Find the Duplicate Number

Every index pointing to arr[i] forms a directed edge in a functional graph. Because every value is between 1 and n while the array has n + 1 cells, the pigeonhole principle guarantees two different indices point to the same node, creating a cycle whose entry point is precisely the duplicate value.

Full solution

Floyd cycle detection over the value-as-pointer graph: it is O(n) time and O(1) space without mutating the input, which is what rules out the hash set and the sign-marking trick in an interview.

def find_duplicate(nums: list[int]) -> int:
    # values are indices, so nums[i] is a "next" pointer; the duplicate is the cycle entrance
    slow = fast = nums[0]
    while True:
        slow = nums[slow]
        fast = nums[nums[fast]]
        if slow == fast:
            break
    # phase 2: the distance from the start to the entrance equals the distance from the meeting point
    slow = nums[0]
    while slow != fast:
        slow = nums[slow]
        fast = nums[fast]
    return slow

function findDuplicate(nums) {
  let slow = nums[0];
  let fast = nums[0];
  do {
    slow = nums[slow];
    fast = nums[nums[fast]];
  } while (slow !== fast);
  slow = nums[0];
  while (slow !== fast) {
    slow = nums[slow];
    fast = nums[fast];
  }
  return slow;
}

The theory behind it

Arrays — the ground this problem stands on. All Arrays problems

What Arrays is

An array is a row of fixed boxes laid side by side in computer memory, like numbered lockers in a hallway. Because each box occupies identical space and sits directly next to its neighbors, jumping to locker zero or locker ten thousand takes the exact same tiny fraction of time. Every box holds an item of the same type, addressed by an offset number called an index.

When to reach for it

Reach for an array when items arrive in a known sequence and need immediate retrieval by position number. Problems asking for running totals, prefix accumulations, cyclic rotations, or in-place rearrangements signal array mechanics. Whenever constraints require constant-time random lookups or contiguous cache scans across fixed collections, a flat sequence is the default container.

How the pattern works

Visualize a tape with zero-indexed slots stretching from start to end. Keep track of write and read cursors when modifying contents without allocating helper buffers. For running computations, maintain an invariant such as having processed all elements left of the current index while pending elements wait to the right. When modifying entries in place, consider scanning backwards from the end so unread data is not overwritten.