Improv Math: Military Clock

A few weeks ago, I went to see PROJECTproject, an improv show in the Toronto area. They had just come back from a tour to Western Canada, and one of the things they brought back with them was a structure called Military Clock. I think they said it came from Winnipeg. I'm always on the look for interesting improv structures, and this has a mathematical tilt to it, so that's why I have written it up.

The way it works is:
- split the team up into two groups
- have each group form a circle on either side of the stage
- the people from either circle who are closest to the audience (two of them) are in the scene
- whenever anyone not in the scene feels that it is getting stale, they yell "Clock!" and both circles rotate
- a new scene starts with the two people who are now at the front
- repeat until you feel you're done, or everyone has had a scene

When I saw PROJECTproject do it, they had one circle of three and another circle of four, like below.

Go ahead! Click on the stage to make the clock run! Also, click on the arrows to add or subtract improvisors to each circle.



However, this only "works" for certain numbers of improvisors in the circles. Ideally, you want to be able to run the military clock around without having any repeats, and have every possible pairing of improvisors show up. When I saw it, the improvisors would keep their character for future scenes, so we saw all possible character combinations (7 choose 2).

So what numbers does it work for? It is possible to find these numbers for any troupe size?

To rotate through every possible combination, the size of the circles must be mutually prime.

Can every troupe size be expressed as a sum of two mutually prime numbers? NO. I'm also assuming we can't have circles of size one, since that isn't the spirit of the game.
It works for
5 - 3&2
7 - 3&4 (PROJECTproject)
8 - 3&5
9 - 4&5
10 - 3&7
11 - 5&6

It doesn't work for 6, sadly.

Does it NOT work for any troupe that is larger than 11? I am not sure. But seriously, who has an improv troupe larger than 11? Think of the egos!