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:

${\displaystyle M={\frac {n(n^{2}+1)}{2}}.}$

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)

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]

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 (禹) was trying to channel the water out to sea, a turtle emerged from it with a curious figure / pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15, which is also the number of days in each of the 24 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.

File:Magic square 4x4 Shams Al-maarif2 (Arabic magics).jpg

File:Magic square 4x4 Shams Al-maarif3 (Arabic magics).jpg

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]

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]

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.

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

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]

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:

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]

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]

The number of different n×n magic squares for n from 1 to 5, not counting rotations and reflections:

1, 0, 1, 880, 275305224 (sequence A006052 in the OEIS).

The number for n = 6 has been estimated to be around 1.7745×10¹⁹.^{[citation needed]}

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.^[22] 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.

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.

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.

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).

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.

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.

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.^[23]

A magic square can be constructed using genetic algorithms.^[24] In this process an initial population of magic squares with random values are generated. The fitness scores of these individual magic squares are calculated based on the degree of deviation in the sums of the rows, columns, and diagonals. The population of magic squares reproduce by exchanging values, together with some random mutations. Those squares with a higher fitness score are more likely to reproduce. The next generation of the magic square population is again calculated for their fitness, and this process continues until a solution has been found or a time limit has been reached.

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.

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.

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:

Other examples of multiplicative magic squares include:

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
27	50	66	84	13	2	32
24	52	3	40	54	70	11
56	9	20	44	36	65	6
55	72	91	1	16	36	30
4	24	45	60	77	12	26
10	22	48	39	5	48	63
78	7	8	18	40	33	60

Still using Ali Skalli's non iterative method, it is possible to produce an infinity of multiplicative magic squares of complex numbers^[25] 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

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^[26] 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}.^[26]

Magic squares may be constructed which contain geometric shapes rather than numbers, as in the geomagic squares introduced by Lee Sallows.^[27]

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.

Over the years, many mathematicians, including Euler, Cayley and Benjamin Franklin have worked on magic squares, and discovered fascinating relations.

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^[28] 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
2087	2633	2803	2753	3389
2843	2729	3347	2099	2647
3359	2113	2687	2819	2687
2663	2777	2699	3373	2153
2713	3413	2129	2621	2789

It is believed that an infinite number of Skalli's magic squares of prime exist, but no demonstration exists to date. However, it is possible to easily produce a considerable number of them, not calculable in the absence of demonstration.

In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa.^[29]

• Charney, Noah The Art Thief Atria (2007), a novel with a key plot point involving a magic square.
• McCranie, Judson (1988). "Magic Squares of All Orders". Mathematics Teacher: 674–78. {{cite journal}}: Unknown parameter |month= ignored (help)
• Semanisinova, Ingrid; Trenkler, Marian (2007). "Discovering the Magic of Magic Squares". Mathematics Teacher. 101 (1): 32–39. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: multiple names: authors list (link)
• King, J. R. (1963). "Magic Square Numbers". {{cite journal}}: Cite has empty unknown parameter: |month= (help); Cite journal requires |journal= (help)

• Eaves, Laurence (2009). "Magic Square". Sixty Symbols. Brady Haran for the University of Nottingham.
• Magic square at Curlie