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++

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// 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;
}

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Java

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// 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)

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Python 3

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# 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

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C#

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// 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)

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PHP

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<?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)

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Output:

4

Time complexity: O(n)



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++

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// 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;
}

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Java

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// 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

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Python 3

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# 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

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C#

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// 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

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PHP

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<?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
?>

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Output:

4

Time complexity: O(log n)

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