Finding Cycle Start

Once you detect a cycle, finding where it starts requires a clever insight.

After fast and slow meet inside the cycle, reset one pointer to the head. Move both pointers one step at a time. They'll meet at the cycle start.

# After detecting cycle (slow == fast)
slow = head
while slow != fast:
    slow = slow.next
    fast = fast.next
return slow # cycle start

Why does this work? Let $a$ = distance from head to cycle start, $b$ = distance from cycle start to meeting point, $c$ = cycle length.

When they first meet: slow traveled $a + b$, fast traveled $a + b + kc$ for some $k$. Since fast moves twice as fast: $2(a + b) = a + b + kc$, so $a = kc - b$. So walking $a$ steps from the meeting point lands exactly at the cycle start.