Tetris, Blokus, and The History of Russia

September 19, 2010

I was playing Blokus with some kids, and I asked: how in the world did they choose these pieces? There are 21 pieces in odd shapes of various sizes. Each one consists of between one and five squares. You can put your pieces down only when they touch one of your existing pieces at a corner (but never a side!). The pieces are, to say the least, quite irregular.

So how did they choose them? It might be good to look at just one set. Here’s an image I found online. I suppose it’s fine, but it doesn’t help that much to see what motivated their choice. Play with them a little, and you might think to look at just the pieces with one square (there’s one), then the pieces with two squares (there’s one), then the pieces with two squares (there are two), and so on. I just did that, and here it is.

Now if we pause for a minute, that middle row–the pieces made of four squares, looks mighty familiar. Almost as if I remember them from some very repetitive game I and 60% of the country used to play obsessively on our computers.

Yes. They’re tetris pieces.
Well, wait a minute. They almost are. Two seem to be missing.

But they’re only sort of missing. You can flip the blokus pieces over and then you’ll be able to get all seven. Of course, in tetris, you can’t flip the pieces over… you can only rotate them.

But it has to mean something that the same shapes show up in both games (except you need slightly different sets depending on whether you’re allowed to flip the pieces over or not).

And indeed, there’s a very simple answer to why they chose what they chose for both these games:

  1. Tetris uses all the different blocks you can make out of four squares (thus, the tetr is tetris). They count as different if you can’t rotate one to make another.
  2. Blokus uses all the different blocks you can make out of 1, 2, 3, 4, or 5 squares. They count as different if you can’t rotate OR reflect one to make another.

So despite looking like a weird choice, they actually made the most natural choice you can make in both cases.

Being such a natural shape, one would expect them to have been studied before, and indeed people have been playing with them for a long time. One very natural thing to do, after seeing how many there are and what they have to be (and deciding whether you want to count ones that are reflections of others as the same or different), is to see how they fit together. Can you make a square out of them? A rectangle? All sorts of fun puzzles are suggested on this page. I recently had a student decide he wanted to see if he could fill in a chess board with tetris pieces, with the added restriction that he use all of them as close to the same number of times as possible. Quite a good puzzle, actually.

The generalization of tetrominoes (tetris pieces) and pentominoes are called polyominoes. The math for them gets slippery pretty quick. For example: how many hexominoes are there? Can you prove you’ve got them all?

And, of course, there are amazing connections to history.
In fact, Tetris explains the entire history of the Soviet Union!

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