Count of x in given range such that bitwise XOR of x, (x+1) and (x+2), (x+3) are equal

Given an integer N, the task is to count the number of integers (say x) in the range [0, 2^N−1] such that x⊕(x+1) = (x+2)⊕(x+3). [where ⊕ represents bitwise Xor]

Examples:

Input: N = 1
Output: 1
Explanation: Only 0 is the valid x, as, 0 ⊕ 1 = 2 ⊕ 3 = 1

Input: N = 3
Output: 4

Naive Approach: The simple approach to solve the given problem is to generate all possible values of x in the range [0, 2^N−1] and check if they satisfy the given criteria i.e x⊕(x+1) = (x+2)⊕(x+3). 

Follow the steps mentioned below to implement the idea:

• Iterate from i = 0 to N.
  • Check if (i ⊕ (i+1)) = ((i+2) ⊕ (i+3))
  • If the condition is satisfied increment the count of such numbers.
• Return the final count as the answer.

Below is the implementation of the above approach:

C++

// C++ code to implement the approach
#include <bits/stdc++.h>

using namespace std;

// Function to find number of integers
// satisfying the condition
int countOfvalues(int N)
{
    // Stores the resultant count of pairs
    int count = 0;

    for (int i = 0; i < (1 << N); i++) {

        // Iterate over the range
        if ((i ^ (i + 1)) == ((i + 2) ^ (i + 3)))
            count += 1;
    }
    return count;
}
// Driver Code
int main()
{
    int N = 3;

    // Function call
    cout << countOfvalues(N);
    return 0;
}

// This code is contributed by Rohit Pradhan

Java

// Java code to implement the approach
class GFG {

  // Function to find number of integers
  // satisfying the condition
  public static int countOfvalues(int N)
  {

    // Stores the resultant count of pairs
    int count = 0;

    for (int i = 0; i < (1 << N); i++) {

      // Iterate over the range
      if ((i ^ (i + 1)) == ((i + 2) ^ (i + 3)))
        count += 1;
    }
    return count;
  }

  // Driver code
  public static void main(String[] args)
  {
    int N = 3;

    // Function call
    System.out.println(countOfvalues(N));
  }
}

// This code is contributed by phasing17

Python3

# Python3 code to implement the approach

# Function to find number of integers
# satisfying the condition
def countOfvalues(N):

    # Stores the resultant count of pairs
    count = 0

    for i in range(0, 2**N):

       # Iterate over the range
        if i ^ (i + 1) == (i + 2) ^ (i + 3):
            count += 1

    return count


# Driver Code
if __name__ == '__main__':
    N = 3

    # Function call
    print(countOfvalues(N))

C#

// C# Program of the above approach
using System;

class GFG
{

  // Function to find number of integers
  // satisfying the condition
  public static int countOfvalues(int N)
  {

    // Stores the resultant count of pairs
    int count = 0;

    for (int i = 0; i < (1 << N); i++) {

      // Iterate over the range
      if ((i ^ (i + 1)) == ((i + 2) ^ (i + 3)))
        count += 1;
    }
    return count;
  }

  // Driver Code
  public static void Main ()
  {
    int N = 3;

    // Function call
    Console.Write(countOfvalues(N));
  }
}

// This code is contributed by sanjoy_62.

Javascript

<script>
       // JavaScript code for the above approach

       // Function to find number of integers
       // satisfying the condition
       function countOfvalues(N)
       {

           // Stores the resultant count of pairs
           let count = 0;

           for (let i = 0; i < (1 << N); i++) {

               // Iterate over the range
               if ((i ^ (i + 1)) == ((i + 2) ^ (i + 3)))
                   count += 1;
           }
           return count;
       }

       // Driver Code
       let N = 3;

       // Function call
       document.write(countOfvalues(N));

   // This code is contributed by Potta Lokesh
   </script>

Output

4

Time Complexity: O(2^N) 
Auxiliary Space: O(1)

Efficient Approach: The problem can be solved efficiently based on the following mathematical idea:

• If x is such that it is a even number then x+2 is also a even number and both (x+1) and (x+3) will be odd numbers. 
• Now two consecutive even and odd number has only a single bit difference only on their LSB position.
• So the bitwise XOR of (x and x+1) and (x+2 and x+3) both will be 1, when x is even.
• Therefore all the even numbers in the given range [0, 2^N−1] is a possible value of x.

So total number of such values are (2^N – 1)/2 = 2^N-1

Follow the illustration below to visualize the idea better:

Illustration:

Consider N = 3. So the numbers are in range [0, 7]

All even numbers in the range are 0, 2, 4 and 6

=> When x = 0: 0 ⊕ 1 = 1 and 2 ⊕ 3 = 1. Therefore the relation holds
=> When x = 2: 2 ⊕ 3 = 1 and 4 ⊕ 5 = 1. Therefore the relation holds
=> When x = 4. 4 ⊕ 5 = 1 and 6 ⊕ 7 = 1. Therefore the relation holds
=> When x = 6: 6 ⊕ 7 = 1 and 8 ⊕ 9 = 1. Therefore the relation holds.

Now for the odd numbers if it is done:

=> When x = 1: 1 ⊕ 2 = 3 and 3 ⊕ 4 = 7. Therefore the relation does not hold.
=> When x = 3: 3 ⊕ 4 = 7 and 5 ⊕ 6 = 3. Therefore the relation does not hold.
=> When x = 5: 5 ⊕ 6 = 3 and 7 ⊕ 8 = 15. Therefore the relation does not hold.
=> When x = 7: 7 ⊕ 8 = 15 and 9 ⊕ 10 = 3. Therefore the relation does not hold.

So total possible values are 4 = 2^3-1

Therefore to get the answer calculate the value of 2^N-1 for given N

Below is the implementation of the above approach.

C++

// C++ code to implement the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to calculate
// the number of possible values
int countValues(int N) { return pow(2, N - 1); }

// Driver Code
int main()
{
    int N = 3;
    // Function call
    cout << countValues(N);
    return 0;
}

// This code is contributed by Rohit Pradhan

Java

// Java code to implement the above approach
class GFG {

  // Function to calculate
  // the number of possible values
  public static int countValues(int N) { return (int)Math.pow(2, N - 1); }

  public static void main(String[] args) {
    int N = 3;

    // Function call
    System.out.println(countValues(N));
  }
}

//This code is contributed by phasing17

Python3

# Python code to implement the above approach

# Function to calculate
# the number of possible values
def countOfValues (N):
    return pow(2, N - 1)

# Driver code
if __name__ == '__main__':
    N = 3

    # Function call
    res = countOfValues(N)
    print(res)

C#

// C# program for above approach
using System;
using System.Collections.Generic;

public class GFG
{

  // Function to calculate
  // the number of possible values
  public static int countValues(int N)
  {
    return (int)Math.Pow(2, N - 1);

  }

  // Driver code
  public static void Main(String[] args)
  {
    int N = 3;

    // Function call
    Console.Write(countValues(N));
  }
}

// This code is contributed by code_hunt.

Javascript

<script>

// JavaScript code to implement the above approach

// Function to calculate
// the number of possible values
function countValues(N)
{
    return Math.pow(2, N - 1);
}

// Driver Code
let N = 3;

// Function call
document.write(countValues(N));

// This code is contributed by shinjanpatra

</script>

Output

4

Time Complexity: O(log(N)) 
Auxiliary Space: O(1)