Sounds like a bit more complicated odds-and-evens. Rather than mess with pi and circles, you could just cast 0-9 fingers and get mod 6 + 1.

I think you need 0-5 to avoid bias. (If you are good at generating random 0-9, then you are biasing 0-4 over 5-6, in the mod-6 regime, and I can preserve that bias by preferring to choose small or large numbers depending on whether I want a small or large result)