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XOR of 1 to n Numbers

Easy Bit Manipulation Tricks to remember Original on GeeksforGeeks

XOR of 1 to n Numbers is a easy bit manipulation problem solved with the tricks to remember pattern. The best approach, optimal (period-4 pattern), runs in O(1) time and O(1) space. Below are 2 approaches in Java, from loop up.

Problem

Return 1 ^ 2 ^ 3 ^ ... ^ n (XOR of all numbers from 1 to n) in O(1).

Examples

Example 1

Input
n = 6
Output
7
Why
1^2^3^4^5^6 = 7.

Example 2

Input
n = 7
Output
0

Constraints

  • 1 <= n <= 10^9

Animated walkthrough

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Concepts first, then the problem and every approach, step by step.

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Solutions

Try it yourself first. Then compare: each approach lists its idea, the steps, its time and space, and the Java code.

ApproachTimeSpace
LoopO(n)O(1)
Optimal (period-4 pattern)O(1)O(1)

1Loop

TimeO(n)
SpaceO(1)

XOR every number from 1 to n.

  1. x = 0; for i in 1..n: x ^= i.
Java
class Solution {
    static int xorUpto(int n) {
        int x = 0;
        for (int i = 1; i <= n; i++) x ^= i;
        return x;
    }
}

2Optimal (period-4 pattern)

TimeO(1)
SpaceO(1)

Every block of four consecutive numbers starting at a multiple of 4 XORs to 0, so the answer depends only on n % 4: 0 → n, 1 → 1, 2 → n + 1, 3 → 0.

  1. switch (n % 4).
Java
class Solution {
    static int xorUpto(int n) {
        switch (n % 4) {
            case 0: return n;
            case 1: return 1;
            case 2: return n + 1;
            default: return 0;
        }
    }
}

Edge cases to test

  • n = 1

Hints

Hint 1

Write out the prefix XORs for n = 1..8. They repeat every 4: n, 1, n + 1, 0.

FAQ

What is the best time complexity for XOR of 1 to n Numbers?

Optimal (period-4 pattern) runs in O(1) time and O(1) extra space.

Which pattern does XOR of 1 to n Numbers use?

It is a bit manipulation problem that uses the tricks to remember pattern. Other problems with the same pattern: Number of 1 Bits, Set the rightmost unset bit.

Is there a brute force solution for XOR of 1 to n Numbers?

Yes. Loop takes O(n) time and O(1) space. XOR every number from 1 to n.

Which edge cases should I test for XOR of 1 to n Numbers?

n = 1.