^{1}The game of nuclear pennies works as follows; we start with a semi-infinite strip of squares, which have pennies on them. We can "split" a penny on site n into two pennies, one on site n-1 and one on site n+1; we can "fuse" pennies on sites n-1 and n+1 into one on site n. We can analyze positions in this game using arithmetic in the ring of integers with a sixth root of unity, ω, adjoined. Given a position with a

_{n}coins in the nth position, associate the number

Σ

_{n}a

_{n}ω

^{n}

with this position; it turns out that two positions can be reached from each other if and only if their associated numbers are the same. (We have the relations ω

^{3}= -1, ω - ω

^{2}= 1.)

In thermonuclear pennies the splitting rule is a bit different; it turns out that "sixth root of unity" can be replaced with "fourth root of unity", which is the imaginary unit. Positions in this game represent Gaussian integers. And there's even a way to multiply the positions, which corresponds to multiplication of Gaussian integers.

Challenge (which may or may not be hard, I haven't thought about it all): develop such a penny-fusing game where the positions can be understood as elements of the ring of integers with a

*fifth*root of unity adjoined. (I don't expect the result to be as nice; for example,

**Z**[e

^{2πi/5}] isn't a lattice in the complex plane. I'm not sure why that's relevant, but it might be.)

1. Is "legitimate" a verb, meaning "to make legitimate"? (The stress in the verb would be on the last syllable.) I think it should be, if it's not already. I've heard "precise" used as a verb (I don't remember how it's pronounced), and "legitimate" strikes me as more

*phonetically*akin to verbs than "precise" is, probably because of the -ate suffix.

## 2 comments:

"Legitimate" is most certainly a verb. "Precise" is not. According to my dictionary, at least.

I have heard "legitimate" as a verb several times (though I think "legitimize" is both better - and, as it happens, more widely used - for "make legitimate").

But "precise" as a verb? Gack!

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