[cjg]

Because there are a finite number of states, it will always loop.

An interesting question might be around how many distinct loops there are and whether there's some pattern to the loop lengths.

[schwartzworld]

> Because there are a finite number of states, it will always loop.

No it wouldn't. The fact that it will pass through every state infinite times doesn't mean it will do it in the same order each time. Each digit of pi has only 10 possible states, but it never loops.

[tedsanders]

If the state evolution:

(a) is deterministic

(b) depends only on the current state, and

(c) can only occupy a finite number of states

then it will loop.

Pi digits do not satisfy this because while the digit space is finite (10), the next digit depends on more than just the prior digit.

Related (but not the same): https://en.wikipedia.org/wiki/Poincaré_recurrence_theorem