Floyd’s tortoise and hare
Figure 1: tortoise and hare
hare and tortoise both starts from point Start, at speed of \(2\) and \(1\) respectively. They’ll meet at some point in the cycle.
when they meet, hare traveled \(S_2 - S_{1}\) more points than tortoise. We have: (\(\lambda\) is the cycle length)
\[
S_{2} - S_{1} = 0 = S_{1} = x + m \pmod{\lambda}
\]
after they meet, one starts over from Start point, the other starts from Meet point, at speed of \(1\), they’ll meet again at Cycle Start point. because:
\[
x = x + m + x \pmod{\lambda}
\]