The Soma Cube

...no time, no leisure ... not a moment to sit down and think - for ever by some unlucky chance such a crevice of time should yawn in the solid substance of their distractions, there is always soma delicious soma...

A. Huxley, Brave New World

The Soma cube is not really a cube and it's also not (really) a drug. It's a puzzle like Tangram. You have to rearrange puzzle pieces to form other figures. In fact there are 7 Soma-pieces like Tangram. But unlike Tangram the pieces from the Soma Cube are three-dimensional.

The Soma-cube was conceived, according to Martin Gardner [1], by the Danish writer Piet Hein in 1936 during a lecture on quantum physics by Werner Heisenberg (You know from that (un-) certain principle). While Heisenberg was speaking of a space sliced into cubes, Piet Hein thought of the following curious geometrical theorem.

first Look at all shapes made of cubes which have a concavity or corner nook in it somewhere. There is only one such shape made of 3 cubes. Note that that shape is "two-dimensional".

Turning to four cubes we find that there are 6 essentially different ways to make such a shape. Note that some of these shapes are essentially "three-dimensional". The curious theorem is now that these 7 pieces can be joined to form a cube again.

rest

A set of pieces was marketed under the trade name Soma and has been a popular all over the world (eventually). But what am i rambling about...

Lets start playing !

Cube The first puzzle is of course to assemble the seven pieces to form a cube. In fact this is one of the most easy Soma-puzzles to solve. I read somewhere that there are 240 essentially different solutions, i.e. not counting rotations and reflections. If you find this hard to do do not dispair since many people, including me, can after a few days playing with Soma solve (some) puzzles in there head. And it's a good and fun way to learn visualizing geometric shapes.

After this first puzzle you might want to try some (+/- 60) more difficult 7-piece structures/puzzles :

Go to the puzzles!

By the way, they are all possible to make.

Of course there is more to Soma than this. At first most people start with just using the trial-and-error tactic. But it's more satisfying to analyze them. A good strategy to adopt is to try to use the 'most irregular' pieces (the three most right of second picture) first and save the simplest (first picture) until last. But these are more rule of thumb than real 'laws'. Soma constructions lend themselves to fascinating theorems and impossibility proofs of combinatorial geometry. For instance look at the following two figures:

12

Question: Which one can not be built with the Soma-pieces?
Of course this question seems to imply that there is exactly one possible to built, which is correct... or is it not?

Be carefull though because there are constructions which are possible to make and just look impossible:

28 cubes?
count and doubt

Last but not least ... you do not need to use all seven pieces to make an interesting puzzle. As a beautifull example try this one:

Two

P.S. Any comments, further puzzles, questions or ... just send an e-mail to me. Oh and..., as an afterthought: How many holes can you fit in a seven-piece soma-construction? Enjoy



Reference:
[1] More Mathematical Puzzles and Diversions
      by Martin Gardner Chapter 6, Penguin Books 1961.

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Author: Christian Eggermont
Created: 26 July 1997
Last updated: 28 July 1997