Lo Shu: A traditional magic square



In a traditional magic square, like the Lo Shu shown here, a square grid of numbers is arranged so that each row, column and diagonal adds up to the same total. Here, that total is 15.

Geomagic square: Inner cell missing



In a geomagic square, each digit is replaced by a "polyomino" made up of different numbers of identical squares. There must be a way to combine the polyominos in each row, column and diagonal to build a single master shape, or target.



This is a "normal" 3 × 3 geomagic square, meaning that the polyominos form the natural progression "1, 2, 3…" It is one of 4370 normal geomagic squares, not including rotations and reflections, that can be formed for which the target is a 4 × 4 square with a missing inner cell. Advertisement

Geomagic square: Corner cell missing



A second normal geomagic square, this time the target is a 4 × 4 square missing one corner cell. It is one of 27,110 normal squares with the same target. By comparison, there are 16,465 normal squares for which the 4 × 4 target is missing an edge cell.

Rare square



In this geomagic square, all the polyominos have the same area. This type is much rarer than those formed using unequal polyominos.

Geomagic squares go 3D



In this square, the target is a 3 × 3 × 3 cube. Note that the polyomino size forms the consecutive series of odd numbers "1, 3, 5, 7, 9, 11, 13, 15, 17" while their shapes are derived from a formula for creating magic squares devised by the 19th-century French mathematician Édouard Lucas.



Unlike the previous geomagic squares in this gallery, this one was not devised by Lee Sallows, who first came up with the concept. Instead, another recreational mathematician, Aad van de Wetering of Driebruggen, the Netherlands, submitted it to the Dutch mathematics periodical Pythagoras.

Impossible geomagic



Some geomagic targets really are 3D. But in this one, the target is an "impossible figure", a two-dimensional object that the visual system interprets as a projection of a 3D object, even though such an object can't actually exist.