Count smaller numbers whose XOR with n produces greater value


Given a positive integer n, count numbers x such that 0 < x <n and x^n > n where ^ is bitwise XOR operation.


Input  : n = 12
Output : 3
Numbers are 1, 2 and 3
1^12 > 12,  2^12 > 12 and 3^12 > 12

Input  : n = 11
Output : 4
Numbers are 4, 5, 6 and 7

A number may x produce a greater XOR value if x has a set bit at a position where n has a 0 bit. So we traverse bits of n, and one by one consider all 0 bits. For every set bit at position k (Considering k = 0 for rightmost bit, k = 1 for second rightmost bit, ..), we add 2 2k to result. For a bit at k-th position, there are 2k numbers with set bit 1.

Below is C++ implementation of the above idea.

// C++ program to count numbers whose XOR with n
// produces a value more than n.
using namespace std;

int countNumbers(int n)
    /* If there is a number like m = 11110000,
    then it is bigger then 1110xxxx. x can either
    0 or 1. So we have pow(2, k) greater numbers
    where k is  position of rightmost 1 in m. Now
    by using XOR bit at each  position can be changed.
    To change bit at any position, it needs to XOR
    it with 1. */
    int k = 0; // Position of current bit in n

    /* Traverse bits from LSB (least significant bit)
       to MSB */
    int count = 0;  // Initialize result
    while (n > 0)
        // If current bit is 0, then there are
        // 2^k numbers with current bit 1 and
        // whose XOR with n produces greater value
        if ((n&1) == 0)
            count += pow(2, k);

        // Increase position for next bit
        k += 1;

        // Reduce n to find next bit
        n >>= 1;

    return count;

// Driver code
int main()
    int n = 11;
    cout << countNumbers(n) << endl;
    return 0;



Time complexity : O(Log n)

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