I will call the hands A, B, and C, from fastest to slowest. So A goes around in 3 minutes, B in 4, and C in 6.
This puzzle only deals with integer minutes, because all hands point directly at numbers only every minute.
Hand B has only three possibilities. It could have moved from 1 to 10, 2 to 11, or 3 to 6. The number of minutes it takes mod 4 is 1 or 3.
Hand C, in that time could have moved 2,6, or 10 numbers over in that time. The only possibility left considering this is from 1 to 11.
Hand B is left only with the possibility of 3 to 6, and hand A goes from 2 to 10.
So this is what happened:
Hand A went from 2 to 10,
Hand B went from 3 to 6,
Hand C went from 1 to 11,
The time this took mod 12 is 5.