# The Soma Puzzle Book

The classical Soma cube is a 3D puzzle invented by the Danish polymath Piet Hein in 1933. A 3 x 3 x 3 cube is partitioned into seven building blocks. Each of these blocks consists of three or four atoms (that are 1 x 1 cubes) glued together on matching faces and they have at least one inside corner. One block has three atoms (this is called the V block and consists of a corner atom with two atoms glued to two adjacent faces). All the others have four atoms, and are obtained by adding a fourth atom to the V. Three of the these 4-blocks are "flat" (the L the S and the T where the fourth atom is added in the same plane as the V) and three where the fourth atom is added on top of the V outside the V-plane: The P (the fourth atom is on top the corner atom of the V) and the remaining ones (A and B) are on the other blocks of the V (these are left and right chiral). There exist commercial versions of 4 x 4 x 4 or 5 x 5 x 5 cubes for the diehards, but these will not be considered here.

There are 240 different ways to put the seven pieces together to form the 3 x 3 x 3 cube. The mathematical background has been fully analysed by John Horton Conway in the *Mathematical Games* column of *Scientific American* way back in 1958. So this cannot be the subject of this book. What is presented are problems (and solutions) of what other kind of challenges can be posed using these same building blocks. Even with one block there are problems to solve like which block can give a hexagonal shadow or how small can a hole in a plane be that allows to get all (or some of) the pieces through, or what is the largest hole that can be filled with every piece.

And then the book continues chapter by chapter posing problems with 2, 3,..., 7 pieces. One or two shapes have to be constructed using a selection of building blocks (possibly with duplicates). Also the chapter involving all the seven blocks adds to the classic problem by asking to construct the cube with constraints or to find non cubic volumes. A nice proposal is to construct fractions where a fraction p/q means that there is a bottom layer of q atoms and a second layer of p atoms resting on the blocks of the bottom layer. One can then construct for example fractions 3/9 and 6/9 (whose sum is 1) and there are other such fractions that sum to 1.

Note that the 3-block with 3 atoms put in one line (that is block I) is excluded and also two 4-blocks (4 in a row which does not fit in the cube and a 2 x 2 square called O) are excluded. If we add the I and O to the set of seven, then new problems can be added to the already extensive list of problems that will now involve 8 or 9 blocks.

There is no mathematical analysis in this book. A challenge is just graphically presenting the blocks that can be used and the required result. A colourful graphical language is defined that is used in the solution sections to explain how to generate the solution in several steps. This it is a book purely for the fun of solving puzzles. It is of course possible to solve the puzzles with pen and paper if one has a well developed 3D imagination, but it is of course the intention that you have a set (sometimes two sets) of seven blocks physically available, which can be bought in most toy shops. These often already come along with non-cube shapes that have to be built. The current book will add new problems to the existing ones. If you happen to have already such a set, then this book will provide new challenges for you.

**Submitted by Adhemar Bultheel |

**5 / Aug / 2019