Find the number of ways of arranging the numbers 1, 2, 3, ... 9 in a circle, so that the sum of any three adjacent numbers 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.
