Regarding Floyd's cycle-finding algorithm

[cyberfrenzy]

Well I understand what is happening - the tortoise moving 1 step, and the hare moving 2 steps and finally coinciding at a point. But I want a mathematical proof of this - ie; when you have 2 counters moving in a cycle, one going 1 step, the other 2 steps, then they finally meet at a point.

 

[matt grime]

Suppose that they start M places apart (suitably oriented) on a cycle of size N. How many places apart are they after k steps?