Soma cube

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The pieces of a Soma cube (with extra coloring)

The same puzzle, assembled into a cube

The Soma cube is a solid dissection puzzle invented by Piet Hein during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3x3x3 cube. The pieces can also be used to make a variety of other interesting 3D shapes.

The pieces of the Soma cube consist of all possible combinations of four or fewer unit cubes, excluding all convex shapes (i.e., the 1x1x1, 1x1x2, 1x1x3, 1x1x4 and 1x2x2 cuboids). This leaves just one three-block piece and six four-block pieces, of which two form an enantiomorphic pair. A similar puzzle consisting solely of all eight four-block pieces (including the cuboids) would contain 32 unit cubes and, thus, could not be assembled into a cube.

The Soma cube is often regarded as the 3D equivalent of polyominos. There are interesting parity properties relating to solutions of the Soma puzzle.

Soma has been discussed in detail by Martin Gardner and John Horton Conway, and the book Winning Ways for your Mathematical Plays contains a detailed analysis of the Soma-cube problem. There are 240 distinct solutions of the Soma-cube puzzle, up to rotations and reflections: these are easily generated by a simple recursive backtracking search computer program similar to that used for the eight queens puzzle.

The seven Soma pieces are all polycubes of order three or four:

The "L" tricube
T tetracube: a row of three blocks with one added below the center
L tetracube: a row of three blocks with one added below the left side
S tetracube: bent triomino with block placed on outside of clockwise side
Left screw tetracube: unit cube placed on top of anticlockwise side. Chiral in 3D.
Right screw tetracube: unit cube placed on top of clockwise side. Chiral in 3D.
Branch tetracube: unit cube placed on bend. Not chiral in 3D.