Geometric magic square

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Figure 1:   A geomagic square showing same-sized pieces Geomagic square - 3x3 decominoes.jpg
Figure 1:   A geomagic square showing same-sized pieces

A geometric magic square, often abbreviated to geomagic square, is a generalization of magic squares invented by Lee Sallows in 2001. A traditional magic square is a square array of numbers (almost always positive integers) whose sum taken in any row, any column, or in either diagonal is the same target number. A geomagic square, on the other hand, is a square array of geometrical shapes in which those appearing in each row, column, or diagonal can be fitted together to create an identical shape called the target shape. As with numerical types, it is required that the entries in a geomagic square be distinct. Similarly, the eight trivial variants of any square resulting from its rotation and/or reflection are all counted as the same square. By the dimension of a geomagic square is meant the dimension of the pieces it uses. Hitherto interest has focused mainly on 2D squares using planar pieces, but pieces of any dimension are permitted.



Figure 1 above shows a 3×3 geomagic square. The 3 pieces occupying each row, column and diagonal pave a rectangular target, as seen at left and right, and above and below. Here the 9 pieces are all decominoes, but pieces of any shape may appear, and it is not a requirement that they be of same size. In Figure 2, for instance, the pieces are polyominoes of consecutive sizes from 1 up to 9 units. The target is a 4×4 square with an inner square hole.

Surprisingly, computer investigations show that Figure 2 is just one among 4,370 distinct 3×3 geomagic squares using pieces with these same sizes and same target. Conversely, Figure 1 is one of only two solutions using similar-sized pieces and identical target. In general, repeated piece sizes imply fewer solutions. However, at present there exists no theoretical underpinning to explain these empirical findings. [1]

Figure 2:   A geomagic square using consecutively-sized pieces. Geomagic square - 3x3 normal sqare target.jpg
Figure 2:   A geomagic square using consecutively-sized pieces.
Figure 3:   A panmagic 3 x 3 geomagic square Geomagic square - 3x3 nasik.jpg
Figure 3:   A panmagic 3×3 geomagic square

The pieces in a geomagic square may also be disjoint, or composed of separated islands, as seen in Figure 3. Since they can be placed so as to mutually overlap, disjoint pieces are often able to tile areas that connected pieces cannot. The rewards of this extra pliancy are often to be seen in geomagics that possess symmetries denied to numerical specimens. [2]

Besides squares using planar shapes, there exist 3D specimens, the cells of which contain solid pieces that will combine to form the same constant solid target. Figure 5 shows an example in which the target is a cube.


A well-known formula due to the mathematician Édouard Lucas characterizes the structure of every 3×3 magic square of numbers. [3] Sallows, already the author of original work in this area, [4] had long speculated that the Lucas formula might contain hidden potential. [5] This surmise was confirmed in 1997 when he published a short paper that examined squares using complex numbers, a ploy leading to a new theorem that correlated every 3 × 3 magic square with a unique parallelogram on the complex plane. [6] Continuing in the same vein, a decisive next step was to interpret the variables in the Lucas formula as standing for geometrical forms, an outlandish idea that led directly to the concept of a geomagic square. [7] It turned out to be an unexpected consequence of this find that traditional magic squares now became revealed as one-dimensional geomagic squares.

Other researchers also took notice. Charles Ashbacher, co-editor of the Journal of Recreational Mathematics, speaks of the field of magic squares being "dramatically expanded" [8] Peter Cameron, winner of the London Mathematical Society's Whitehead Prize and joint winner of the Euler Medal, called geomagic squares "a wonderful new piece of recreational maths, which will delight non-mathematicians and give mathematicians food for thought." [1] Mathematics writer Alex Bellos said, "To come up with this after thousands of years of study of magic squares is pretty amazing." [9] It may be asked whether geomagic squares might have applications outside the study of puzzles. Cameron is convinced of it, saying, "I can immediately see a lot of things I'd like to do with this." [9]

Methods of construction

Trivial examples excepted, there are no known easy methods for producing geomagic squares. To date, two approaches have been explored. [10] Where the pieces to be used are polyforms , or shapes built up from repeated units, an exhaustive search by computer becomes possible.

In the case of Figure 1, for instance, a first step would be to decide on the piece sizes to be used (in this case all the same), and the shape of the desired target. An initial program would then be able to generate a list L corresponding to every possible tiling of this target shape by 3 distinct decominoes (polyominoes of size 10). Each decomino is represented by a unique integer, so that L will consist of a list of integer triads. A subsequent routine can then run through and test every combination of three different triads in turn. The test will consist in treating the candidate triads as the row entries in a 3×3 square, and then checking to see whether the columns and diagonals thus formed each contain 3 integers that are also in L—which is to say, are also target-tiling triads. If so, a 3×3 geomagic square using 9 decominoes and selected target has been identified. If this fails, alternative target shapes can be tried. An elaborated version of the same method can be used to search for larger squares, or for squares including differently-sized pieces.

An alternative method of construction begins with a trivial geomagic square showing repeated pieces, the shapes of which are then modified so as to render each distinct, but without disrupting the square's magic property. This is achieved by means of an algebraic template such as seen below, the distinct variables in which are then interpreted as different shapes to be either appended to or excised from the initial pieces, depending on their sign.

Figure 4:   A 'self-interlocking' geomagic square Geomagic square - Magic Jigsaw.jpg
Figure 4:   A 'self-interlocking' geomagic square

Figure 4 illustrates such a geometrical interpretation of the template in which k is interpreted as a small square shape, while a, b, c and d represent the protrusions (+) and/or indentations (-) by means of which it becomes modified so as to result in 16 distinct jigsaw pieces.

Relation to traditional magic squares

Contrary to the impression made at first sight, it is a misunderstanding to regard the term 'geomagic square' as referring to some category of magic square. In fact the exact opposite is the case: every (additive) magic square is a particular instance of a geomagic square, but never vice versa. The point is made clear by the example below that appears in a wide-ranging article on geomagic squares by Jean-Paul Delahaye in Pour la Science , the French version of Scientific American . [11] In this case the target "shape" for the geomagic square at right is simply a one dimensional line segment 15 units long, the pieces again being no more than straight line segments. As such, the latter is obviously a straightforward translation into geometrical terms of the numerical magic square at left.

Target is   15
Target is   •••••••••••••••

As Delahaye says, "This example shows that the geomagic square concept generalizes magic squares. The result here is hardly spectacular, but happily there are other geomagic squares that are not the result of such a translation." [11] [12]

The point being that every numerical magic square can be understood as a one-dimensional geomagic square as above. Or as Sallows himself puts it, "Traditional magic squares featuring numbers are then revealed as that particular case of 'geomagic' squares in which the elements are all one-dimensional." [2] This however does not exhaust the 1D case, because there exist 1D geomagic squares whose components are disconnected line segments, and which do not correspond to any numerical magic square. Thus, even in dimension one, the traditional types correspond to only a tiny subset of all geometric magic squares.

Special types

The richer structure of geomagic squares is reflected in the existence of specimens showing a far greater degree of 'magic' than is possible with numerical types. Thus a panmagic square is one in which every diagonal, including the so-called broken diagonals , shares the same magic property as the rows and columns. However, it is easily shown that a panmagic square of size 3×3 is impossible to construct with numbers, whereas a geometric example can be seen in Figure 3. No comparable example using connected pieces has yet been reported. [2]

Figure 5:   A 3D geomagic square with cubic target shapes Geomagic square - Maancubes.jpg
Figure 5:   A 3D geomagic square with cubic target shapes
Figure 6:   A geomagic square whose pieces comprise a self-tiling tile set Geomagic square - self-tiler.jpg
Figure 6:   A geomagic square whose pieces comprise a self-tiling tile set

In addition to being geomagic, there exist squares with auxiliary properties making them even more distinctive. In Figure 6, for example, which is magic on rows and columns only, the 16 pieces form a so-called self-tiling tile set . Such a set is defined as any set of n distinct shapes, each of which can be tiled by smaller replicas of the complete set of n shapes. [13]

A second example is Figure 4, which is a so-called 'self-interlocking' geomagic square. Here the 16 pieces are no longer contained within separate cells, but define the square cell shapes themselves, so as to mesh together to complete a square-shaped jigsaw.

Macau stamp featuring geometric magic square Macau stamp featuring geometric magic square.jpg
Macau stamp featuring geometric magic square

On October 9, 2014 the post office of Macau issued a series of stamps based on magic squares. [14] The stamp below, showing one of the geomagic squares created by Sallows, was chosen to be in this collection. [15]

Related Research Articles


A pentomino is a polyomino of order 5, that is, a polygon in the plane made of 5 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 12 different free pentominoes. When reflections are considered distinct, there are 18 one-sided pentominoes. When rotations are also considered distinct, there are 63 fixed pentominoes.

Magic square Sums of each row, column, and main diagonals are equal

In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers , the magic square is said to be normal. Some authors take magic square to mean normal magic square.

Rectangle Quadrilateral with four right angles

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as  ABCD.


A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.

Shape Form of an object or its external boundary

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.

Tessellation Tiling of a plane in mathematics

A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

Wallpaper group Classification of a two-dimensional repetitive pattern

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles and tiles as well as wallpaper.

A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

Abstract polytope

In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope without specifying purely geometric properties such as angles or edge lengths. A polytope is a generalisation of polygons and polyhedra into any number of dimensions.

Elementary mathematics

Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels.

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.

Tiling puzzle

Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps. Some tiling puzzles ask you to dissect a given shape first and then rearrange the pieces into another shape. Other tiling puzzles ask you to dissect a given shape while fulfilling certain conditions. The two latter types of tiling puzzles are also called dissection puzzles.

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish mathematician Tadeusz Banachiewicz in 1938.

Pythagorean theorem Equation relating the side lengths of a right triangle

In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:

Penrose tiling Non-periodic tiling of the plane

A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.

Lee Sallows British electronics engineer (born 1944)

Lee Cecil Fletcher Sallows is a British electronics engineer known for his contributions to recreational mathematics. He is particularly noted as the inventor of golygons, self-enumerating sentences, and geomagic squares.

Kellers conjecture Conjecture in geometry about hypercube tiling

In geometry, Keller's conjecture is the conjecture that in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other. For instance, in any tiling of the plane by identical squares, some two squares must share an entire edge, as they do in the illustration.


In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

An alphamagic square is a magic square that remains magic when its numbers are replaced by the number of letters occurring in the name of each number. Hence 3 would be replaced by 5, the number of letters in "three". Since different languages will have a different number of letters for the spelling of the same number, alphamagic squares are language dependent. Alphamagic squares were invented by Lee Sallows in 1986.

Self-tiling tile set

A self-tiling tile set, or setiset, of order n is a set of n shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of n shapes. That is, the n shapes can be assembled in n different ways so as to create larger copies of themselves, where the increase in scale is the same in each case. Figure 1 shows an example for n = 4 using distinctly shaped decominoes. The concept can be extended to include pieces of higher dimension. The name setisets was coined by Lee Sallows in 2012, but the problem of finding such sets for n = 4 was asked decades previously by C. Dudley Langford, and examples for polyaboloes and polyominoes were previously published by Gardner.



  1. 1 2 "Magic squares are given a whole new dimension", by Alex Bellos, The Observer , April 3, 2011
  2. 1 2 3 Geometric Magic Squares by Lee Sallows, The Mathematical Intelligencer, Vol 23, No. 4 Winter 2011, pp 25-31
  3. "Alphamagic Squares", of Mathematics
  4. "New advances with 4 × 4 magic squares" by Lee Sallows
  5. Sallows, pp 3 and 91
  6. "The Lost Theorem" by Lee Sallows The Mathematical Intelligencer Vol 19, No. 4, pp 51-4, 1997
  7. Complex Projective 4-Space Where exciting things happen: Geomagic squares
  8. Geometric Magic Squares reviewed by Charles Ashbacher Mathematical Association of America, September 24, 2013
  9. 1 2 "Ancient puzzle gets new lease of 'geomagical' life" by Jacob Aron, New Scientist , January 24, 2011
  10. Sallows, pp 1–12
  11. 1 2 Les carrés magiques géométriques by Jean-Paul Delahaye, Pour La Science No. 428, June 2013
  12. Cet exemple montre que la notion de carré géomagique généralise celle de carré magique. Le résultat n’est ici guère spectaculaire, mais heureusement, il existe d’autres carrés géomagiques ne provenant pas d’une telle traduction directe.
  13. On Self-Tiling Tile Sets by Lee Sallows, Mathematics Magazine, December 2012
  14. Macau Post Office web site Archived 2014-11-11 at the Wayback Machine
  15. Macau's magic square stamps just made philately even more nerdy The Guardian Science, November 3, 2014