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.)
Each number is a multiple of 3, or 1 more than a multiple of 3, or one less than a multiple of 3. These number we will call x's, y's, and z's.
If we add x + y + z, then we get x, because y and z will even each other out, since y is 1 more than multiple of 3 and z is one less.
Because of this, we can conclude that we have to arrange the numbers in a repeating pattern, like x, y, z, x, y, z, ... or z, y, x, z, y, x, ...
In this set 1, 2, 3, ..., 9, there are 3 x's, 3 y's, and 3 z's.
To fill up a pattern, there are 3 choices for the first x, y, z, 2 choices for second, then 1 choice.
3^3 * 2^3 * 1^3 = 27*8*1 = 216
This is equal for both patterns, so there are 216 * 2 = 432 ways to arrange the numbers.