The Soma cube is seven pieces made from unit cubes glued face to face, and their combined volume is exactly 27, which is the volume of a three by three by three cube. That is the whole setup, and it is one of the few puzzles where the goal is obvious to a five year old and the underlying combinatorics is still interesting to a mathematician.
The seven pieces
One piece is a tricube, an L-shaped run of three cubes. The other six are tetracubes, each made of four cubes. The set is not arbitrary: it is defined as every irregular shape you can build from three or four unit cubes, which is why it feels complete rather than designed.
Three plus six times four gives 27. Nothing is left over and nothing is missing, which means every valid arrangement uses every piece exactly once with no gaps.
Where 240 comes from
The famous number is the count of essentially different solutions to the three by three by three cube. Two solutions that differ only by rotating the whole finished cube are counted as one, which is what makes the count meaningful rather than inflated.
A teaching document on the Soma cube from Fergusson College records that the puzzle was invented by Piet Hein in Denmark in 1933 and that there are 240 distinct solutions to the three by three by three cube, crediting the enumeration work published in Winning Ways for your Mathematical Plays.
The count was originally produced by hand rather than by machine, which is worth pausing on. Someone worked through the cases systematically, on paper, and got it right. The result has since become a standard benchmark. A recent arXiv paper by Kardes, Krakauer and Grochow on the physical complexity of a cognitive artifact uses the Soma cube and its 240 solutions as a worked example of how a physical object can carry genuine computational difficulty.
Piet Hein’s wider work
Hein was not primarily a puzzle designer. He worked across mathematics, design, and poetry, and his best known contribution outside games is the superellipse.
An explainer from HKUST’s Science Focus describes the problem he was handed in 1959: Stockholm planners needed a roundabout for Sergels Torg that fitted a rectangular space between buildings. Circles wasted space, ellipses had awkward pointed ends, and shapes built from joined arcs produced jarring changes in curvature. Hein took the Lame curve, formulated by Gabriel Lame in 1818, and picked an exponent of 2.5 with a width to height ratio of 6 to 5. He called the result a superellipse.
The same instinct shows up in the Soma cube. Both are attempts to find the shape that sits exactly between two obvious options, and to define it precisely rather than approximately.
Beyond the cube
The cube is the starting point, not the endpoint. Soma sets usually come with a booklet of other figures: a tower, a wall, steps, a bathtub, a dog. Some of these are considerably harder than the cube because they have long thin sections that only a couple of pieces can occupy.
A useful thing to know is that not every plausible looking figure is possible. Volume alone is not sufficient. A shape can have exactly 27 cells and still be unbuildable, because of how the pieces are forced to interact around narrow regions. Colouring arguments work here just as they do for flat tiling puzzles: colour the cells in a repeating pattern, work out what each piece must cover, and check whether the totals can be reconciled.
Why it is a good first polycube puzzle
- Fast feedback. With 240 solutions, a beginner stumbles into one within a reasonable session. That matters more than people admit.
- Small piece count. Seven pieces is enough to be interesting, few enough to keep in your head.
- No orientation traps. The pieces are chunky and easy to distinguish by feel, so you spend your time thinking rather than identifying.
- It scales. Once the cube is easy, the alternative figures give you years of material, and the ideas transfer directly to pentacube and larger polycube sets.
- Durable in wood. Glued unit cubes are structurally sound and tolerate handling far better than thin notched sticks.
Buy one with crisp cubes and a tight glue-up. A Soma set with slightly rounded or mismatched cubes will fight you at every assembly.
