to Dürer's magic square, the Sagrada Familia's magic square can also be extended to a magic cube.[39]

This magic square has 24 groups of four fields with the sum of 139 and in the first row - shown at bottom-right - Ramanujan's date of birth.

### Srinivasa Ramanujan's magic square

The Indian mathematician Srinivasa Ramanujan created a square where - in addition to several groups of four squares - the first row shows his date of birth, 22nd Dec. 1887.

### Magic Square using calculator digit shapes

Professor Roger Bowley has shown a fascinating 4x4 magic square using calculator digits shapes. All rows, columns and diagonals total 176. Then the whole paper is turned upside down and still all rows, columns, diagonal of new numbers formed by turning paper upside down is 176. Next reflection of this magic square also displays the same property.

## Types of 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.[40]

 Unsolved problem in mathematics:How many n×n magic squares, and how many magic tori of order n, are there for n>5?(more unsolved problems in mathematics)

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 (sequence in the OEIS). The number for n = 6 has been estimated to be [41][42]

Cross-referenced to the above sequence, a new classification enumerates the magic tori that display these magic squares. The numbers of magic tori of order n from 1 to 5, are: 1, 0, 1, 255, 251449712 (sequence in the OEIS).

### Method for constructing a magic square of order 3

In the 19th century, Édouard Lucas devised the general formula for order 3 magic squares. Consider the following table made up of positive integers a, b and c:

 c - b c + (a + b) c - a c - (a - b) c c + (a - b) c + a c - (a + b) c + b

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 3×3 square of distinct positive integers is of this form.

### Method for constructing a magic square of odd order

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.[43] 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.

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.

The following formulae help construct magic squares of odd order

Order ${\displaystyle n}$
Squares (n) Last no. Middle no. Sum (M) Ith row and Jth column no.
${\displaystyle n}$ ${\displaystyle {n^{2}}}$ ${\displaystyle {\frac {n^{2}+1}{2}}}$ ${\displaystyle \left({\frac {n^{2}+1}{2}}\right)n}$ ${\displaystyle n((I+J-1+\left\lfloor {\frac {n}{2}}\right\rfloor )\,{\bmod {\,}}n)+((I+2J-2)\,{\bmod {\,}}n)+1}$

Example:

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.
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 4p (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 (This is reflection of Albrecht Dürer's square.) 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.

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 n2 (left-to-right, top-to-bottom), and a '0' indicates selecting from the square where the numbers are written in reverse order n2 to 1. For M = 4, the "truth" table is as shown below, (third matrix from left.)

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.

### Euler's method

Euler's method for constructing the magic square is similar to Narayana-De la Hire's method. It consists of breaking the magic square into two primary squares, which when added gives the magic square. As a running example, we will consider a 3×3 magic square. We can uniquely label each number of the 3×3 square by a pair of numbers as

where every pair of Greek and Latin alphabets, e.g. ?a, are meant to be added together, i.e. ?a = ? + a. Here, (?, ?, ?) = (0, 3, 6) and (a, b, c) = (1, 2, 3). An important general constraint to note here is that

• a Greek letter is paired with a Latin letter only once.

Thus, the original square can now be split into two simpler squares:

A magic square can be constructed by ensuring that the Greek and Latin squares are magic squares too. The converse of this statement is also true: A magic square can always be decomposed into a Greek and a Latin square, which are themselves magic squares. However, the Greek and Latin squares with just three unique terms are much easier to deal with than the original square with nine different terms. The row sum and the column sum of the Greek square will be the same, ? + ? + ?, if

• each letter appears only once in a given column and a row.

This can be achieved by cyclic permutation of ?, ?, and ?.

For the odd square, since ?, ?, and ? are in arithmetic progression, their sum is equal to the product of the square's order and the middle term, i.e. ? + ? + ? = 3 ?. Thus, the diagonal sums will be equal if we have ?s in the main diagonal and ?, ?, ? in the skew diagonal. Similarly, for the Latin square. The resulting Greek and Latin squares and their combination will be as below. Note that the Latin square is just a rotation of the Greek square with the corresponding letters interchanged. Substituting the values of the Greek and Latin letters will give the 3×3 magic square.

For the odd squares, this method explains why the Siamese method (method of De la Loubere) and its variants work. This basic method can be used to construct odd ordered magic squares of higher orders. To summarise:

• For odd ordered squares, to construct Greek square, place the middle term along the main diagonal, and place the rest of the terms along the skew diagonal. The remaining empty cells are filled by diagonal moves. The Latin square can be constructed by rotating or flipping the Greek square, and replacing the corresponding alphabets. The magic square is obtained by adding the Greek and Latin squares.

A peculiarity of the construction method given above for the odd magic squares is that the middle number (n2 + 1)/2 will always appear at the centre cell of the magic square.

As another example, the construction of 5×5 magic square is given. Numbers are directly written in place of alphabets. The numbers are placed about the skew diagonal such that the middle column of the resulting square has 0, 5, 10, 15, 20 (from bottom to top). The second primary square is obtained by rotating the first square counter-clockwise by 90 degrees, and replacing the numbers. The resulting square is an associative magic square. In this case, 1 is placed at center cell of bottom row, and successive numbers are placed via elongated knight's move (two cells right, two cells down). When a collision occurs, the break move is to move one cell up. Also note that all the odd numbers occur inside the central diamond formed by 1, 5, 25 and 21, while the even numbers are placed at the corners. We have re-recreated the so called lozenge method.

A variation of the above example, where the skew diagonal sequence is taken in different order, is given below. The resulting magic square is an associative magic square and is the same as that produced by Moschopoulos's method. Here the resulting square starts with 1 placed in the cell which is to the right of the centre cell, and proceeds as De la Loubere's method, with downwards-right move. When a collision occurs, the break move is to shift two cells to the right.

Another possible method of creating the Greek squares is to shift a row by two places to form the next row. The second square is made by flipping the first along the main diagonal. This essentially re-creates the knight's move (two cells up, one cell right). The break move is to move one cell up, one cell left. In this particular case, the square has been constructed such that 1 is at the center cell. The resulting square is a pandiagonal magic square. The knight's move method to create a pandiagonal magic square is valid for all orders that are prime numbers greater than three.

We can also combine the Greek and Latin squares constructed by different methods. In the example below, the second square is made using knight's move. We have re-created the magic square obtained by De la Loubere's method.

We can also construct even ordered squares in this fashion, although it takes a bit of trial and error. An example of a 4×4 square is given below:

Euler's method has given rise to the study of Graeco-Latin squares. Euler's method is valid for squares of any order except 2 and 6.

### Narayana-De la Hire's method

Narayana-De la Hire's method for odd square is the same as that of Euler's. However, for even squares, we drop the second requirement that each Greek and Latin letter appear only once in a given row or column. This allows us to take advantage of the fact that the sum of an arithmetic progression with an even number of terms is equal to the sum of two opposite symmetric terms multiplied by half the total number of terms. Thus, when constructing the Greek or Latin squares,

• for even ordered squares, a letter can appear n/2 times in a column but only once in a row, or vice versa.

As a running example, if we take a 4×4 square, where the Greek and Latin terms have the values (?, ?, ?, ?) = (0, 4, 8, 12) and (a, b, c, d) = (1, 2, 3, 4), respectively, then we have ? + ? + ? + ? = 2 (? + ?) = 2 (? + ?). Similarly, a + b + c + d = 2 (a + d) = 2 (b + c). This means that the complementary pair ? and ? (or ? and ?) can appear twice in a column (or a row) and still give the desired magic sum. Thus, we can build a Greek magic square by first placing ?, ?, ?, ? along the main diagonal in some order. The skew diagonal is then filled in the same order or by picking the terms that are symmetric to the terms in the main diagonal. Finally, the remaining cells are filled column wise. Given a column, we use the complementary terms in the diagonal cells intersected by that column, making sure that they appear only once in a given row but n/2 times in the given column.

In the example given below, the main diagonal (from top left to bottom right) is filled with sequence ordered as ?, ?, ?, ?, while the skew diagonal (from bottom left to top right) filled in the same order. The remaining cells are then filled column wise such that the complementary letters appears only once within a row, but twice within a column. In the first column, since ? appears on the 1st and 4th row, the remaining cells are filled with its complementary term ?. Similarly, the empty cells in the 2nd column are filled with ?; in 3rd column ?; and 4th column ?. Each Greek letter appears only once along the rows, but twice along the columns. As such, the row sums are ? + ? + ? + ? while the column sums are either 2 (? + ?) or 2 (? + ?). Likewise for the Latin square, which is obtained by flipping the Greek square along the main diagonal and interchanging the corresponding letters.

The above example explains why the "criss-cross" method for doubly even magic square works. Another possible 4×4 magic square, which is also pan-diagonal as well as most-perfect, is constructed below using the same rule. However, the diagonal sequence is chosen such that all four letters ?, ?, ?, ? appear inside the central 2×2 sub-square. Remaining cells are filled column wise such that each letter appears only once within a row. In the 1st column, the empty cells need to be filled with one of the letters selected from the complementary pair ? and ?. Given the 1st column, the entry in the 2nd row can only be ? since ? is already there in the 2nd row; while, in the 3rd row the entry can only be ? since ? is already present in the 3rd row. We proceed similarly until all cells are filled. The Latin square given below has been obtained by flipping the Greek square along the main diagonal and replacing the Greek alphabets with corresponding Latin alphabets.

As another example, below is a construction of a 6×6 magic square, the numbers are directly given, rather than the alphabets.

### 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+9y where x is the original number and y is a number from 0 to 3, following the pattern of the medjig-square.

Example:

Similarly, for any larger integer N, a magic square of order 2N can be constructed from any N × N medjig-square with each row, column, and long diagonal summing to 3N, and any N × N magic square (using the four numbers from 1 to 4N2 that equal the original number modulo N2).

### Construction of panmagic squares

Any number p in the order-n square can be uniquely written in the form , with r chosen from 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 and one for the r-numbers This requires a lot of puzzling, but can be done. When successful, combine them into one 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.

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. The square attributed to Euler is in fact due to William Beverley, published in the Philosophical Magazine 1848. In this 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.[44]

### The construction of a magic square using genetic algorithms

A magic square can be constructed using genetic algorithms.[45] 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.

## Solving partially completed magic squares

Similar to the Sudoku and KenKen puzzles, solving partially completed has become a popular mathematical puzzle. Puzzle solving centers on analyzing the initial given values and possible values of the empty squares. One or more solution arises as the participant uses logic and permutation group theory to rule out all unsuitable number combinations.

## Variations of the magic square

### Extra constraints

Euler diagram of requirements of some types of 4×4 magic squares. Cells of the same colour sum to the magic constant. * In 4×4 most-perfect magic squares, any 2 cells that are 2 cells diagonally apart (including wraparound) sum to half the magic constant, hence any 2 such pairs also sum to the magic constant.

Certain extra restrictions can be imposed on magic squares. If not only the main diagonals but also the broken diagonals sum to the magic constant, the result is a panmagic square.

If raising each number to the nth 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 heterosquares and antimagic squares, the 2n + 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, because the logarithm of the product of 2 numbers is the sum of logarithm of each. Alternatively, if any 3 numbers in a line are 2a, 2b and 2c, their product is 2a+b+c, which is constant if a+b+c is constant, as they would be if a, b and c were taken from ordinary (additive) magic square.[46] For example, the original Lo-Shu magic square becomes:

M = 32768
16 512 4
8 32 128
256 2 64

Other examples of multiplicative magic squares include:

### 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[47] belonging to ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \mathbb {R} }$. The product is: -352,507,340,640 - 400,599,719,520 i.

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

### Additive-multiplicative magic and semimagic squares

Additive-multiplicative magic squares and semimagic squares satisfy properties of both ordinary and multiplicative magic squares and semimagic squares, respectively.[48]

 First known additive-multiplicative magic square 8×8 found by W. W. Horner in 1955 Sum = 840 Product = 2 058 068 231 856 000 162 207 51 26 133 120 116 25 105 152 100 29 138 243 39 34 92 27 91 136 45 38 150 261 57 30 174 225 108 23 119 104 58 75 171 90 17 52 216 161 13 68 184 189 50 87 135 114 200 203 15 76 117 102 46 81 153 78 54 69 232 175 19 60
 Smallest known additive-multiplicative semimagic square 4×4 found by L. Morgenstern in 2007 Sum = 247 Product = 3 369 600 156 18 48 25 30 144 60 13 16 20 130 81 45 65 9 128

It is unknown if any additive-multiplicative magic squares smaller than 8×8 exist, but it has been proven that no 3×3 or 4×4 additive-multiplicative magic squares and no 3×3 additive-multiplicative semimagic squares exist.[49]

### 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 dodecahedrons, magic triangles[50]magic stars, and magic hexagons. Going up in dimension results in magic cubes and other magic hypercubes.

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}.[50]

A geometric magic square.

### Other component elements

Magic squares may be constructed which contain geometric shapes instead of numbers. Such squares, known as geometric magic squares, were invented and named by Lee Sallows in 2001.[51]

### 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.

The first linear area magic square

### Area Magic Squares

In 2017, following initial ideas of William Walkington and Inder Taneja, the first linear area magic square (L-AMS) was constructed by Walter Trump.[52]

## Related problems

Over the years, many mathematicians, including Euler, 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 primes, in this case nine Chen primes:

 17 89 71 113 59 5 47 29 101

The Green-Tao theorem implies that there are arbitrarily large magic squares consisting of primes.

### n-Queens problem

In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa.[53]

### Enumeration of magic squares

Semi-log plot of Pn, the probability of magic squares of dimension n

As mentioned above, the set of normal squares of order three constitutes a single equivalence class-all equivalent to the Lo Shu square. Thus there is basically just one normal magic square of order 3. But the number of distinct normal magic squares rapidly increases for higher orders.[54] There are 880 distinct magic squares of order 4 and 275,305,224 of order 5.[55] These squares are respectively displayed on 255 magic tori of order 4, and 251,449,712 of order 5.[56] The number of magic tori and distinct normal squares is not yet known for any higher order.[57]

Algorithms tend to only generate magic squares of a certain type or classification, making counting all possible magic squares quite difficult. Traditional counting methods have proven unsuccessful, statistical analysis using the Monte Carlo method has been applied. The basic principle applied to magic squares is to randomly generate n × n matrices of elements 1 to n2 and check if the result is a magic square. The probability that a randomly generated matrix of numbers is a magic square is then used to approximate the number of magic squares.[58]

More intricate versions of the Monte Carlo method, such as the exchange Monte Carlo, and Monte Carlo Backtracking have produced even more accurate estimations. Using these methods it has been shown that the probability of magic squares decreases rapidly as n increases. Using fitting functions give the curves seen to the right.

## Magic squares in popular culture

In Goethe's Faust, the witch's spell used to make a youth elixir for Faust, the Hexen-Einmal-Eins (de), has been interpreted as a construction of a magic square.

Macau stamps featuring magic squares

On October 9, 2014 the post office of Macao in the People's Republic of China issued a series of stamps based on magic squares.[59] The figure below shows the stamps featuring the nine magic squares chosen to be in this collection.[60]

The metallic artifact at the center of The X-Files episode "Biogenesis" is alleged by Chuck Burks to be a magic square.[61][62]

Mathematician Matt Parker attempted to create a 3x3 magic square using square numbers in a YouTube video on the Numberphile channel. His failed attempt is known as the Parker Square.

The first season Stargate Atlantis episode "Brotherhood" involves completing a magic square as part of a puzzle guarding a powerful Ancient artefact.

## Notes

1. ^ Miller, Jeff (September 3, 2016). "Earliest Known Uses of Some of the Words of Mathematics (M)".
2. ^ Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. p. 130.
3. ^ Hawley, Del (February 2011) [September 1998]. "Magic Squares II". NRICH.
4. ^ Allenby, R. B. J. T. (1995). Linear Algebra. p. 97.
5. ^ Richardson, William H. "Non-Normal Magic Squares". Wichita State University.
6. ^ Adler, Allan; Alejandre, Suzanne. "Why there are no 2x2 magic squares". mathforum.org.
7. ^ Sallows, Lee (Fall 1997) [09 January 2009]. "The lost theorem". The Mathematical Intelligencer. 19 (4): 51-54. doi:10.1007/BF03024415.
8. ^ Swetz, Frank (2008). Legacy of the Luoshu: The 4,000 Year Search for the Meaning of the Magic Square of Order Three. CRC Press. p. 67.
9. ^ a b c Loly, Peter (March 2004) [1 August 2016]. "The invariance of the moment of inertia of magic squares" (PDF). Mathematical Gazette. 88 (511): 151-153. doi:10.1017/S002555720017456X.
10. Swaney, Mark. "Mark Swaney on the History of Magic Squares". Archived from the original on 2004-08-07.
11. ^ The most famous Arabic book on magic, named "Shams Al-ma'arif (Arabic: ? ??), for Ahmed bin Ali Al-boni, who died about 1225 (622 AH). Reprinted in Beirut in 1985
12. ^ Swetz, Frank J. (2008). The Legacy of the Luoshu (2nd ed.). A.K. Peters/CRC Press.
13. Yoke, Ho Peng (2008). Magic Squares in China. Encyopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2 ed.). Springer. pp. 1252-1259.
14. ^ Andrews, William Symes (1917). Magic Squares and Cubes (2nd ed.). Open Court Publishing Company. p. 122.
15. Cammann, Schuyler (April 1960). "The Evolution of Magic Squares in China" (PDF). Journal of the American Oriental Society. 80 (2): 116-124.
16. ^ 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.
17. ^ Helaine Selin, Ubiratan D'Ambrosio, Mathematics Across Cultures: The History of Non-Western Mathematics, Published by Springer, 2001, ISBN 1-4020-0260-2, p. 160.
18. ^ Sesiano, J., Ab?al-Waf?\rasp's treatise on magic squares (French), Z. Gesch. Arab.-Islam. Wiss. 12 (1998), 121-244.
19. ^ Peter, J. Barta, The Seal-Ring of Proportion and the magic rings (2016), pp. 6-9.
20. ^ J?bir ibn Hayy?n, Book of the Scales. French translation in: Marcelin Berthelot (1827-1907), Histoire de sciences. La chimie au moyen âge, Tom. III: L'alchimie arabe. Paris, 1893. [rprt.. Osnabruck: O. Zeller, 1967], pp. 139-162, in particular: pp. 150-151
21. ^ al-Ghaz?l?, Deliverance From Error (al-munqidh min al-?al?l ) ch. 145. Arabic: al-Munkidh min al-dalal. ed. J. Saliba - K. Ayyad. Damascus: Maktab al-Nashr al-'Arabi, 1934, p. 79. English tr.: Richard Joseph McCarthy, Freedom and Fulfillment: An annotated translation of al-Ghazali's al-Munkidh min al-Dalal an other relevant works of al-Ghazali. Boston, Twayer, 1980. He refers a book titled 'The Marvels of Special Properties' as his source. This square was named in the Orient as the ,,Seal of Ghazali" after him.
22. ^ This tradition about a series of magic squares from order three to nine, which are associated with the seven planets, survives in Greek, Arabic, and Latin versions. The Latin version is Liber de septem figuris septem planetarum figurarum Geberi regis Indorum. This treatise is the identified source of Dürer and Heinrich Cornelius Agrippa von Nettesheim. Cf. Peter, J. Barta, The Seal-Ring of Proportion and the magic rings (2016), pp. 8-9, n. 10
23. ^ a b c d Datta, Bibhutibhusan; Singh, Awadhesh Narayan (1992). Magic Squares in India (PDF). Indian Journal of History of Science. 27. pp. 51-120.
24. Hayashi, Takao (2008). Magic Squares in Indian Mathematics. Encyopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2 ed.). Springer. pp. 1252-1259.
25. ^ a b Hayashi, Takao (1987). "Varahamihira's Pandiagonal Magic Square of the Order Four" (PDF). Historia Mathematica. 14 (2): 159-166.
26. ^ Andrews, William Symes (1917). Magic Squares and Cubes (2nd ed.). Open Court Publishing Company. pp. 124-126.
27. ^ Magic Squares of Manuel Moschopoulos
28. ^ See Alfonso X el Sabio, Astromagia (Ms. Reg. lat. 1283a), a cura di A.D'Agostino, Napoli, Liguori, 1992
29. ^ 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)
30. ^ 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."
31. ^ 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
32. ^ 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)
33. ^ 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)
34. ^ 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.
35. ^ Drury, Nevill (1992). Dictionary of Mysticism and the Esoteric Traditions. Bridport, Dorset: Prism Press. ISBN 1-85327-075-X.
36. ^ "The Book of Power: Cabbalistic Secrets of Master Aptolcater, Mage of Adrianople", transl. 1724. In Shah, Idries (1957). The Secret Lore of Magic. London: Frederick Muller Ltd.
37. ^ "Archived copy". Archived from the original on 2005-11-09. Retrieved .
38. ^ "Magic cube with Dürer's square" Ali Skalli's magic squares and magic cubes
39. ^ "Magic cube with Gaudi's square" Ali Skalli's magic squares and magic cubes
40. ^ Structure of Magic and Semi-Magic Squares, Methods and Tools for Enumeration
41. ^ Pinn, K.; Wieczerkowski, C. (1998). "Number of Magic Squares From Parallel Tempering Monte Carlo". Int. J. Mod. Phys. C. 9: 541. arXiv:. doi:10.1142/s0129183198000443.
42. ^ "Number of Magic Squares From Parallel Tempering Monte Carlo, arxiv.org, April 9, 1998. Retrieved November 2, 2013.
43. ^ Mathematical Circles Squared By Phillip E. Johnson, Howard Whitley Eves, p.22
44. ^ Hartley, M. "Making Big Magic Squares".
45. ^ Evolving a Magic Square using Genetic Algorithms
46. ^ Stifel, Michael (1544), Arithmetica integra (in Latin), pp. 29-30.
47. ^ "8x8 multiplicative magic square of complex numbers" Ali Skalli's magic squares and magic cubes
48. ^
49. ^ "MULTIMAGIE.COM - Smallest additive-multiplicative magic square". Retrieved 2015.
50. ^ a b Magic Designs, Robert B. Ely III, Journal of Recreational Mathematics volume 1 number 1, January 1968
51. ^ Magic squares are given a whole new dimension, The Observer, April 3, 2011
52. ^ "Area Magic Squares". Futility Closet. 2017-01-19. Retrieved .
53. ^ Demirörs, O.; Rafraf, N.; Tanik, M. M. "Obtaining n-queens solutions from magic squares and constructing magic squares from n-queens solutions". Journal of Recreational Mathematics. 24 (272-280): 1992.
54. ^ How many magic squares are there? by Walter Trump, Nürnberg, January 11, 2001
55. ^
56. ^
57. ^ Anything but square: from magic squares to Sudoku by Hardeep Aiden, Plus Magazine, March 1, 2006
58. ^ Kitajima, Akimasa; Kikuchi, Macoto; Altmann, Eduardo G. (14 May 2015). "Numerous but Rare: An Exploration of Magic Squares". PLOS ONE. 10 (5): e0125062. doi:10.1371/journal.pone.0125062. PMC . PMID 25973764.
59. ^ Macau Post Office web site Archived 2014-11-11 at the Wayback Machine.
60. ^ Macau's magic square stamps just made philately even more nerdy The Guardian Science, November 3, 2014
61. ^ Michelle Erica Green (June 15, 1997). "Biogenesis on The X-Files". littlereview.com. The Little Review. Retrieved 2017. Moreover, it's a magic square, a pattern in which God supposedly instructed the early Hebrews to gain power from names or their numeric equivalents.
62. ^ Zack Handlen (November 17, 2012). "The X-Files: "Biogenesis" / Millennium: "Goodbye To All That"". The A.V. Club. The Onion, Inc. Retrieved 2017. I love when they bring the nerdy FBI guy in to explain the concept of "the magic square," which he does by telling us that magic squares have been around for a while, and then nothing else. Unless I missed something, all I have at this point is that magic squares are squares that people once thought were magic.