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Parity and Centerness Applied to the SOMA Cube

Twelve different shapes can be formed by combining not more than four cubes, all the same size and joined at their faces. The seven of these shapes, one 3-cube and six 4-cubes, left after eliminating all straight shapes (rectangular parallelepipeds) constitute the SOMA puzzle invented by Piet Hein. The challenge of the puzzle is to use the seven pieces to construct any one of a profusion of specified forms.