Count of integers having difference with its reverse equal to D

Given an integer D, the task is to find the count of all possible positive integers N such that reverse(N) = N + D.

Examples:

Input: D = 63 
Output: 2 
Explanation:
For N = 18, 18 + 63 = 81, which satisfies the condition N + D = reverse(N). 
For N = 29, 29 + 63 = 92, which satisfies the condition N + D = reverse(N). 
Therefore, the required output is 2

Input: D = 75 
Output: 0

Approach: The problem can be solved based on the following observations:

N + D = reverse(N) => N – reverse(N) = D 
=> D = ?_{(i=0 to [L/2]-1)}(10^L-1-i -10ⁱ )*( n_i – n_L-1-i), L = Count of digits in N.

If d_i = n_i ? n_L?1?i (0 ? i ? ?L/2? ? 1) 
reverse(N) ? N = ?_{(i=0 to [L/2]-1)}  (10^L-1-i -10ⁱ )*d_i

Follow the steps below to solve the problem:

• Let f(L, D) = reverse(N) ? N. Finding the count of N that satisfy the given formula is almost equivalent to enumerating the pairs of L and a sequence of integers in the range [?9, 9], { d₀, d₁, . . ., d_?L/2??1 } (take into account that there is more than one N that corresponds to a sequence of d_i, though). Here, for any i such that 0 ? i < ?L/2? ? 1, the following holds:

10^L?1?i ? 10ⁱ  > ?_{(j=i+1 to [L/2 – 1])}  ((10^L?1?j ? 10^j) · 9) + 10^L??L/2?

• Let L_D be the number of digits of D in decimal notation. Then, when L > 2L_D and f(L, d) > 0, it can be seen that f(L, d) > D.
• Therefore, it is enough to consider the values between L_D and 2L_D as the value of L.
• For a fixed value of L, consider enumerating the sequences { d₀, d₁, …, d_?L/2??1 } such that f(L, d) = D (and finding the count of N that satisfy the given formula), by performing Depth First Search starting from d₀.
• When the values up to d_i?1 are already decided, it can be seen that there are at most two candidates of the value of d_i that have to be considered: the maximum value such that (10ⁱ ? 10^L?1?i )d_i ? dif , and the minimum value such that (10ⁱ ? 10^L?1?i )d_i > dif . (Intuitively, if the “halfway” value of f(L, d) during the search gets too far from D, it is not possible to “get back”, and thus it should “stay close” to D.)
• Therefore, for a fixed value of L, find the count of N that satisfy the given formula in O(2^?L/2?) time.

Below is the implementation of the above approach :

C++

// Cpp program for the
// above approach
#include <bits/stdc++.h>

using namespace std;

// Maximum digits in N
int MAXL = 17;
int N;
vector<int> v;

// Function to find count
// of possible values
// of N such that N + D = reverse(N)
int count(int D, int l, int t, int x[])
{

    // Base Case
    if (t == N) {

        // If D is not equal to 0
        if (D != 0)
            return 0;

        // Stores count of possible values
        // of N such that N + D = reverse(N)
        long ans = 1;

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

            // Update ans
            ans *= (i == 0 ? 9 : 10) - abs(x[i]);
        }

        // If l is even
        if (l % 2 == 0) {

            // Update ans
            ans *= 10;
        }

        return ans;
    }

    // Stores count of possible values
    // of N such that N + D = reverse(N)
    long ans = 0;

    // Iterate over the range [-9, 9]
    for (int m = -9; m <= 9; m++) {

        if (-v[t] < D + v[t] * m && D +
                      v[t] * m < v[t]) {

            x[t] = m;

            // Update ans
            ans += count(D + v[t] * m, l,
                                 t + 1, x);
        }
    }

    return ans;
}

// Utility function to find count of N
// such that N + D = reverse(N)
int findN(int D)
{

    // If d is a multiple of 9,
    // no such value N found
    if (D % 9 != 0)
        return 0;

    // Divide D by 9 check reverse
    // of number and its sum
    D /= 9;

    // B[i]: Stores power of (10, i)
    vector<int> B(MAXL);
    B[0] = 1;

    // Calculate power(10, i)
    for (int i = 1; i < MAXL; i++) {

        // Update B[i]
        B[i] = B[i - 1] * 10;
    }

    // Stores count of N such
    // that N + D = reverse(N)
    int ans = 0;

    // Iterate over the range [1, MAXL]
    for (int i = 1; i <= MAXL; i++) {

        N = (i + 1) / 2;
        v.clear();
        v.resize(N);
        for (int j = 0; j < N; j++)
            for (int k = j; k < i - j; k++)
                v[j] += B[k];

        int arr[N];
        ans += count(D, i, 0, arr);
    }

    return ans;
}

// Driver Code
int main()
{
    int D = 63;

    // Function call
    cout << findN(D);
}

Java

// Java program of the above approach
import java.util.*;

public class Main {

    // Maximum digits in N
    static final int MAXL = 17;
    static int N;
    static long[] v;

    // Utility function to find count of N
    // such that N + D = reverse(N)
    static long findN(int D)
    {

        // If d is a multiple of 9,
        // no such value N found
        if (D % 9 != 0)
            return 0;

        // Divide D by 9 check reverse
        // of number and its sum
        D /= 9;

        // B[i]: Stores power of (10, i)
        long[] B = new long[MAXL];
        B[0] = 1;

        // Calculate power(10, i)
        for (int i = 1; i < MAXL; i++) {

            // Update B[i]
            B[i] = B[i - 1] * 10;
        }

        // Stores count of N such
        // that N + D = reverse(N)
        long ans = 0;

        // Iterate over the range [1, MAXL]
        for (int i = 1; i <= MAXL; i++) {

            N = (i + 1) / 2;
            v = new long[N];
            for (int j = 0; j < N; j++)
                for (int k = j; k < i - j; k++)
                    v[j] += B[k];

            // Update ans
            ans += count(D, i, 0, new int[N]);
        }

        return ans;
    }

    // Function to find count of possible values
    // of N such that N + D = reverse(N)
    static long count(long D, int l,
                      int t, int[] x)
    {

        // Base Case
        if (t == N) {

            // If D is not equal to 0
            if (D != 0)
                return 0;

            // Stores count of possible values
            // of N such that N + D = reverse(N)
            long ans = 1;

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

                // Update ans
                ans *= (i == 0 ? 9 : 10)
                       - Math.abs(x[i]);
            }

            // If l is even
            if (l % 2 == 0) {

                // Update ans
                ans *= 10;
            }

            return ans;
        }

        // Stores count of possible values
        // of N such that N + D = reverse(N)
        long ans = 0;

        // Iterate over the range [-9, 9]
        for (int m = -9; m <= 9; m++) {

            if (-v[t] < D + v[t] * m
                && D + v[t] * m < v[t]) {

                x[t] = m;

                // Update ans
                ans += count(D + v[t] * m,
                             l, t + 1, x);
            }
        }

        return ans;
    }

    // Driver Code
    public static void main(String[] args)
    {
        Scanner sc = new Scanner(System.in);
        int D = 63;

        // Function call
        System.out.println(findN(D));

        sc.close();
    }
}

Python3

# Python program of the above approach

# Maximum digits in N
MAXL = 17;
N = 0;
v = 0;

# Utility function to find count of N
# such that N + D = reverse(N)
def findN(D):
    global N;
    global v;

    # If d is a multiple of 9,
    # no such value N found
    if (D % 9 != 0):
        return 0;

    # Divide D by 9 check reverse
    # of number and its sum
    D //= 9;

    # B[i]: Stores power of (10, i)
    B = [0]*MAXL;
    B[0] = 1;

    # Calculate power(10, i)
    for i in range(1, MAXL):

        # Update B[i]
        B[i] = B[i - 1] * 10;

    # Stores count of N such
    # that N + D = reverse(N)
    ans = 0;

    # Iterate over the range [1, MAXL]
    for i in range(1, MAXL + 1):
        N = (i + 1) // 2;
        v = [0]*N;
        for j in range(N):
            for k in range(j, i - j):
                v[j] += B[k];

        # Update ans
        temp = [0]*N;
        ans += count(D, i, 0, temp);
        return ans;

# Function to find count of possible values
# of N such that N + D = reverse(N)
def count(D, l, t, x):
    global N;
    global v;

    # Base Case
    if (t == N):

        # If D is not equal to 0
        if (D != 0):
            return 0;

        # Stores count of possible values
        # of N such that N + D = reverse(N)
        ans = 1;
        for i in range(N):

            # Update ans
            ans *= (9 if i == 0 else 10) - abs(x[i]);

        # If l is even
        if (l % 2 == 0):

            # Update ans
            ans *= 10;
        return ans;

    # Stores count of possible values
    # of N such that N + D = reverse(N)
    ans = 0;

    # Iterate over the range [-9, 9]
    for m in range(-9, 10):
        if (-v[t] < D + v[t] * m and D + v[t] * m < v[t]):
            x[t] = m;

            # Update ans
            ans += count(D + v[t] * m, l, t + 1, x);
    return ans;

# Driver Code
if __name__ == '__main__':
    D = 63;

    # Function call
    print(findN(D));

    # This code is contributed by 29AjayKumar

C#

// C# program for the above approach
using System;
class GFG
{

    // Maximum digits in N
    static int MAXL = 17;
    static int N;
    static long[] v;

    // Utility function to find count of N
    // such that N + D = reverse(N)
    static long findN(int D)
    {

        // If d is a multiple of 9,
        // no such value N found
        if (D % 9 != 0)
            return 0;

        // Divide D by 9 check reverse
        // of number and its sum
        D /= 9;

        // B[i]: Stores power of (10, i)
        long[] B = new long[MAXL];
        B[0] = 1;

        // Calculate power(10, i)
        for (int i = 1; i < MAXL; i++)
        {

            // Update B[i]
            B[i] = B[i - 1] * 10;
        }

        // Stores count of N such
        // that N + D = reverse(N)
        long ans = 0;

        // Iterate over the range [1, MAXL]
        for (int i = 1; i <= MAXL; i++)
        {
            N = (i + 1) / 2;
            v = new long[N];
            for (int j = 0; j < N; j++)
                for (int k = j; k < i - j; k++)
                    v[j] += B[k];

            // Update ans
            ans += count(D, i, 0, new int[N]);
        }
        return ans;
    }

    // Function to find count of possible values
    // of N such that N + D = reverse(N)
    static long count(long D, int l,
                      int t, int[] x)
    {

        // Base Case
        if (t == N)
        {

            // If D is not equal to 0
            if (D != 0)
                return 0;

            // Stores count of possible values
            // of N such that N + D = reverse(N)
            long ans = 1;
            for (int i = 0; i < N; i++)
            {

                // Update ans
                ans *= (i == 0 ? 9 : 10)
                       - Math.Abs(x[i]);
            }

            // If l is even
            if (l % 2 == 0)
            {

                // Update ans
                ans *= 10;
            }
            return ans;
        }

        // Stores count of possible values
        // of N such that N + D = reverse(N)
        long anss = 0;

        // Iterate over the range [-9, 9]
        for (int m = -9; m <= 9; m++)
        {
            if (-v[t] < D + v[t] * m
                && D + v[t] * m < v[t])
            {
                x[t] = m;

                // Update ans
                anss += count(D + v[t] * m,
                             l, t + 1, x);
            }
        }
        return anss;
    }

  // Driver code
  public static void Main(String[] args)
  {
        int D = 63;

        // Function call
        Console.WriteLine(findN(D));
  }
}

// This code is contributed by code_hunt.

Javascript

<script>
// javascript program of the above approach

    // Maximum digits in N
    let MAXL = 17;
    let N;
    let v = []

    // Utility function to find count of N
    // such that N + D = reverse(N)
    function findN(D)
    {

        // If d is a multiple of 9,
        // no such value N found
        if (D % 9 != 0)
            return 0;

        // Divide D by 9 check reverse
        // of number and its sum
        D /= 9;

        // B[i]: Stores power of (10, i)
        let B = new Array(MAXL).fill(0);

        B[0] = 1;

        // Calculate power(10, i)
        for (let i = 1; i < MAXL; i++) {

            // Update B[i]
            B[i] = B[i - 1] * 10;
        }

        // Stores count of N such
        // that N + D = reverse(N)
        let ans = 0;

        // Iterate over the range [1, MAXL]
        for (let i = 1; i <= MAXL; i++) {

            N = Math.floor((i + 1) / 2);
            v = new Array(N).fill(0);
            for (let j = 0; j < N; j++)
                for (let k = j; k < i - j; k++)
                    v[j] += B[k];

            // Update ans
            ans += count(D, i, 0, new Array(N));
        }

        return ans;
    }

    // Function to find count of possible values
    // of N such that N + D = reverse(N)
    function count(D, l, t, x)
    {

        // Base Case
        if (t == N) {

            // If D is not equal to 0
            if (D != 0)
                return 0;

            // Stores count of possible values
            // of N such that N + D = reverse(N)
            let ans = 1;

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

                // Update ans
                ans *= (i == 0 ? 9 : 10)
                       - Math.abs(x[i]);
            }

            // If l is even
            if (l % 2 == 0) {

                // Update ans
                ans *= 10;
            }

            return ans;
        }

        // Stores count of possible values
        // of N such that N + D = reverse(N)
        let ans = 0;

        // Iterate over the range [-9, 9]
        for (let m = -9; m <= 9; m++)
        {

            if (-v[t] < D + v[t] * m
                && D + v[t] * m < v[t])
            {

                x[t] = m;

                // Update ans
                ans += count(D + v[t] * m,
                             l, t + 1, x);
            }
        }

        return ans;
    }

    // Driver Code
        let D = 63;

        // Function call
        document.write(findN(D));

 // This code is contributed by target_2.
</script>

Output:

2

Time Complexity: O(2^L_D), where L_D denotes the number of digits of D in decimal notation.
Auxiliary Space: O( L_D)