The earliest known magic square is Chinese, recorded around 2800 B.C. Fuh-Hi described the "Loh-Shu", or "scroll of the river Loh". It is a typical 3x3 magic square except that the numbers were represented by patterns not numerals.
Although this may be the first record, it seems likely that others played with numbers to make the "first" magic square. Probably, many early humans discovered them independently. They may have played with piles of stones on a pattern in the sand or they may have stacked nuts on leaves layed out as a square grid. It seems somewhat improbable that it required a single mathematical genius in 2800 B.C. to develop the first simple 3x3 magic square.
Since then, certainly, many people in many nations have enjoyed, studied, and recorded magic squares. For further details of early records see See Mark Farrar's Website, and David Singmaster's Chronology of Recreational Mathematics
The best known early square is probaby the 4x4 magic square depicted in 1514 in Albrecht Dürer's woodcut "Melancholia". The square is magic but not pan-magic. Only two of the broken diagonals are magic.
Amongst our forebears, some very distinguished names have played with magic squares. Benjamin Franklin is one example. He played with the construction of magic squares in 1736-37 when he was a clerk of the Pennsylvania Assembly. These two order eight squares are reproduced on pages 394 and 395 of the Papers of Benjamin Franklin Volume 3.
Neither of these squares is Pan-magic, but a Spreadsheet Analysis of his squares allowed the above examples of his squares to be colored. The resulting pattern makes it clear that he created them with a logical and structured underlying scheme. Benjamin Franklin started his squares conventionally at 1, so the magic sum for his squares is 260.
Writing in 1917 Dudeney said: "Of recent years many ingenious methods have been devised for the construction of magics (magic squares), and the law of their formation is so well understood that all the ancient mystery has evaporated and there is no longer any difficulty in making squares of any dimensions. Almost the last word has been said on this subject." (1) Almost, apparently, but not quite! He promptly invalidated his own prediction because he continued, himself, to write sufficient additional words on magic squares to cover several more pages.
He cites Frenicle who, in 1693, first described all 880 possible magic squares of order 4; these results, Dudeney assures us, "have been verified over and over again." He also describes Bergholt's "general solution" published in Nature in 1910 (May 26) as being "of the greatest importance to students of this subject." He also refers to his own article (The Queen, January 15, 1910) in which he explains his method for creating all 880 different magic squares of order 4.
Dudeney is not, however, especially impressed with the value of classifying magic squares: "A man once said that he divides the human race into two great classes: those that take snuff and those who do not. I am not sure that some of our classifications of magic squares are not almost as valueless." Dudeney does, however, explain one method of classification and explains one of the earlier names given to pan-magic squares. Apparently a Mr. Frost gave the name "Nasik" to pan-magic squares after the town in India in which he lived.
Dudeney describes this type of square as "the most perfect of all" and he employs the adjective "broken" to describe the diagonals which are interrupted by the edge of the square. He beautifully describes the properties of a pan-magic square: "...its properties are such that if you repeat the square in all directions you may mark off a square, 4 x 4, wherever you please, and it will be magic." This is this property which I have called a Magic Carpet. It is a valuable concept and can be applied to complete squares, Latin squares and, when appropriate, to the even smaller component squares.
Dudeney goes on to provide a complete enumeration of the varieties of order 4 squares in which he reports there being 48 "Nasik" (pan-magic) squares. In view of his earlier remarks about marking off a 4 x 4 square wherever you please, this implies that he might have realized that there are actually only three fundamentally unique squares, although he does not say this explicitly. He does go on to clearly explain that each square "will produce seven others by mere reversals and reflections, which we do not count as different."
In his book on mathematical diversion, Martin Gardner (2) devoted a chapter to magic squares. He cites a custom started by Euler (pronounced "Oiler") of employing Latin characters in magic squares (as distinct from Greek characters) and attributes the name "Latin Squares" to this custom. A Latin square may be superimposed on a Greek Square with the property that "...when each Latin letter combines once and once only with each Greek letter.....they are said to be orthogonal squares. The combined square is known as a 'Graeco-Latin' square."
Numerical substitution in such a square results in a family of magic squares. Gardener gives four examples of order 5 Latin squares, including two which have pan-magic properties and are the basis for all the order 5 pan magic squares. He makes no mention of this, however, and goes on to discuss how the achievement of Parker, Bose, and Shrikhande in the late 1950s of creating two orthogonal Latin Squares of order ten had spoiled a conjecture made by Euler that there were no magic squares of order 4n + 2. This team became known as Euler's spoilers for their success.
Gardener also pays tribute to the value of randomly chosen Latin squares in research., e.g., when testing agricultural fertility experiments- the square ensures that each fertilizer is tested in a way which removes bias due to soil conditions.
Some years later, Parker worked with a much faster computer and showed that Euler's conjecture was, in fact, very wide of the mark. The computer found an orthogonal pair for more than half of the Latin squares of order 10 that were fed in to it.
In his book on Magic Squares and Cubes, Andrews (3) describes the construction of pan-magic squares of order 5. He describes the use of the reflected pair of "primitive squares" - identical to the "Latin Squares" of Gardner except that Andrews uses a second series of "Latin" characters instead of Greek characters. He also accurately predicts (4) that the total number of possible pan-magic squares of order five will be 28800.
Andrews (5) also describes De la Hire's method of constructing magic squares. This consists, essentially, of creating two orthogonal Latin squares, one of which is the reflection or the rotation of the other. Appropriate numerical substitution produces the resulting magic squares. Throughout his book, Andrews shows examples where this technique is applied to both odd, even, and "oddly-even" (4n+2) squares. In these examples the underlying squares reflect the size of the square. Thus for an order four square, the Latin squares use the digits 1 through 4 - each digit being used four times. The oddly-even squares were broken down further - to the underlying bit structure. This excellent idea was not applied to the even squares, i.e., not to magic carpets.
All magic squares have at least eight variations: the square can be rotated into four positions and each of these rotations can be reflected - for a total of eight variations of any one unique design. Most magic squares do not remain magic if one border is moved to the opposite edge - the change leaves the main diagonal no longer magic. This process, translocation - repeatedly moving one edge across to the opposite side or the top to the bottom - does not affect pan-magic squares which have, therefore, additional variations. In a 5 x 5 square this is equivalent to moving the starting square through all twenty-five positions - for a total of 25 x 8 = 200 variations. For the order 7 square, each pan-magic square has 49 x 8 = 392 variations and for the size 11 square there are 121 x 8 = 968 variations, i.e., 8 * N^2 where N is the order of the square.
This reduces the problem to determining the number of distinct patterns for each size. This is calculated by knowing the possible number of pan-magic Latin squares for each order. Martin Gardner (2) states that "for any order n there are never more than n - 1 (Latin) squares that are mutually orthogonal." This total includes, however, two squares which cannot be pan-magic, - mutually orthogonal Latin squares constructed using a 1 x 1 knight's move - or the original diagonal technique of De La Hire. Thus when looking for mutually orthogonal Latin squares to make pan-magic squares, there should be N - 3 squares for each order.
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Updated Mar 6, 2010 |