When was magic squares invented

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 when he was a clerk of the Pennsylvania Assembly.

These two order eight squares are reproduced on pages and 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 Writing in 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. 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 , first described all possible magic squares of order 4; these results, Dudeney assures us, "have been verified over and over again.

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. 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: " 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. The Lo Shu is the first known magic square, and has served as a cosmic map uniting people with heaven, a talisman for good fortune, a key to medieval alchemists' attempts at transmuting lead into gold, an aid to childbirth, an astrological means of communicating with the planet Saturn, and so on.

Other magic squares can easily be derived from the Lo Shu: Through a series of rotations and reflections about its major axes, seven other third-order magic squares can be derived from the Lo Shu. Find them. If all the terms of the Lo Shu are multiplied by a constant or if the same number is either added to, or subtracted from, each term, a new third-order magic square results.

When two third-order magic squares are added term by term, a new magic square is formed. If the Lo Shu is considered a matrix and is muliplied by itself three times, a new magic square is formed. Durer's etching is shown below with a blow up of the magic square beneath it. A famous magic square using palindromes was found on an ancient Roman ruin in Cirencester, England see figure below. Loosely translated, it is "Arepo, the mechanic, guides the work of the wheels.

Peter Cameron, professor of mathematics at Queen Mary, University of London, calls geomagic squares "a wonderful new piece of recreational maths, which will delight non-mathematicians and give mathematicians food for thought". Despite the inherent whimsy of the field, he said: "I have no doubt that there is serious maths to be done here, too.

Perhaps no other area of non-practical mathematics has been so popular for so long as magic squares. Their story begins 4, years ago in China, where, according to legend, a turtle crept out of the Yellow River.

The reptile is said to have had dots on its underside positioned in such a way as to make the 3x3 square described above. The Chinese called this square the lo shu , and gave it spiritual importance, believing that it encapsulated the harmonies of the universe. Feng shui, the Chinese system of arranging objects, such as furniture in homes, is in part based on the lo shu.

But veneration of magic squares was not confined to the Chinese. In India amulets with magic squares were worn as protective charms, in Turkey virgins embroidered magic squares on the shirts of warriors and in western Europe Renaissance astrologers equated them with planets.

This square is particularly amazing. Not only do the rows, lines and diagonals add up to 34, but the four corners, the four digits in the central square, and the four digits in the top left, top right, bottom left and bottom right quarters do too. There are many other combinations of four numbers in the square that add up to 34, and it is fun looking for them. The most notable aficionado was United States founding father Benjamin Franklin, who liked to spend his spare time constructing particularly innovative variations.

In one evening in his 40s he composed a 16x16 square that he claimed was "the most magically magical of any magic square ever made by any magician". Sallows, aged 66, is very much an enthusiastic amateur. He left school at 17 with no qualifications and his mathematics is entirely self-taught. He was an electronics engineer in the non-academic staff at the University of Nijmegen until he retired two years ago.

While it is certainly remarkable that a non-mathematician has given this established field a new lease of life, it is perhaps only to be expected, since most academic mathematicians would now consider magic squares as too frivolous to occupy their time.

So, as above, the 3x3 has every number from one to nine and the 4x4 every number from one to Sallows says he was instantly "turned on by the symmetries" of magic squares, and once he had got the bug he began to invent new rules and modifications. He is celebrated for inventing the "alphamagic square", which is doubly magic:.

The rows, columns and diagonals add up to 45 when considering the meaning of the words. But when considering the number of letters in each word — so five is 4 and twenty-two is 9 — this also generates a magic square, whose rows, columns and diagonals add up to Another fantastic property of this square is that the word lengths consist of the consecutive numbers from 3 to Sallows gained a certain amount of fame in the recreational maths world for his alphamagic square, but dismisses it as "zany, a bit silly".

He was more excited by looking at the algebra behind magic squares. Sallows was playing around with the Lucas formula: "And then it came to me — why don't I represent the variables by shapes? A 'plus' could be appending, and a 'minus' excising. From that everything else followed. It is absolutely incredible that nobody else had thought of it before.

I knew at once I had landed on a really important idea. My mind was crowded with possibilities. As we saw above, the rows, columns and diagonals of a magic square add up to the same number. If each number in the square is represented by a line of that length, then it follows that these line segments can be joined head to tail to form a larger line — and this line will have the same length whichever row, column or diagonal you choose.

Similarly, if each number n is replaced by a shape that has area n, it follows that the shapes on each row, column and diagonal when put together will have the same combined area. Yet Sallows wanted the extra condition that the shapes fit together so that each row, column and diagonal made the exact same target shape.