Geometric magic square

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.

A well-known formula due to the mathematician Édouard Lucas characterizes the structure of every 3 × 3 magic square of numbers.

[4] Sallows, already the author of original work in this area,[5] had long speculated that the Lucas formula might contain hidden potential.

[7] 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.

Charles Ashbacher, co-editor of the Journal of Recreational Mathematics, speaks of the field of magic squares being "dramatically expanded"[9] 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.

"[2] Mathematics writer Alex Bellos said, "To come up with this after thousands of years of study of magic squares is pretty amazing.

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).

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.

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.

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.

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.

In Figure 6, for example, which is magic on rows and columns only, the 16 pieces form a so-called self-tiling tile set.

On October 9, 2014 the post office of Macau issued a series of stamps based on magic squares.

Figure 1: A geomagic square with same-sized pieces ( decominoes )
Figure 2: A geomagic square using consecutively-sized pieces.
Figure 3: A panmagic 3 × 3 geomagic square
Figure 4: A 'self-interlocking' geomagic square
Figure 5: A 3D geomagic square with cubic target shapes
Figure 6: A geomagic square whose pieces comprise a self-tiling tile set
Macau stamp featuring geometric magic square