C-x_C-f

I believe GP is using the stats/data science meaning of the word "censor" which is a neutral term simply indicating missing/obscured data (without implication of intent) [0]

V - E + F = 2

where V, E, F are the number of vertices, edges, and faces, respectively.

This holds for any polyhedron (and other shapes have similar equations possibly with a different right hand side) and the left hand side is called the Euler characteristic of the soccer ball (or any polyhedron).

Spoiler warning: the Wikipedia article for the Euler characteristic [0] has a worked out example specifically for the soccer ball.

FWIW the generalization of binomial coefficient which allows you to express an n-dimensional solution is called a multinomial coefficient [0]. So in a 3d 20x20x20 box we would have (60 multichoose 20, 20, 20) paths.

Also, the wiki article doesn't mention this but the growth rate of (n multichoose k1, k2, ..., km) as we increase n but fix the ratios p1 = k1 / n, ..., pm = km / n is precisely the Shannon entropy of the categorical distribution with probabilities p1, ..., pm . The wiki article for entropy [1] states the result for the binomial coefficient, which can be written as (n choose k) = (n multichoose k, (n - k)) .

Actually a lot of basic information-theoretic results about entropy and related quantities (e.g. the properties of the Boltzmann distribution/softmax function) can be derived from similar discrete counting problems after taking a large-n limit. I don't have links at the ready but I might edit this comment if I remember places which explain this stuff.

ETA: I realize this sounds nitpicky and stickler-y so I just want to point out that I loved the book (and Greg Egan's work in general) and figuring out the automaton stuff was genuinely some of the most fun I've had out of a book.

Many interesting systems (like springs) are near equilibrium, which means that the potential energy is at a local minimum. A spring is an example, but also a pendulum.

When the potential is at a local minimum, its gradient is zero. So if you Taylor expand it you only get second-order contributions. For a spring, the potential energy looks like V(x) = V(0) + k * x * 2 where x is the displacement and k is a constant.

Differentiating, you get harmonic motion: F(x) = k * x

Broadly speaking, this applies to all systems near equilibrium, simply from Taylor expanding the energy. And it's not only in classical mechanics, but in all branches of physics. Sydney Coleman [0] is often quoted as saying something like "QFT is simple harmonic motion taken to increasing levels of abstraction." [1]