By Sheldon Greaves

The mind must play (especially during a lockdown).

We’ve heard the old saw that, “All work and no play makes Jack a dull boy”, but what isn’t generally recognized is that the word “dull” in this context and usage most likely refers to mental dullness. Anyone who has had to spend a long time doing dull work knows exactly what I’m talking about. As an antidote, allow me to introduce some favorite mind games.

A Classic Puzzle

When I was eleven, my parents gave my brothers and me a puzzle game that totally captivated us. It was called “Soma” and manufactured by Parker Brothers. I didn’t know it at the time, but the puzzle itself first appeared on the pages of (where else?) Martin Gardner’s column on recreational mathematics in Scientific American. The inventor of the puzzle was a Danish inventor, physicist, and poet named Piet Hein who, while listening to a lecture on quantum physics by Werner Heisenberg when the subject turned to a space sliced into cubes. Hein found himself puzzling over the idea of combining shapes made from smaller cubes into a larger one and suddenly arrived at the following theorem:

If you take all the irregular shapes that can be formed by combining no more than four cubes, all the same size and joined at their faces, these shapes can be assembled to form a larger cube.

Hein realized that there were only seven such shapes containing a total of 27 smaller cubes, and they would form a larger 3 x 3 x 3 cube. According to the documentation that came with the game, there are 240 simple ways and 1,105,920 mathematically different ways to make the cube.

My brothers and I spent many happy hours playing with this “toy”. One extra twist was that the cubes can also be assembled into whimsical shapes and sculptures. For a long time, the game was out of print. Our old one had long since disappeared and over the years I grew nostalgic about the fun I had playing with Soma. Then one day I wandered into a Tap Plastics store and discovered a bin filled with clear acrylic cubes about one inch on a side. I realized that with these and some superglue I could make my own set, and I did. One thing I didn’t anticipate was that the glue “frosted” the plastic, but I remembered an old bottle of acrylic floor finish under the sink. I brushed a coat of this over the pieces and the frosting vanished.

If you have access to a 3D printer, there are several versions of Soma available online to download and print.

All three cubes have continued to be a pleasant diversion, especially when I try to compare my approaches to solving the puzzle as a young boy to how I think about it as an adult. As a kid, the idea was to just find a way to put the pieces together, and usually (in my case at least) that meant fiddling with them in a fairly uncoordinated and unsophisticated way until the cube emerged. One of my younger brothers took especial interest in reproducing the non-cube shapes and sculptures and kept a notebook describing how to make the ones he had figured out. When I started playing with Soma as an adult, I tried to be more deliberative about the exercise. I like to take two pieces, combine them in some fashion, and then try to build the cube around that nucleus. One thing I have not taken the time to research is how one might mathematically describe a solution as some kind of equation.

As luck would have it, a few months after I made my own cube I was in a game store and happened to walk past their used game shelf. There I saw two Soma sets, in excellent condition, complete with the documentation. It had literally been decades since I had seen one in the flesh. I handed over my lunch money bus fare to buy them and walked happily home. A few years later a game called Block By Block appeared in toy and game stores that turned assembling the Soma cube and making the other shapes into a competitive game. I bought one mostly for the version of the cube it contains, which is smaller but quite robust. For some reason, the idea of turning this kind of puzzle into a competitive activity didn’t appeal to me at all. But they did have some interesting new shapes to make.

There is a sense in parts of our society that playing games is fun, but frivolous. I’ve known a couple of elderly folks who would not indulge in crossword puzzles or other games and puzzles in their leisure time because they grew up with this attitude. They saw it as a waste of time in spite of the clear benefits of giving your mind the occasional workout. As I said, and as more and more leading educators now insist, the mind must play. Setting problems for yourself has many benefits. Perhaps Piet Hein himself said it best:

“Problems worthy of attack,
prove their worth by hitting back.”

Pentominoes and Polycubes

Another favorite puzzle game of mine also gained notoriety on the pages of Martin Gardner’s inimitable column, and that was pentominoes. This is a variation on the Soma cube, but with added complexity. The game involves manipulating pieces formed by all the shapes you can make by joining no more than five squares at their faces. Gardner introduced them in 1957, calling them “super dominoes” at first. But polyominoes–defined as shapes formed from more than two (do-minoes) squares–goes back at least as far as 1907. There are 12 pieces in a set of pentominoes. Of course, one can make pieces with more squares: there are 35 possible hexominoes, and 108 heptominoes. Beyond that, it starts to get a little nuts. I doubt anyone wants to play octominoes with 369 pieces, or nonominoes (or enneominoes) with 1,285 pieces.

Pentominoes make for interesting explorations of tessellation problems. Two classic problems are enclosing as much space as possible with a border made up of the pieces. Another is to take one piece, and using it as a model, try to replicate a larger copy of that shape using the rest of the pieces. You can get into some pretty heavy (well, heavy for me) math pretty quickly. The Wikipedia article on polyominoes has a lot to chew on.

A fun variation on pentominoes is to add a third dimension. The result is a set of pentacubes. I made a set using wooden cubes from Michael’s. I made a simple jig to hold the pieces in line, and used ordinary white glue. To facilitate playing with tesselation problems, I used a drawing program to make a grid the same size as the cube faces. Taking two pages of grid, I taped them together and had them laminated at a local copy center.

Now keep in mind that these are planar pentacubes. A true set of pentacubes would have shapes that extend into three dimensions as the Soma pieces do. In such a case, you have 29 pieces instead of the twelve shown here. David Goodger has a remarkable web site that delves into all kinds of polycubes and polyominoes based on squares, hexagons, who knows what else.

My set of planar pentacubes will also form “boxes”, but not cubes. More specifically, you can assemble the pieces to form rectangular prisms of 2 x 3 x 10, 2 x 5 x 6 and 3 x 4 x 5. Again, the Wikipedia page on Pentominoes can give you lots of possibilities for playing with the pieces.

I find that these kinds of puzzles are a great way to introduce kids to science problems, not as problems per se, but in getting used to the idea that you have to go down some blind alleys sometimes before you find a solution. They help develop both tenacity and confidence; two indispensable traits for any scientist, amateur or otherwise.

Finally, I’d like to close with another ditty by Piet Hein, which I think may be the most compact description of how science works that I’ve ever encountered:

The road to wisdom? Well, it’s plain
and simple to express:
Err
and err
and err again
but less
and less
and less.

Have fun!