page_title: Geometric magic square
section_title: Special types
hierarchical_section_title: Geometric magic square / Special types
caption_attribution_description: English: A 3D geomagic square with target a perfect cube.
context_page_description: 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 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.
context_section_description: 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.
  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.
  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.
caption: Figure 5:   A 3D geomagic square with cubic target shapes
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