Determine the number of ways of placing the numbers $1, 2, 3, \dots, 9$ in a circle, so that the sum of any three numbers in consecutive positions is divisible by $3.$ (Two arrangements are considered the same if one arrangement can be rotated to obtain the other.)
I've experimented with possible combinations and found that it works when we put a multiple of 3 next to a number one more than a multiple of three beside a number that is two more than a multiple of 3. If we continue with this pattern around the circle, it works.
However, I'm curious in finding a more systematic approach than listing out all different combinations.
Thanks in advance!