How would you identify the size of a circular turing tape?
This answer was inspired by YuvalFilums's comment.



I will denote values in the cells with $t$ and $f$ and locations on the tape with $0, 1, 2, \dots$ to avoid confusion.

First, check that the length $\neq 1$. To do this, set the cell under the counter to $f$. Then, move right $1$ cell and set that cell to $t$, and return to $0$, checking if that is $t$. If so you have your answer, otherwise move to $2$ and set that to $f$, and return to $0$.

Second, check that the length $\neq 2$ similarly.

Then repeat the following procedure indefinitely:


Move right until encountering an $f$. Set this to $t$.
Move right one cell and set it to $f$.
Return to $0$. If it is $t$, find the first $f$ after $1$, and move left one cell. This is your answer. Otherwise, repeat.