Evil and odious

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.

Morse-Thue_sequence

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.

Notes

  1. Cf. R. W. Armour