Number of solutions of n = x + n ⊕ x

Given a number n, we have to find the number of possible values of X such that n = x + n ? x. Here ? represents XOR

Examples:

Input : n = 3
Output : 4
The possible values of x are 0, 1, 2, and 3.

Input : n = 2
Output : 2
The possible values of x are 0 and 2.

Brute force approach: We can see that x is always equal to or less than n, so we can iterate over the range [0, n] and count the number of values that satisfy the required condition. The time complexity of this approach is O(n).

C++

// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the number of
// solutions of n = n xor x
int numberOfSolutions(int n)
{
    // Counter to store the number
    // of solutions found
    int c = 0;

    for (int x = 0; x <= n; ++x)
        if (n == x + n ^ x)
            ++c;

    return c;
}

// Driver code
int main()
{
    int n = 3;
    cout << numberOfSolutions(n);
    return 0;
}

Java

// Java implementation of above approach
import java.util.*;
import java.lang.*;

class GFG
{
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Counter to store the number
    // of solutions found
    int c = 0;

    for (int x = 0; x <= n; ++x)
        if (n == x + (n ^ x))
            ++c;

    return c;
}

// Driver code
public static void main(String args[])
{
    int n = 3;
    System.out.print(numberOfSolutions(n));
}
}

// This code is contributed
// by Akanksha Rai(Abby_akku)

Python3

# Python 3 implementation
# of above approach

# Function to find the number of
# solutions of n = n xor x
def numberOfSolutions(n):

    # Counter to store the number
    # of solutions found
    c = 0

    for x in range(n + 1):
        if (n ==( x +( n ^ x))):
            c += 1

    return c

# Driver code
if __name__ == "__main__":
    n = 3
    print(numberOfSolutions(n))

# This code is contributed
# by ChitraNayal

C#

// C# implementation of above approach
using System;

class GFG
{
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Counter to store the number
    // of solutions found
    int c = 0;

    for (int x = 0; x <= n; ++x)
        if (n == x + (n ^ x))
            ++c;

    return c;
}

// Driver code
public static void Main()
{
    int n = 3;
    Console.Write(numberOfSolutions(n));
}
}

// This code is contributed
// by Akanksha Rai(Abby_akku)

PHP

<?php
// PHP implementation of above approach

// Function to find the number of
// solutions of n = n xor x
function numberOfSolutions($n)
{
    // Counter to store the number
    // of solutions found
    $c = 0;

    for ($x = 0; $x <= $n; ++$x)
        if ($n == $x + $n ^ $x)
            ++$c;

    return $c;
}

// Driver code
$n = 3;
echo numberOfSolutions($n);

// This code is contributed
// by Akanksha Rai(Abby_akku)

Javascript

<script>

    // Javascript implementation of above approach

    // Function to find the number of
    // solutions of n = n xor x
    function numberOfSolutions(n)
    {

        // Counter to store the number
        // of solutions found
        let c = 0;

        for(let x = 0; x <= n; ++x)
            if (n == x + n ^ x)
                ++c;

        return c;
    }

    let n = 3;
    document.write(numberOfSolutions(n));

    // This code is contributed by divyesh072019.
</script>

Output:

4

Time complexity: O(n)

Auxiliary Space: O(1)

Efficient approach: We can solve this problem in a more efficient way if we consider n in its binary form. If a bit of n is set, i.e. 1, then we can deduce that there must be a corresponding set bit in either x or n ? x (but not both). If the corresponding bit is set in x, then it is not set in n ? x as 1 ? 1 = 0. Otherwise the bit is set in n ? x as 0 ? 1 = 1. Therefore for every set bit in n, we can have either a set bit or an unset bit in x. However, we cannot have a set bit in x corresponding to an unset bit in n. By this logic, the number of solutions comes out to be 2 raised to the power of the number of set bits in n. The time complexity of this approach is O(log n). 

C++

// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the number of
// solutions of n = n xor x
int numberOfSolutions(int n)
{
    // Number of set bits in n
    int c = 0;

    while (n) {
        c += n % 2;
        n /= 2;
    }

    // We can also write (1 << c)
    return pow(2, c);
}

// Driver code
int main()
{
    int n = 3;
    cout << numberOfSolutions(n);
    return 0;
}

Java

// Java  implementation of above approach
import java.io.*;

class GFG {
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Number of set bits in n
    int c = 0;

    while (n>0) {
        c += n % 2;
        n /= 2;
    }

    // We can also write (1 << c)
    return (int)Math.pow(2, c);
}

// Driver code

    public static void main (String[] args) {
        int n = 3;
    System.out.println( numberOfSolutions(n));
    }
}
//This code is contributed by anuj_67

Python3

# Python3 implementation of above approach

# from math lib. import everything
from math import *

# Function to find the number of
# solutions of n = n xor x
def numberOfSolutions(n) :

    # Number of set bits in n
    c = 0

    while(n) :
        c += n % 2
        n //= 2

    # We can also write (1 << c)
    return int(pow(2, c))


# Driver code
if __name__ == "__main__" :

    n = 3
    print(numberOfSolutions(n))

# This code is contributed by ANKITRAI1

C#

// C# implementation of above approach
using System;

class GFG
{
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Number of set bits in n
    int c = 0;

    while (n > 0)
    {
        c += n % 2;
        n /= 2;
    }

    // We can also write (1 << c)
    return (int)Math.Pow(2, c);
}

// Driver code
public static void Main ()
{
    int n = 3;
    Console.WriteLine(numberOfSolutions(n));
}
}

// This code is contributed by anuj_67

PHP

<?php
// PHP implementation of above approach

// Function to find the number of
// solutions of n = n xor x
function numberOfSolutions($n)
{
    // Number of set bits in n
    $c = 0;
    while ($n)
    {
        $c += $n % 2;
        $n /= 2;
    }

    // We can also write (1 << c)
    return pow(2, $c);
}

// Driver code
$n = 3;
echo numberOfSolutions($n);

// This code is contributed by jit_t
?>

Javascript

<script>

    // Javascript implementation of above approach

    // Function to find the number of
    // solutions of n = n xor x
    function numberOfSolutions(n)
    {
        // Number of set bits in n
        let c = 0;

        while (n > 0) {
            c += n % 2;
            n = parseInt(n / 2, 10);
        }

        // We can also write (1 << c)
        return Math.pow(2, c);
    }

    let n = 3;
    document.write(numberOfSolutions(n));
</script>

Output:

4

Time complexity: O(log n)
Auxiliary Space: O(1)