There are seven of these critters, six of them composed of 4 cubes each, while one only has 3 cubes. There is a mathematical criteria for selecting these special pieces:
1. The number of cubes for each piece cannot exceed 4.
2. Each piece has to be irregular, with a nook or a turn somewhere. All regular pieces are eliminated. Another way of defining an irregular piece: a piece that is NOT a hexahedron.
Starting off with the first rule, we have 8 tetra-cubed pieces, 2 tri-cubes, 1 double-cube and 1 single-cube. The single-cube and double-cube pieces are automatically eliminated. You just can't get more regular with those pieces, as one is just a sugar-cube, while the other is a simple domino. One of the tri-cube pieces forms a 3x1x1 stick, which is a hexahedron, so it goes away, too. Finally, there are two regular tetra-cubes that disappear, the 4x1x1 stick, and the 2x2x1 square-like thing. After the smoke clears, there are only 7 pieces remaining, the most irregular of the litter.
The question is: can these oh-so-very irregular pieces combine together to form a simple 3x3x3 cube? Of course they can! How do you think they packed and shipped all those Soma cubes?
There are many conflicting statements on how many ways these seven pieces can do the cube. One number is said to be merely 240, while another is as high as 1,105,920. The truth is there are only 240 standard solutions if you do not include rotations or mirror images. There are 24 ways to rotate an assembled cube, and each rotation has a mirror-image solution. This brings the total up to 11,520. In addition to that, five of the pieces (#'s 1, 3, 4, 5 & 6) can be picked out of the cube, rotated so it "looks like itself again", and placed back in for a perfect fit. Piece #7 can be rotated in 3 such ways, cranking the number of solutions up by a factor of 96. This gives us the very large answer of 1,105,920 combinations, the same number that appears in your Soma book (you still have that thing, don't you?).
Basically, a Soma figure is a structure that has 27 cubes. It is easy enough to invent a structure like this; just get 27 baby blocks, slap them together, and what do you know... instant Soma figure. The question is whether this figure can be built using the seven special Soma pieces.
Some people invent figures by merely putting the seven pieces together into a structure that is sort-of-kind-of decent looking. Others dream up a 27-cube structure, and try to see if their seven pieces can do the job. The frustration comes when the dreamers cannot accomplish this, forever pondering whether the structure is impossible to solve; or maybe it is possible, and they just didn't figure it out yet. Is it worth wandering aimlessly down this path of darkness?
Even though Peit Hein invented the Soma Cube in 1936, it did not become a reality in the 'States until 1969 when Parker Brothers manufactured and sold them. Actually, it was a nicely made. The pieces were hollow plastic and came in three colors: red, blue and disco-gold. The size of an individual cube was 9/8 inches on the edge, which is kind of large, but not enough to clutter up the coffee table.
The P.B. Soma cube also came with a black base, a sort of platform to store your favorite puzzle. Inside the base was a tiny 56-page booklet with about 50 figures to solve; some of them for double-sets only (yet another shameless and cheezy marketing tactic). It had other goodies such as history of the cube and a couple of interesting articles to read. There were 36 standard figures in the booklet, so after burning through the 33 possible figures, you could try to take a stab at solving the 3 impossible ones, and dream about winning a Nobel prize for solving the infamous 'W-Wall' figure (which, by the way, is definitely impossible).