find the duplicate number

def findDuplicate(nums):
    slow, fast = nums[0], nums[nums[0]]
 
    while slow != fast:
        slow = nums[slow]
        fast = nums[nums[fast]]
    
    slow = 0
    while slow != fast:
        slow = nums[slow]
        fast = nums[fast]
    return slow
 
# Example usage:
nums = [3, 1, 3, 4, 2]
duplicate = findDuplicate(nums)
print(f"The duplicate number is: {duplicate}")
 
 
nums = [1, 3, 4, 2, 2]
duplicate = findDuplicate(nums)
print(f"The duplicate number is: {duplicate}")

public class FindDuplicatesUsing2Pointers {
    public int findDuplicate(int[] nums) {
        int slow = nums[0];
        int fast = nums[nums[0]];
 
        // Phase 1: find a meeting point inside the cycle
        while (slow != fast) {
            slow = nums[slow];
            fast = nums[nums[fast]];
        }
 
        // Phase 2: find the entrance to the cycle
        slow = 0;
        while (slow != fast) {
            slow = nums[slow];
            fast = nums[fast];
        }
 
        return slow;
    }
 
    public static void main(String[] args) {
        FindDuplicatesUsing2Pointers solution = new FindDuplicatesUsing2Pointers();
        int[] nums = {3, 1, 3, 4, 2}; // Example input
        int duplicate = solution.findDuplicate(nums);
        System.out.println("The duplicate number is: " + duplicate);
    }
}

ELI5 Version:

Imagine 5 treasure chests in a row, numbered 0 to 4. Each chest has a note inside it saying "go open chest number ___."
You (the slow kid) open chest 0. The note says "go to chest 3." So now you're standing at chest 3 - you moved, based on the note.
Your friend (the fast kid) does the same thing but takes two notes at once before stopping to wait for you.
You keep following notes: 0 → 3 → 4 → 2 → 3 → 4 → 2 → 3 → 4 → 2... forever. You never run out of chests to visit because eventually the notes just send you in a loop between chest 2, 3, and 4 forever. You'll never reach chest 1 no matter how long you walk, because nothing points there.
Because your friend moves faster, they lap you inside that loop - you two bump into each other at some chest. That bump is Phase 1. But the chest where you bump into each other isn't special - it's just wherever your speeds happened to line up.
To find the actual doorway into the loop - the first chest that's part of the repeating cycle - you send yourself back to chest 0 and walk one step at a time, while your friend also now only takes one step at a time from where you bumped. Walk, walk, walk - you meet again, and this meeting spot is guaranteed to be the doorway. That's Phase 2.