You might be interested to know that, if you change the wording to the more general "every nth person" instead of "every other person," this is called the Josephus problem. To the best of my knowledge, it is still unsolved.
> A similar probelem is the locker problem but the lockers are in a row, > not in a circle. This turned out to be a version of mod math but we are > unsure of how to write it up due to the powers of 2 that are involved.
If you're talking about this locker problem:
There's a row of 100 lockers. The first student runs down the hall and closes every locker. The second student runs down the hall and opens every other locker. The third student runs down the hall and CHANGES every third locker (opens it if it was closed, closes it if it was open). Etc. After the 100th student heads down the hall, which of the lockers are open and which are closed?
The answer depends on facts about factorizations rather than modular arithmetic or powers of two. Perhaps there's another "locker problem"?