Soma Cubes
Contents of this page

What is a Soma Cube? 
Main Problem
Positions of the Soma Pieces 3 and 2
Figures of Soma Cubes
Magnification Problem
Making of Soma Cubes
Three Variants:
...Soma schräg
...Chequered Pattern
...Coloured Cube
Similar Cube problems
Soma Cubes on the Internet
More 3x3x3-Puzzles 
To the Main Page       "Mathematische Basteleien"

What is a Soma Cube? 
The seven pieces of cubes are named Soma cubes.  Altogether you have 1x3 + 6x4 = 27 separate smallcubes. 

Main Problem  top
...... The main problem of the Soma "research" is to assemble the seven Soma pieces or 27 cubes to make a 3x3x3 cube.
The chance of solving this three-dimensional puzzle  is good, because there are 240 possibilities to put  the cube together, not counting symmetries. 

If you try to solve the puzzle for the first time, you need approximately 15 min. You have a better chance if you start with the three-dimensional pieces 5,6,7.

Three solutions
Solution 1 was my first solution, which I kept in my mind.
Solution 2 is easy to remember: You start with the three 3D pieces 5,6, and 7. Piece 4 follows.
Solution 3 is one of the few solutions without piece 7 forming a corner.

All 240 solutions
Loesung Nr: 1 
  000  100  000  011  000  000  000 
  000  000  000  001  110  000  000 
  000  000  011  000  100  000  000 

  000  100  000  001  000  010  000 
  000  110  001  000  000  000  000 
  000  000  001  000  000  000  110 

  000  000  000  000  000  111  000 
  111  000  000  000  000  000  000 
  100  000  000  000  000  000  011

Jochen Wermuth makes available all 240 solutions found by a program in C++.

He is using the notation on the left.

The pattern has seven rows. The whole cube and the position of a soma cube is given in a row.

The three blocks in the lines are assigned to the three layers on top, centre and bottom.

I offer all 240 solutions for downloading, likewise the source code in C++ with the author's agreement. 

Positions of the Soma Pieces 3 and 2     top
Piece 2 and 3 contain a 1x1x3 bar. 

The shapes of the Soma cubes result in the following statements.

Piece 2 forms 0,1 or 2 corners.
Piece 3 forms either 0 or 2 corners.

Now there are two possibilities:
(1) Piece 3 forms no corners. The other six pieces form at most 2+5 corners. This case is not possible, because you don't have eight corners.
(2) Piece 3 forms two corners. The other 6 pieces form 6 corners. Therefore piece 2 must form at least one corner. 

Piece 3 forms two corners.

Soma piece 2 must always form a corner and must not lie in the middle.
This can be the first steps to prove that there are 240 solution.

Figures of Soma Cubes   top
It is interesting to look for new 27-cube-shapes. This is one example, a "car".

More well known figures are sofa, bed, bathtub, gate, gravestone, and tower. 

There are innumerable figures of Soma cubes. This is proved by the following thoughts about designing new ones.
...... All Soma cubes have 27 cubes. If you give a mat of  5x4 = 20 cubes, there are [20 above 7] = 20!/13!/7! = 77520 places for the remaining seven cubes, so 77520 figures. There is an example on the left. 
...... If you form a bar with five cubes, there are two left. So you get [15 above 2] =15!/13!/2! = 105 figures. There is an example on the left. If you demand symmetry, you only get 18 cube figures. You can see them as top view on the right and in perspective below. ......

Two figures are insoluble. 

Magnification Problem top
Can you imitate a somacube with magnification? It is a productive problem with pentominos and tetracubes, but not with Soma cubes. Only the cube with three cubes is possible. You need the rest of the Soma cubes to build it with double magnification. 

Making of Soma Cubes   top

If you want to play with Soma cubes you have to construct them by hand.

The simpliest way is to buy a length of wood which is square in cross-section, cut it into cubes and glue the cubes together.

Another method is glueing dice. The best idea is to use a two component glue, because it needs time to harden. Then you can form the Soma pieces without having to hurry.

A cheap but difficult method is constructing them from a sheet of paper. You have to design the base of every Soma cube and fold it. 

You can buy Soma cubes. 

Parker Brothers sold the Soma cubes in 1969. Maybe they protected the name "Soma cube" by copyright.

In Germany you can buy them on Christmas fairs, mostly without a name or with other names like Ostfriesenwürfel ;-). Another version is Babylon. This is the "Rehmsche Spielsatz".

In the United States there is a game using Soma cubes, called Block by Block, which I bought in Las Vegas in 1996.

Three variants

1st Variant: Soma schräg (Soma crooked)   top
There is a curiosity: You don't use a cube as the basic piece, but a parallelepiped. This is a figure bordered by parallelograms. In this case it is a rhombus with an 50° angle. 
You should think you can simply apply the solutions of the 3x3x3 cube problem to this figure. You seldom succeed.
...... If you add three or four parallelepipeds, you get two possibilities of each Soma cube form. I only found one solution. 

(Soma schräg, HOLZINSEL 56290 Beltheim, Art.-Nr. 019/1)

2nd Variant: Chequered Pattern    top
You can produce this Soma puzzle, if you build a cube with method 2 and colour the cubes black and white alternately. This pattern is transfered to the Soma pieces. 

The problem is to form again a 3x3x3 cube of all pieces.

3rd Variant: Coloured Cube top
You can design new Soma puzzles, if you build a 3x3x3 cube and colour the surface all in one colour. You can also make a pattern (Example: die). The Soma pieces will adopt these changings, too.

If you like you can transfer these changings to inner squares in order to make the puzzle more difficult. 

The problem is to form again a 3x3x3 cube of all pieces.

History    top
There are several forerunners of Soma Cubes.
In 1936 the Dane Piet Hein took seven polycubes, so that they formed a 3x3x3-cube. 
There are ten pieces with 2,3, and 4 cubes. He chose the polycubes, which don't form a cuboid.

Maybe he obtained the name from the book "Brave New World" by Aldous Huxley. Soma was the name of an addictive drug used in the fictional state in Huxley's book which is set in the year 2600.

Piet Hein (1905-1996) was a poet and scientist with wide ranging interests. 

J.H. Conway and M.J.T. Guy worked out in 1961 that there are 240 possible ways to form a 3x3x3-cube. 
Computers confirmed this result later. 

The Soma cubes became famous by its publication in the magazine Scientific American (1958).  In Germany the magazine Bild der Wissenschaften circulated this puzzle (1967).

Similar Cube problems top
Theodore Katsani suggested a game with all polycubes of four cubes in the 1950s. 

It is impossible to build a 3x3x3 cube, but  8*4=32 leads to cuboids 2x4x4 or 2x2x8 (4). You find more on my page tetracube
Rehm's Cube Set

There is the problem: Which polycubes with 3 or 4 cubes form a 3x3x3 cube? 
There are 2 tricubes and 8 tetracubes. 
The numbers are used in book 5.
There are 14 possibilities for a 3x3x3 cube because of  27=3+6*4.
I345678 (138)
I245678 (000)
I235678 (031)
I234678 (099)
I234578 (245)
I234568 (245)
I234567 (027)

II345678 (240)
II245678 (047)
II235678 (261)
II234678 (221)
II234578 (337)
II234568 (337)
II234567 (039)

All sets except the combination II2..45678 (3 is missing) leads to a cube (5). 
The numbers inside the brackets is the number of solutions (Torsten Sillke, Rehm's 3-Cubes, URL below).
Hollow Cube
You can build a hollow cube from the tricubes I and II and the tetracubes 3,4,6,7 and 8. 
There is 2*3+5*4=26
You can ask for more solutions with other tetracubes of Rehm's cube set and for the number of solutions. 
(This idea had Volker Latussek from Würzburg.)

More 3x3x3-Puzzles top
All soma cubes together contain 27 cubes. The distribution is 27=3+4+4+4+4+4+4. 

You can imagine that there are more sums between 27=1+1+...+1 and 27=9+9+9, which are suitable for puzzles.

I introduce some puzzles in this chapter, which I found more at random. 

Diabolical Cube - Hoffmann (27=2+3+4+5+6+7)

... some solvable symmetric figures

Mikusinski's Cube (27=4+4+4+5+5+5)
... There are two solutions only.

... I only found a 4x5 mat with 7 cubes on top, which is worth to show, after playing around. 

Der vertrackte Würfel- The tricky cube  (27=4+4+4+5+5+5), "Toys pure", Art.-Nr.:HS 630

... You can form a closed ring with all pieces.

Loops cube (27=4+4+4+5+5+5), designed and made by René Dawir

... Order in the solution: 564321

Soma Cube on the Internet top

C. Meier - Warenhandel
Holzwürfel - Somawürfel - Dezimalrechenwürfel

Matroid Matheplanet
Der Soma Würfel

Peter Kenter   (Mathematik und Kunst)
Die 41 Polykuben der Ordnung 1 bis 5  (pdf.-Datei)

Sascha Preisegger
Der Soma-Würfel

Stromberg-Gymnasium, Vaihingen an der Enz
Der Somawürfel

Thimo Rosenkranz



Balmoral Software
Soma Site

Bill McKeeman
Soma Cube Solutions

Binary Arts Corporation (Binary Arts' Block by Block® puzzle) 
Millennium Puzzles: Soma cube

Ed Vogel
A very large Soma puzzle
at the Minnesota State Fair 2006

Erich Friedman
Erich's 3-D Jigsaw Puzzles

Eric W. Weisstein, (MathWorld)
Soma Cube

Stewart T. Coffin
The Puzzling World of Polyhedral Dissections

Thorleif  Bungård 
Thorleif's SOMA page

Torsten Sillke
cube-secrets, Rehm's 3-Cubes (1980)

Soma cube, Bedlam cube, Conway puzzle

SOMA CUBE puzzle solutionPuzzle Cube SolutionHow to solve the Bedlam Cube Retro
SOMA CUBE ANIMATION (12 figures, 3D puzzle solution)

References  top
(1) "Bild der Wissenschaft" November 1967
(2) Pieter van Delft /Jack Botermans: Denkspiele der Welt, München 1980 (1998 neu aufgelegt) [ISBN 3-685-1998)
(3) Martin Gardner: Bacons Geheimnis, Frankfurt am Main, 1986   [ISBN 3-8105-0800-4]
Fast 30 Jahre nach Erscheinen einer Kolumne in der Septemberausgabe 1958 von <Scientific American> zieht der Autor eine Bilanz der "Somaforschung".
(4) R.Thiele, K.Haase: Der verzauberte Raum, Leipzig, 1991  [ISBN3-332-00480-8]
(5) R.Thiele, K.Haase: Teufelsspiele, Leipzig, 1991 [ISBN 3-332-00116-7]

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©  1999 Jürgen Köller