## Friday, 24 January 2014

### Magic Square

In recreational mathematics, a magic square is an arrangement of numbers (usually integers) in a 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 n2 numbers, and its size (the number of rows [and columns] it has) is described as being "of order n".[1] A magic square that contains the integers from 1 to n2 is called a normal magic square. (The term "magic square" is also sometimes used to refer to any of various types of word squares.)
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).
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:
$M = \frac{n(n^2+1)}{2}.$
For example, if n = 3, the formula says M = [3 (32 + 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 A006003 in the OEIS)

## History

Iron plate with an order 6 magic square in Arabic numbers from China, dating to the Yuan Dynasty (1271–1368).
Magic squares were known to Chinese mathematicians as early as 650 BCE,[2] and to Arab mathematicians possibly as early as the 7th century CE, when the Arabs conquered northwestern parts of the Indian subcontinent and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics.[citation needed] 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.[2] Some of these squares were later used in conjunction with magic letters, as in (Shams Al-ma'arif), to assist Arab illusionists and magicians.[3]

### Lo Shu square (3×3 magic square)

Chinese literature dating from as early as 650 BCE tells the legend of Lo Shu or "scroll of the river Lo".[2] 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 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 cycles of the 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.
 4 9 2 3 5 7 8 1 6
The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.
The Square of Lo Shu is also referred to as the Magic Square of Saturn.

### 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.[4] 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.[5] 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.[6]

### Arabia

Magic squares were known to Islamic mathematicians in Arabia as early as the 7th century CE. 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 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.[2]
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.[2]

### India

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.[7]
 7 12 1 14 2 13 8 11 16 3 10 5 9 6 15 4
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.
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.
          Alternate representation of Chautisa Yantra

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

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

### 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
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.[8] 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.[9] In that text, each magic square is assigned to the respective planet, as in the Islamic literature.[10] 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.[11] 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.[12] 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.[13] 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).[14] The magical operations involve engraving the appropriate square on a plate made with the metal assigned to the corresponding planet,[15] 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 immortalizes a 4×4 square in his famous engraving "Melancholia I".
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.[2][16]
Saturn=15
492
357
816
Jupiter=34
414151
97612
511108
162313
Mars=65
11247203
41225816
17513219
101811422
23619215
Sol=111
632334351
7112728830
191416152324
182022211713
25291092612
365334231
Venus=175
2247164110354
5234817421129
3062449183612
1331725431937
3814321264420
213983322745
461540934328
Mercury=260
858595462631
4915145253111056
4123224445191848
3234352928383925
4026273736303133
1747462021434224
955541213515016
642361606757
Luna=369
37782970216213545
63879307122631446
47739803172235515
16488408132642456
57174994173336525
26581850142743466
67275910512437535
36681960115234476
77286920611253445

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;[17] 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:
 S A T O R A R E P O T E N E T O P E R A R O T A S
 6 66 848 938 8 11 544 839 1 11 383 839 2 73 774 447
 H E S E B E Q A L S E G B
 A D A M D A R A A R A D M A D A

### Albrecht Dürer's magic square

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 queens in the two solutions of the 4 queens puzzle [18]), 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.
 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.[19]
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.[20]

## 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.[21]
 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 (sequence A006052 in OEIS). The number for n = 6 has been estimated to be (0.17745 ± 0.00016) × 1020.[22][23]

### 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:
 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

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.[24] 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.
step 1
1

step 2
1

2
step 3
1
3
2
step 4
1
3
42
step 5
1
35
42
step 6
16
35
42
step 7
16
357
42
step 8
816
357
42
step 9
816
357
492
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.

Order 3
816
357
492
Order 5
17241815
23571416
46132022
101219213
11182529
Order 9
47586980112233445
57687991122334446
67788102132435456
77718203142535566
61719304152636576
16272940516264755
26283950617274415
36384960717331425
37485970812132435
The following formulae help construct magic squares of odd order
Order $n$
Squares (n)Last no.Middle no.Sum (M)Ith row and Jth column no.
$n$${n^2}$$\frac{n^2+1}{2}$$\left(\frac{n^2+1}{2}\right) n$$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)
5251365
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 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.
M = Order 4
14
67
1011
1316
M = Order 4
115144
12679
810115
133216
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.)
M = Order 4
1234
5678
9101112
13141516
M = Order 4
16151413
1211109
8765
4321
M = Order 4
1001
0110
0110
1001
M = Order 4
115144
12679
810115
133216
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+9y where x is the original number and y is a number from 0 to 3, following the pattern of the medjig-square.
Example:
Order 3
816
357
492
Medjig 3 × 3
230202
103131
311220
020331
322002
013113
Order 6
2635119624
17828103315
30121423257
3215323416
3122279220
41336181129
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 p = an + r, with r chosen from {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 (0,..., n−1), and one for the r-numbers (1,...,n). 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.
Order 8, sum 260
624135146202935
559541221433828
521436136301945
115360627374422
642154948183133
757561023414026
501616334321747
95558825394224
Order 12, sum 870
138817127114324110390566579
191251406431011163067779254
128187137104423111380665589
513912620291151024453917868
1361015129112343910588586381
21123142445991182869759452
130169135106403311182645787
314112422271171004651937670
1341213131110363710786606183
23121144247971202671739650
1321411133108383510984625985
11431222425119984849957472
Order 12, sum 870
11201214885727360972425144
14227229958757087461231183
1111013138956283501071435134
136331610552816493401291129
811312841926580531041732137
138311810354796691421271147
511612544896877561012029140
139301910255786790431261156
1210913237966184491081336133
1353415106518263943913011110
21191224786717459982326143
141282110057766988451241174
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.[25]

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

A magic square can be constructed using genetic algorithms.[26] 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 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. For example, the original Lo-Shu magic square becomes:
M = 32768
165124
832128
256264
Other examples of multiplicative magic squares include:
M = 216
2912
3661
3418
M = 6720
162056
402823
145244
128710
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.
Skalli multiplicative 7 × 7
2750668413232
2452340547011
569204436656
5572911163630
4244560771226
1022483954863
787818403360

### 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[27] belonging to $\mathbb C$ 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 $\mathbb R$. 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−105−217i16+50i4−14i14−8i
63−35i28+114i−14i2+6i3−11i211+357i−123−87i
31−15i13−13i−103+69i−261−213i49−49i−46+2i−6+2i
102−84i−28−14i43+247i−10−2i5+9i31−27i−77+91i
−22−6i7+7i8+14i50+20i−525−492i−28−42i−73+17i
54+68i138−165i−56−98i−63+35i4−8i2−4i70−53i
24+22i−46−16i6−4i17+20i110+160i84−189i42−14i

### 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 dodecahedrons, magic triangles[28] magic stars, and magic hexagons. Going up in dimension results in magic cubes, magic tesseracts 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}.[28]

### Other component elements

Geomagic square

Geomagic square
Magic squares may be constructed which contain geometric shapes rather than numbers, as in the geomagic squares introduced by Lee Sallows.[29]

### 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, 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.
Using Ali Skalli's non-iterative method of magic squares construction, it is easy to create magic squares of primes[30] 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.
Skalli Primes 5 × 5
20872633280327533389
28432729334720992647
33592113268728192687
26632777269933732153
27133413212926212789
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 n-queens solutions, and vice versa.[31]

## Notes

1. Jump up ^ "Magic Square" by Onkar Singh, Wolfram Demonstrations Project.
2. Swaney, Mark. [1].
3. Jump up ^ The most famous Arabic book on magic, named "Shams Al-ma'arif (Arabic: كتاب شمس المعارف‎), for Ahmed bin Ali Al-boni, who died about 1225 (622 AH). Reprinted in Beirut in 1985
4. Jump up ^ 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.
5. Jump up ^ Helaine Selin, Ubiratan D'Ambrosio, Mathematics Across Cultures: The History of Non-western Mathematics, Published by Springer, 2001, ISBN 140200260, p. 160.
6. Jump up ^ Sesiano, J., Abūal-Wafā\rasp's treatise on magic squares (French), Z. Gesch. Arab.-Islam. Wiss. 12 (1998), 121–244.
7. Jump up ^ Magic Squares and Cubes By William Symes Andrews, 1908, Open court publish company
8. Jump up ^ Manuel Moschopoulos – Mathematics and the Liberal Arts
9. Jump up ^ See Alfonso X el Sabio, Astromagia (Ms. Reg. lat. 1283a), a cura di A.D'Agostino, Napoli, Liguori, 1992
10. Jump up ^ 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)
11. Jump up ^ 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."
12. Jump up ^ 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
13. Jump up ^ 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)
14. Jump up ^ 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)
15. Jump up ^ 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.
16. Jump up ^ Drury, Nevill (1992). Dictionary of Mysticism and the Esoteric Traditions. Bridport, Dorset: Prism Press. ISBN 1-85327-075-X.
17. Jump up ^ "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.
19. Jump up ^ "Magic cube with Dürer's square" Ali Skalli's magic squares and magic cubes
20. Jump up ^ "Magic cube with Gaudi's square " Ali Skalli's magic squares and magic cubes
21. Jump up ^ Structure of Magic and Semi-Magic Squares, Methods and Tools for Enumeration
22. Jump up ^ Pinn K. and Wieczerkowski C., (1998) "Number of Magic Squares From Parallel Tempering Monte Carlo", Int. J. Mod. Phys. C 9 541
23. Jump up ^ "Number of Magic Squares From Parallel Tempering Monte Carlo, arxiv.org, April 9, 1998. Retrieved November 2, 2013.
24. Jump up ^ Mathematical Circles Squared By Phillip E. Johnson, Howard Whitley Eves, p.22
25. Jump up ^ Hartley, M. "Making Big Magic Squares".
26. Jump up ^ Evolving a Magic Square using Genetic Algorithms
27. Jump up ^ "8x8 multiplicative magic square of complex numbers" Ali Skalli's magic squares and magic cubes
28. Magic Designs,Robert B. Ely III, Journal of Recreational Mathematics volume 1 number 1, January 1968
29. Jump up ^ Magic squares are given a whole new dimension, The Observer, April 3, 2011
30. Jump up ^ "magic square of primes" Ali Skalli's magic squares and magic cubes
31. Jump up ^ 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