The intended solution was seemingly to calculate the length of the cycle (as in, until you were on a "Z" node) for each starting node separately. Once you'd done this, you could find the LCM of these cycle lengths to find the overall period of the cycle (~10^13).
This solution might not work if your cycles weren't constant length -- for example imagine when walking your graph you found your i_th Z-node after {6, 7, 6, 7, 6, 7, 6, 7 ...} steps; or perhaps {6, 7, 7, 7, ...}. In the test data, this didn't occur - it was always {n, n, n, n, ...}.
I can give another example perhaps - consider you're asked to find the last 3 digits of 2^n for n >= 1. You start off calculating: 002, 004, 008, ... eventually you get to 2^103 which ends in 008. This means that the cycle length would be {103, 101, 101, 101, 101, ...} since it'll never get back to 002 or 004. Solving this is a bit more difficult than the constant cycle lengths, since it's not a simple LCM.