How many ways can you arrange two square tiles so that they share at least edge? It doesn’t take much to show that there’s only one way: form a rectangle, sometimes called a “domino.” If you have three squares that have to be joined so that each shares an edge with one of the others, you can have them all in a straight line, or forming a right angle. Accounting for rotation, all of these lines are identical, and so are the right angles, thus ‘straight’ and ‘angled’ are really the only two options for “triominoes.” Using the same rules, a tetromino has five varieties, and a pentomino twelve. It’s that last set that has captivated puzzlers for several decades, and earned its place in our collection beside other classics.
As with Burr puzzles and many others, the invention of pentominoes is frequently misattributed to the person who named them. In this case, that’s Solomon W. Golomb, who gave a talk at the Harvard Mathematics Club in 1953 about his mathematical study of “polyominoes”: figures composed of some number of contiguous squares like in the example above. His talk was published the following year in American Mathematical Monthly, and parts of it went on to be reprinted by Scientific American after the great Martin Gardner became a fan. Nevertheless, the twelve major units had been previously identified in antiquity by some unknown master of Go, and pentomino problems were a favorite of the Fairy Chess Review in the 1830’s and 40’s, as a kind of dissection puzzle for chessboards.
While Golomb may not have invented pentominoes, he certainly deserves credit for popularizing them. His 1965 book “Polyominoes” continued the forward momentum from the Scientific American article, as did Gardner’s Mathematical Puzzles and Diversions, which included several chapters on the subject. Along with the name, Golomb’s works are notable for their rigorous approach, as befits his original mathematical study, and for giving us the nomenclature, where each of the twelve shapes was named for the letter of the alphabet they most resemble. That’s T, U, V, W, X, Y, Z, F, I, L, P, and N respectively; most enthusiasts use the convenient mnemonic of “Filipino” to recall the last five letters.
Like other mathematical games, pentominoes have their devoted fans, and it’s easy to see why. Mathematically speaking, they are a closed set: there’s only twelve ways to arrange five contiguous squares on a plane. This sets them apart from tangrams, which are mathematically arbitrary and only form a set due to ancient tradition. Where tangrams are good for making paradoxes, pentominoes are uniquely suited to creating tessellations, and illustrating the mathematical principles of fractals. This, then, is perhaps the greatest advantage of pentominoes: they are less a single game or puzzle and more a platform that allows you to run a variety of games: scaling, tiling, taking turns with a partner, checkerboard challenges, or pursuing more complex designs.
I was in elementary school when a teacher introduced me to pentominoes. These were the three dimensional kind made with cubes rather than squares, and my fellow students and I learned to arrange them into a variety of shapes, with all sorts of transformations and variations. As part of our curriculum, we came up with our own names for each piece based on what we thought they resembled, which was a fun exercise in abstract thinking to go along with the geometry. I remember learning how to make them into a cube, and then into a long rectangular shape with lumps on the ends that we called a “bed.” One of my classmates discovered that there was a way to make the bed turn into the cube by rolling it up carefully, or removing the lower section and inverting it. I remember being stunned because my own cube solution didn’t unroll to make a bed, and this meant there were at least two ways to make the cube. I spend hours trying to figure out the relationship between the two, and shared what I learned with my classmates. It’s the sharing I remember best: there’s something about a good puzzle that begs to be shared.
Many years later, I happened upon a travel game that used the flat variety of polyominoes. Players took turns laying pieces down so that their own pieces touched at the corners only, trying to cover enough of the board that their opponent couldn’t make a legal move. This puzzle became very popular among our children for a while, and still challenges us to this day. Having an opponent to sharpen your brain against is a wonderful feeling, and a counterpart to that desire to share. Even as we work against them, our opponents make us better, and that mutual self-improvement can be as fulfilling as sharing our individual discoveries.
If you want to try your hand at the cube puzzle from my elementary school, it’s available as our own Pentomino Puzzle, or the next level up with the Pentomino Chess Puzzle. It’s unlikely those will ever get boring, but if you want an extra challenge, check out our 3D Pentomino Cube Chess, where you have to turn twenty-five versions of the “y” pentomino into a 5x5x5 cube with proper checkerboard arrangement.
On the other hand, if the competition is more your speed, our Pentomino Puzzle Set can be played solo or with another player selecting your card and setting a time limit. This set is also a fun alternative to tangrams, although we still have plenty of those as well. If you want to try the blocking game we got for our kids, you can make it yourself with two sets of pentominoes that have different markings (i.e. plain and checkerboard). You’ll also need a gameboard to limit the space--any of the boxes they come in should do the trick. Once you have that, the rules I describe above should be enough for hours of competitive fun.
Thanks as always for puzzling with me. If you like sharing puzzles as much as we do, please share this blog post with puzzle lover in your life, and consider us for your holiday gift-giving.
By Matthew Barrett