Soma cube

# Soma cube

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Encyclopedia
The Soma cube is a solid dissection puzzle
Mechanical puzzle
A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces.- History :The oldest known mechanical puzzle comes from Greece and appeared in the 3rd century BC....

invented by Piet Hein
Piet Hein (Denmark)
Piet Hein was a Danish scientist, mathematician, inventor, designer, author, and poet, often writing under the Old Norse pseudonym "Kumbel" meaning "tombstone"...

in 1933 during a lecture on quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

conducted by Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

. 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 3D
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

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 cuboid
Cuboid
In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

s). This leaves just one three-block piece and six four-block pieces, of which two are mirror images of each other (see Chirality
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

). 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 polyomino
Polyomino
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 with a connected interior....

es. There are interesting parity properties relating to solutions of the Soma puzzle.

Soma has been discussed in detail by Martin Gardner
Martin Gardner
Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...

and John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

, and the book Winning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games...

contains a detailed analysis of the Soma cube problem. There are 240 distinct solutions of the Soma cube puzzle, excluding rotations and reflections: these are easily generated by a simple recursive
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

backtracking search computer program similar to that used for the eight queens puzzle
Eight queens puzzle
The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens attack each other. Thus, a solution requires that no two queens share the same row, column, or diagonal...

.

The seven Soma pieces are all polycube
Polycube
thumb|200px|right|The seven free tetracubesthumb|200px|right|A [[Chirality |chiral]] pentacubethumb|200px|right|Puzzle with a unique solution...

s 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
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

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.

## Trivia

Similar to Soma cube is the 3D pentomino
Pentomino
A pentomino is a polyomino composed of five congruent squares, connected along their edges ....

puzzle, which can fill boxes of 2×3×10, 2×5×6 and 3×4×5 units.

Piet Hein (Denmark)
Piet Hein (Denmark)
Piet Hein was a Danish scientist, mathematician, inventor, designer, author, and poet, often writing under the Old Norse pseudonym "Kumbel" meaning "tombstone"...

authorized a finely crafted rosewood
Rosewood
Rosewood refers to any of a number of richly hued timbers, often brownish with darker veining, but found in many different hues. All rosewoods are strong and heavy, taking an excellent polish, being suitable for guitars, marimbas, turnery , handles, furniture, luxury flooring, etc.In general,...

version of the Soma cube manufactured by Theodor Skjøde Knudsen's company Skjøde Skjern (of Denmark). Beginning in about 1967, it was marketed in the U.S. for several years by the game manufacturer Parker Brothers
Parker Brothers
Parker Brothers is a toy and game manufacturer and brand. Since 1883, the company has published more than 1,800 games; among their best known products are Monopoly, Cluedo , Sorry, Risk, Trivial Pursuit, Ouija, Aggravation, and Probe...

. Plastic Soma cube sets were also commercially produced by Parker Brothers in two colors (blue and red) during the 1970s.