Count of Numbers in Range where first digit is equal to last digit of the number

Given a range represented by two positive integers L and R. Find the count of numbers in the range where the first digit is equal to the last digit of the number.

Examples: 

Input : L = 2, R = 60
Output : 13
Explanation : Required numbers are 
2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44 and 55

Input : L = 1, R = 1000
Output : 108

Prerequisites: Digit DP
There can be two approaches to solve this type of problem, one can be a combinatorial solution and others can be a dynamic programming based solution. Below is a detailed approach to solving this problem using digit dynamic programming. 

Dynamic Programming Solution: Firstly, if we are able to count the required numbers up to R i.e. in the range [0, R], we can easily reach our answer in the range [L, R] by solving for from zero to R and then subtracting the answer we get after solving for from zero to L – 1. Now, we need to define the DP states. 

DP States:  



In each recursive call, we set the last digit as the digit we placed in the last position, and we set the first digit as the first non-zero digit of the number. In the final recursive call, when we are at the last position if the first digit is equal to the last digit, return 1, otherwise 0.

Below is the implementation of the above approach. 

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// C++ Program to find the count of
// numbers in a range where the number
// does not contain more than K non
// zero digits
  
#include <bits/stdc++.h>
  
using namespace std;
  
const int M = 20;
  
// states - position, first digit,
// last digit, tight
int dp[M][M][M][2];
  
// This function returns the count of
// required numbers from 0 to num
int count(int pos, int firstD, int lastD,
        int tight, vector<int> num)
{
    // Last position
    if (pos == num.size()) {
  
        // If first digit is equal to
        // last digit
        if (firstD == lastD)
            return 1;
        return 0;
    }
  
    // If this result is already computed
    // simply return it
    if (dp[pos][firstD][lastD][tight] != -1)
        return dp[pos][firstD][lastD][tight];
  
    int ans = 0;
  
    // Maximum limit upto which we can place
    // digit. If tight is 1, means number has
    // already become smaller so we can place
    // any digit, otherwise num[pos]
    int limit = (tight ? 9 : num[pos]);
  
    for (int dig = 0; dig <= limit; dig++) {
        int currFirst = firstD;
  
        // If the position is 0, current
        // digit can be first digit
        if (pos == 0)
            currFirst = dig;
  
        // In current call, if the first
        // digit is zero and current digit
        // is nonzero, update currFirst
        if (!currFirst && dig)
            currFirst = dig;
  
        int currTight = tight;
  
        // At this position, number becomes
        // smaller
        if (dig < num[pos])
            currTight = 1;
  
        // Next recursive call, set last
        // digit as dig
        ans += count(pos + 1, currFirst,
                    dig, currTight, num);
    }
    return dp[pos][firstD][lastD][tight] = ans;
}
  
// This function converts a number into its
// digit vector and uses above function to compute
// the answer
int solve(int x)
{
    vector<int> num;
    while (x) {
        num.push_back(x % 10);
        x /= 10;
    }
    reverse(num.begin(), num.end());
  
    // Initialize dp
    memset(dp, -1, sizeof(dp));
    return count(0, 0, 0, 0, num);
}
  
// Driver Code
int main()
{
    int L = 2, R = 60;
    cout << solve(R) - solve(L - 1) << endl;
  
    L = 1, R = 1000;
    cout << solve(R) - solve(L - 1) << endl;
      
    return 0;
}
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// Java program to find the count of 
// numbers in a range where the number 
// does not contain more than K non 
// zero digits
import java.util.Collections;
import java.util.Vector;
  
class GFG 
{
    static int M = 20;
  
    // states - position, first digit,
    // last digit, tight
    static int[][][][] dp = new int[M][M][M][2];
  
    // This function returns the count of
    // required numbers from 0 to num
    static int count(int pos, int firstD, 
                     int lastD, int tight, 
                     Vector<Integer> num) 
    {
  
        // Last position
        if (pos == num.size())
        {
  
            // If first digit is equal to
            // last digit
            if (firstD == lastD)
                return 1;
            return 0;
        }
  
        // If this result is already computed
        // simply return it
        if (dp[pos][firstD][lastD][tight] != -1)
            return dp[pos][firstD][lastD][tight];
        int ans = 0;
  
        // Maximum limit upto which we can place
        // digit. If tight is 1, means number has
        // already become smaller so we can place
        // any digit, otherwise num[pos]
        int limit = (tight == 1 ? 9 : num.elementAt(pos));
  
        for (int dig = 0; dig <= limit; dig++)
        {
            int currFirst = firstD;
  
            // If the position is 0, current
            // digit can be first digit
            if (pos == 0)
                currFirst = dig;
  
            // In current call, if the first
            // digit is zero and current digit
            // is nonzero, update currFirst
            if (currFirst == 0 && dig != 0)
                currFirst = dig;
  
            int currTight = tight;
  
            // At this position, number becomes
            // smaller
            if (dig < num.elementAt(pos))
                currTight = 1;
  
            // Next recursive call, set last
            // digit as dig
            ans += count(pos + 1, currFirst, 
                         dig, currTight, num);
        }
        return dp[pos][firstD][lastD][tight] = ans;
    }
  
    // This function converts a number into its
    // digit vector and uses above function to 
    // compute the answer
    static int solve(int x) 
    {
        Vector<Integer> num = new Vector<>();
        while (x > 0
        {
            num.add(x % 10);
            x /= 10;
        }
  
        Collections.reverse(num);
  
        // Initialize dp
        for (int i = 0; i < M; i++)
            for (int j = 0; j < M; j++)
                for (int k = 0; k < M; k++)
                    for (int l = 0; l < 2; l++)
                        dp[i][j][k][l] = -1;
  
        return count(0, 0, 0, 0, num);
    }
  
    // Driver Code
    public static void main(String[] args)
    {
        int L = 2, R = 60;
        System.out.println(solve(R) - solve(L - 1));
  
        L = 1;
        R = 1000;
        System.out.println(solve(R) - solve(L - 1));
    }
}
  
// This code is contributed by
// sanjeev2552
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# Python3 code for above approach
  
# Returns the count of numbers in range
# if the first digit is equal to last digit of number
def count(l, r):
    cnt = 0       # Initialize counter
    for i in range(l, r):
          
        # If number is less than 10
        # then increment counter
        # as number has only one digit 
        if(i < 10):     
            cnt += 1
              
        else:
            n = i % 10     # Find the last digit
            k = i
  
            # Find the first digit
            while(k >= 10):
                k = k // 10
  
            # If first digit equals last digit
            # then increment counter
            if(n == k):
                cnt += 1
                  
    return(cnt)     # Return the count
  
# Driver Code
L = 2; R = 60;
print(count(L, R)) 
  
L = 1; R = 1000;
print(count(L, R))
  
# This code is contributed by Raj
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// C# program to find the count of 
// numbers in a range where the number 
// does not contain more than K non 
// zero digits
using System;
using System.Collections.Generic;             
      
class GFG 
    static int M = 20; 
  
    // states - position, first digit, 
    // last digit, tight 
    static int[,,,] dp = new int[M, M, M, 2]; 
  
    // This function returns the count of 
    // required numbers from 0 to num 
    static int count(int pos, int firstD, 
                     int lastD, int tight, 
                     List<int> num) 
    
  
        // Last position 
        if (pos == num.Count) 
        
  
            // If first digit is equal to 
            // last digit 
            if (firstD == lastD) 
                return 1; 
            return 0; 
        
  
        // If this result is already computed 
        // simply return it 
        if (dp[pos, firstD, lastD, tight] != -1) 
            return dp[pos, firstD, lastD, tight]; 
        int ans = 0; 
  
        // Maximum limit upto which we can place 
        // digit. If tight is 1, means number has 
        // already become smaller so we can place 
        // any digit, otherwise num[pos] 
        int limit = (tight == 1 ? 9 : num[pos]); 
  
        for (int dig = 0; dig <= limit; dig++) 
        
            int currFirst = firstD; 
  
            // If the position is 0, current 
            // digit can be first digit 
            if (pos == 0) 
                currFirst = dig; 
  
            // In current call, if the first 
            // digit is zero and current digit 
            // is nonzero, update currFirst 
            if (currFirst == 0 && dig != 0) 
                currFirst = dig; 
  
            int currTight = tight; 
  
            // At this position, number becomes 
            // smaller 
            if (dig < num[pos]) 
                currTight = 1; 
  
            // Next recursive call, set last 
            // digit as dig 
            ans += count(pos + 1, currFirst, 
                         dig, currTight, num); 
        
        return dp[pos, firstD, lastD, tight] = ans; 
    
  
    // This function converts a number into its 
    // digit vector and uses above function to 
    // compute the answer 
    static int solve(int x) 
    
        List<int> num = new List<int>(); 
        while (x > 0) 
        
            num.Add(x % 10); 
            x /= 10; 
        
  
        num.Reverse();
  
        // Initialize dp 
        for (int i = 0; i < M; i++) 
            for (int j = 0; j < M; j++) 
                for (int k = 0; k < M; k++) 
                    for (int l = 0; l < 2; l++) 
                        dp[i, j, k, l] = -1; 
  
        return count(0, 0, 0, 0, num); 
    
  
    // Driver Code 
    public static void Main(String[] args) 
    
        int L = 2, R = 60; 
        Console.WriteLine(solve(R) - solve(L - 1)); 
  
        L = 1; 
        R = 1000; 
        Console.WriteLine(solve(R) - solve(L - 1)); 
    
  
// This code is contributed by 29AjayKumar
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Output
13
108

Time Complexity : O(18 * 10 * 10 * 2 * 10), if we are dealing with the numbers upto 1018
 

Alternative approach:

We can also solve this problem by recursion, for every position we can fill any number which satisfies the given condition except for the first and last position because they will be paired together, and for this, we will use recursion and in every call just check if the first number is smaller or larger than the last number if it turns out to be greater than we will add 8 otherwise 9 and call for number / 10, once the number becomes smaller than 10 first and the last digit becomes same so return the number itself.

Below is the implementation of the above approach

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// C++ program to implement
// the above approach
#include <iostream>
using namespace std;
  
int solve(int x)
{
  
    int ans = 0, first, last, temp = x;
  
    // Base Case
  
    if (x < 10)
        return x;
  
    // Calculating the last digit
    last = x % 10;
  
    // Calculating the first digit
    while (x) {
        first = x % 10;
        x /= 10;
    }
  
    if (first <= last)
        ans = 9 + temp / 10;
    else
        ans = 8 + temp / 10;
  
    return ans;
}
  
// Drivers Code
int main()
{
  
    int L = 2, R = 60;
    cout << solve(R) - solve(L - 1) << endl;
  
    L = 1, R = 1000;
    cout << solve(R) - solve(L - 1) << endl;
  
    return 0;
}
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// Java program to implement
// the above approach
class GFG{
     
public static int solve(int x)
{
  int ans = 0, first = 0
      last, temp = x;
  
  // Base Case
  if (x < 10)
    return x;
  
  // Calculating the 
  // last digit
  last = x % 10;
  
  // Calculating the 
  // first digit
  while (x != 0
  {
    first = x % 10;
    x /= 10;
  }
  
  if (first <= last)
    ans = 9 + temp / 10;
  else
    ans = 8 + temp / 10;
  
  return ans;
}
      
// Driver code
public static void main(String[] args) 
{
  int L = 2, R = 60;
  System.out.println(solve(R) - 
                     solve(L - 1));
  
  L = 1; R = 1000;
  System.out.println(solve(R) - 
                     solve(L - 1));
}
}
  
// This code is contributed by divyeshrabadiya07
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// C# program to implement
// the above approach
using System;
  
class GFG{
      
public static int solve(int x)
{
    int ans = 0, first = 0, 
        last, temp = x;
          
    // Base Case
    if (x < 10)
        return x;
      
    // Calculating the 
    // last digit
    last = x % 10;
      
    // Calculating the 
    // first digit
    while (x != 0) 
    {
        first = x % 10;
        x /= 10;
    }
      
    if (first <= last)
        ans = 9 + temp / 10;
    else
        ans = 8 + temp / 10;
      
    return ans;
}
      
// Driver code
public static void Main(String[] args) 
{
    int L = 2, R = 60;
    Console.WriteLine(solve(R) - 
                      solve(L - 1));
      
    L = 1; R = 1000;
    Console.WriteLine(solve(R) - 
                      solve(L - 1));
}
}
  
// This code is contributed by shivanisinghss2110
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Output
13
108





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