# Ancient puzzle gets new lease of 'geomagical' life

##### 点击量： 时间：2019-03-07 07:20:01

By Jacob Aron An ancient mathematical puzzle that has fascinated mathematicians for centuries has found a new lease of life. The magic square is the basis for Sudoku, pops up in Chinese legend and provides a playful way to introduce children to arithmetic. But all this time it has been concealing a more complex geometrical form, says recreational mathematician Lee Sallows. He has dubbed the new kind of structure the “geomagic square”, and recently released dozens of examples online. Click here to see a gallery of geomagic squares. “To come up with this after thousands of years of study of magic squares is pretty amazing,” blogged Alex Bellos, author of the book Alex’s Adventures in Wonderland. Peter Cameron, a mathematician at Queen Mary, University of London, believes that an even deeper structure may lie hidden beyond geomagic squares. “I can immediately see a lot of things I’d like to do with this,” he says. The traditional magic square is a square grid of numbers arranged so that each row, column and diagonal adds up to the same total. For example, in this magic square, that total is 15: 4 9 2 3 5 7 8 1 6 In fact, this particular magic square has been around for thousands of years – Chinese legend calls it the Lo Shu and claims it was found carved into the shell of a turtle. Now Sallows, who lives in the Netherlands and says he has spent about 10 years playing around with geometric versions of magic squares, has shown how it is possible to extend the idea in entirely new ways. In his geomagic square, each digit of the grid is replaced by a Tetris-like shape called a polyomino, which is made up of different numbers of identical squares. Crucially, there must be a way to combine the polyominos in each row, column and diagonal to build a single master shape (see picture). The bricks can be in two, three, or, in theory, even more dimensions – though visually representing a 4D geomagic square would be challenging. Sallows’s first attempt at a geomagic square was based on a formula for creating magic squares devised by the 19th-century French mathematician Édouard Lucas. Sallows found that applying the formula to shapes didn’t quite produce the result he was looking for, but it inspired him to develop a series of computer programs to help him build dozens of true geomagic squares. Among those he’s generated is a geomagic version of the Lo Shu, in which each digit is represented by a polyomino consisting of that number of smaller squares (see picture). But while any magic square can be represented geometrically, he explains, the reverse is not true. “Magic squares are not numerical, they’re geometrical objects. They were only seen like that before because we always represented them with numbers.” Sallows describes his discovery as a “Copernican revolution in our understanding of magic squares”. So could geomagic squares have applications outside the study of puzzles? Cameron certainly thinks so. “You can ask these questions in much more general terms,” he says. For example, the concepts behind geomagic squares might be applied in a more abstract way in the fields of set and group theory, where you can examine the mathematical properties of hypothetical objects without reference to their physical form. He outlines this idea in more detail on his blog. Geomagic squares might even be put to work in the real world. A variant of the magic square known as the Latin square is already used to help create codes for transmitting information and in the design of drug trials, where it can be used to check that participants receive the right combination of treatments. Sudoku is also a particular type of Latin square. Cameron speculates that a “geolatin” square – if such a thing exists – might also have applications. In the meantime, Sallows is happy to keep exploring the geomagic world. Placing the work online has already helped him achieve his long-standing goal of finding a 2 × 2 geomagic square. Smaller squares are harder than larger ones, because the larger squares give you more options – and Sallows had been drawing a blank. But shortly after his site went live, fellow square-hunter Frank Tinkelenberg sent him an example. And the search doesn’t end there, Sallows is now looking for a geomagic square in which the master shape is smooth, without any gaps for missing cells. “As soon as you find one that has the properties you’re looking for, you’re on to the next challenge,” he explains. See more: