Single Number: the idea to remember.

The core insight

XOR every value together so paired values cancel and the unique value remains.

Why it works

XOR is associative and commutative, with x XOR x equal to zero and x XOR zero equal to x.

The mistake to avoid

This invariant depends on pairs; it does not directly handle values occurring three times.

Commented Java solution

Use java.util.* imports and a surrounding solution class. Tree snippets assume the standard TreeNode definition; linked lists use ListNode and Clone Graph uses a Node with val and neighbors; Copy Random List uses Node with val, next and random. Read the input assumptions above.

// Java reference adapted from neetcode-gh/leetcode (MIT).
// XOR every value together so paired values cancel and the unique value remains.
public int singleNumber(int[] nums) {
    int ans = nums[0];
    for (int i = 1; i < nums.length; i++) ans ^= nums[i];
    return ans;
}

Java reference adapted from neetcode-gh/leetcode · MIT license

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