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What is a Soma Cube?
The seven pieces of cubes are named Soma cubes.
Altogether you have 1x3 + 6x4 = 27 separate cubelets.
Main Problem top
... ... |
The main problem of the Soma "research" is to
assemble the seven Soma pieces or 27 cubelets 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
1
... ...
|
2
 ...
|
3
 ...
|
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.
Results:
........ ..
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-cubelet-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 cubelets. If you give a mat of 5x4 = 20
cubelets, there are [20 above 7] = 20!/13!/7! = 77520 places for the remaining
seven cubelets, so 77520 figures. There is an example on the left. |
... ... |
If you form a bar with five cubelets, 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 cubelets 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.
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. Basically I only found one solution. Use solution
2, see above. |
(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 cubelets 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 cubelets. 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
Tetracubes
Theodore Katsani suggested a game with all polycubes of four cubelets
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 cubelets 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)
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I245678 (000)
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I235678 (031)
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I234678 (099)
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I234578 (245)
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I234568 (245)
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I234567 (027)
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II345678 (240)
Somawürfel
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II245678 (047)
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II235678 (261)
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II234678 (221)
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II234578 (337)
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II234568 (337)
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II234567 (039)
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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 cubelets. 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.
[There is more systematic on the pages of David Singmaster, Stewart
T. Coffin und Erich Friedman (URL below).]
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 cubelets 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 |
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Soma Cube on the Internet
top
German
C. Meier - Warenhandel
Holzwürfel - Somawürfel
- Dezimalrechenwürfel
http://www.zahlenjagd.at/
Würfelpuzzles
Peter Kenter (Mathematik
und Kunst)
Die
41 Polykuben der Ordnung 1 bis 5 (pdf.-Datei)
Wikipedia
Somawürfel
English
Balmoral Software
Soma Site
Bill McKeeman
Soma
Cube Solutions
Binary Arts Corporation (Binary Arts' Block by Block® puzzle)
Millennium
Puzzles: Soma cube
Christian Eggermont
The
Soma Cube ("Go to the puzzles!")
Courtney McFarren (Mathematica)
The Soma
Cube
David Singmaster
CLASSIFICATION OF
DISSECTED CUBES (.doc file)
Dennis Nehen
The Soma Figure Gallery
Erich Friedman
Erich's
3-D Jigsaw Puzzles
Lee Stemkoski (Mathematrix)
The
Soma Cube
Thorleif Bungård
Thorleif's SOMA
page
Torsten Sillke
cube-secrets
, Rehm's
3-Cubes (1980)
Wikipedia
Soma cube
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]
Feedback: Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
1999 Jürgen Köller
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