Years ago, when I was a youngster in Bigamy City, Utah1, I played games stepping on the cracks between sidewalk blocks while walking home from the park. My favorite was as follows:
Step on a crack with the right foot.
Step on a crack with the left foot.
Reverse that: left, then right.
Take the entire sequence so far and reverse it: left, right, right, left.
Now take the entire eight-step sequence, and reverse that: left, right, right, left, right, left, left, right.
And so on.
I could usually keep the sequence straight for 64 cracks or so, but eventually I would either be distracted by something or run out of sidewalk. Later I found that if you arranged black and white boxes in a square grid using the sequence, the sides of which square were a power of two, you’d get a pattern with a high degree of symmetry. You could also use “0” and “1” to represent the elements of the sequence, which led to other sorts of games.
Last night, while browsing around online, I discovered that my game does in fact have a name: the Thue-Morse sequence. I was amused to see that one of the ways of generating it involves the notion of “evil” and “odious” numbers.
- Cf. R. W. Armour