Why Floyd Works

Let $\mu$ be the tail length and $\lambda$ be the cycle length. When they first meet, slow traveled $\mu + a$ and fast traveled $\mu + b$ where $a$ and $b$ differ by a multiple of $\lambda$. Since fast moves twice as fast, $\mu + b = 2(\mu + a)$, so $\mu = b - 2a$.

This means $\mu$ is a multiple of $\lambda$ steps from $b$. Resetting slow and moving both one step puts them at cycle start when they meet.