All possible binary numbers of length n with equal sum in both halves

Given a number n, we need to print all n-digit binary numbers with equal sum in left and right halves. If n is odd, then mid element can be either 0 or 1.

Examples:

Input  : n = 4
Output : 0000 0101 0110 1001 1010 1111 
Input : n = 5
Output : 00000 00100 01001 01101 01010 01110 10001 10101 10010 10110 11011 11111 

The idea is to recursively build left and right halves and keep track of the difference between counts of 1s in them. We print a string when the difference becomes 0 and there are no more characters to add.  

C++

// C++ program to generate all binary strings with
// equal sums in left and right halves.
#include <bits/stdc++.h>
using namespace std;

/* Default values are used only in initial call.
   n is number of bits remaining to be filled
   di is current difference between sums of
      left and right halves.
   left and right are current half substrings. */
void equal(int n, string left="", string right="",
                                        int di=0)
{
    // TWO BASE CASES
    // If there are no more characters to add (n is 0)
    if (n == 0)
    {
        // If difference between counts of 1s and
        // 0s is 0 (di is 0)
        if (di == 0)
            cout << left + right << " ";
        return;
    }

    /* If 1 remains than string length was odd */
    if (n == 1)
    {
        // If difference is 0, we can put remaining
        // bit in middle.
        if (di == 0)
        {
            cout << left + "0" + right << " ";
            cout << left + "1" + right << " ";
        }
        return;
    }

    /* If difference is more than what can be
       be covered with remaining n digits
       (Note that lengths of left and right
        must be same) */
    if ((2 * abs(di) <= n))
    {

         /* add 0 to end in both left and right
         half. Sum in both half will not
               change*/
         equal(n-2, left+"0", right+"0", di);

         /* add 0 to end in both left and right
         half* subtract 1 from di as right
         sum is increased by 1*/
         equal(n-2, left+"0", right+"1", di-1);

        /* add 1  in end in left half and 0 to the
        right half. Add 1 to di as left sum is
        increased by 1*/
        equal(n-2, left+"1", right+"0", di+1);

        /* add 1 in end to both left and right
          half the sum will not change*/
        equal(n-2, left+"1", right+"1", di);
    }
}

/* driver function */
int main()
{
    int n = 5; // n-bits
    equal(n);
    return 0;
}

Java

// Java program to generate all binary strings
// with equal sums in left and right halves.
import java.util.*;

class GFG
{

// Default values are used only in initial call.
// n is number of bits remaining to be filled
// di is current difference between sums of
// left and right halves.
// left and right are current half substrings.
static void equal(int n, String left,
                         String right, int di)
{
    // TWO BASE CASES
    // If there are no more characters to add (n is 0)
    if (n == 0)
    {
        // If difference between counts of 1s and
        // 0s is 0 (di is 0)
        if (di == 0)
            System.out.print(left + right + " ");
        return;
    }

    /* If 1 remains than string length was odd */
    if (n == 1)
    {
        // If difference is 0, we can put
        // remaining bit in middle.
        if (di == 0)
        {
            System.out.print(left + "0" + right + " ");
            System.out.print(left + "1" + right + " ");
        }
        return;
    }

    /* If difference is more than what can be
    be covered with remaining n digits
    (Note that lengths of left and right
     must be same) */
    if ((2 * Math.abs(di) <= n))
    {

            // add 0 to end in both left and right
            // half. Sum in both half will not
            // change
            equal(n - 2, left + "0", right + "0", di);

            // add 0 to end in both left and right
            // half* subtract 1 from di as right
            // sum is increased by 1
            equal(n - 2, left + "0", right + "1", di - 1);


        // add 1 in end in left half and 0 to the
        // right half. Add 1 to di as left sum is
        // increased by 1*
        equal(n - 2, left + "1", right + "0", di + 1);

        // add 1 in end to both left and right
        // half the sum will not change
        equal(n - 2, left + "1", right + "1", di);
    }
}

// Driver Code
public static void main(String args[])
{
    int n = 5;

    // n-bits
    equal(n, "", "", 0);
}
}

// This code is contributed
// by SURENDRA_GANGWAR

Python3

# Python program to generate all binary strings with
# equal sums in left and right halves.

# Default values are used only in initial call.
# n is number of bits remaining to be filled
# di is current difference between sums of
# left and right halves.
# left and right are current half substrings.
def equal(n: int, left = "", right = "", di = 0):

    # TWO BASE CASES
    # If there are no more characters to add (n is 0)
    if n == 0:

        # If difference between counts of 1s and
        # 0s is 0 (di is 0)
        if di == 0:
            print(left + right, end = " ")
        return

    # If 1 remains than string length was odd
    if n == 1:

        # If difference is 0, we can put remaining
        # bit in middle.
        if di == 0:
            print(left + "0" + right, end = " ")
            print(left + "1" + right, end = " ")
        return

    # If difference is more than what can be
    # be covered with remaining n digits
    # (Note that lengths of left and right
    # must be same)
    if 2 * abs(di) <= n:

        # add 0 to end in both left and right
        # half. Sum in both half will not
        # change
        equal(n - 2, left + "0", right + "0", di)

        # add 0 to end in both left and right
        # half* subtract 1 from di as right
        # sum is increased by 1
        equal(n - 2, left + "0", right + "1", di - 1)

        # add 1 in end in left half and 0 to the
        # right half. Add 1 to di as left sum is
        # increased by 1
        equal(n - 2, left + "1", right + "0", di + 1)

        # add 1 in end to both left and right
        # half the sum will not change
        equal(n - 2, left + "1", right + "1", di)

# Driver Code
if __name__ == "__main__":
    n = 5 # n-bits
    equal(5)

# This code is contributed by
# sanjeev2552

C#

// C# program to generate all binary strings
// with equal sums in left and right halves.
using System;

class GFG
{

// Default values are used only in initial call.
// n is number of bits remaining to be filled
// di is current difference between sums of
// left and right halves.
// left and right are current half substrings.
static void equal(int n, String left,
                         String right, int di)
{
    // TWO BASE CASES
    // If there are no more characters
    // to add (n is 0)
    if (n == 0)
    {
        // If difference between counts of 1s
        // and 0s is 0 (di is 0)
        if (di == 0)
            Console.Write(left + right + " ");
        return;
    }

    /* If 1 remains than string length was odd */
    if (n == 1)
    {
        // If difference is 0, we can put
        // remaining bit in middle.
        if (di == 0)
        {
            Console.Write(left + "0" +
                          right + " ");
            Console.Write(left + "1" +
                          right + " ");
        }
        return;
    }

    /* If difference is more than what can be
    be covered with remaining n digits
    (Note that lengths of left and right
    must be same) */
    if ((2 * Math.Abs(di) <= n))
    {


            // add 0 to end in both left and right
            // half. Sum in both half will not
            // change
            equal(n - 2, left + "0", right + "0", di);

            // add 0 to end in both left and right
            // half* subtract 1 from di as right
            // sum is increased by 1
            equal(n - 2, left + "0",
                  right + "1", di - 1);


        // add 1 in end in left half and 0 to the
        // right half. Add 1 to di as left sum is
        // increased by 1*
        equal(n - 2, left + "1",
              right + "0", di + 1);

        // add 1 in end to both left and right
        // half the sum will not change
        equal(n - 2, left + "1", right + "1", di);
    }
}

// Driver Code
public static void Main(String []args)
{
    int n = 5;

    // n-bits
    equal(n, "", "", 0);
}
}

// This code is contributed by 29AjayKumar

Javascript

<script>

// JavaScript program to generate all binary strings
// with equal sums in left and right halves.

// Default values are used only in initial call.
// n is number of bits remaining to be filled
// di is current difference between sums of
// left and right halves.
// left and right are current half substrings.
function equal(n,left,right,di)
{
    // TWO BASE CASES
    // If there are no more characters to add (n is 0)
    if (n == 0)
    {
        // If difference between counts of 1s and
        // 0s is 0 (di is 0)
        if (di == 0)
            document.write(left + right + " ");
        return;
    }

    /* If 1 remains than string length was odd */
    if (n == 1)
    {
        // If difference is 0, we can put
        // remaining bit in middle.
        if (di == 0)
        {
            document.write(left + "0" + right + " ");
            document.write(left + "1" + right + " ");
        }
        return;
    }

    /* If difference is more than what can be
    be covered with remaining n digits
    (Note that lengths of left and right
     must be same) */
    if ((2 * Math.abs(di) <= n))
    {

            // add 0 to end in both left and right
            // half. Sum in both half will not
            // change
            equal(n - 2, left + "0", right + "0", di);

            // add 0 to end in both left and right
            // half* subtract 1 from di as right
            // sum is increased by 1
            equal(n - 2, left + "0", right + "1", di - 1);


        // add 1 in end in left half and 0 to the
        // right half. Add 1 to di as left sum is
        // increased by 1*
        equal(n - 2, left + "1", right + "0", di + 1);

        // add 1 in end to both left and right
        // half the sum will not change
        equal(n - 2, left + "1", right + "1", di);
    }
}

// Driver Code
let n = 5;

// n-bits
equal(n, "", "", 0);


// This code is contributed by rag2127

</script>

Output

00000 00100 01001 01101 01010 01110 10001 10101 10010 10110 11011 11111 

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