# Minimum number of palindromes required to express N as a sum | Set 1

Given a number N, we have to find the minimum number of palindromes required to express N as a sum of them. **Examples:**

Input:N = 11Output:1

11 is itself a palindrome.Input:N = 65Output:3

65 can be expressed as a sum of three palindromes (55, 9, 1).Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the

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**Approach:**

We can use Dynamic Programming to solve this problem. The idea is to first generate all the palindromes up to N in a sorted fashion, and then we can treat this problem as a variation of the subset sum problem, where we have to find the size of the smallest subset such that its sum is N.**Below is the implementation of above approach:**

## CPP

`// C++ implementation of above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Declaring the DP table as global variable` `vector<vector<` `long` `long` `> > dp;` `// A utility for creating palindrome` `int` `createPalindrome(` `int` `input, ` `bool` `isOdd)` `{` ` ` `int` `n = input;` ` ` `int` `palin = input;` ` ` `// checks if number of digits is odd or even` ` ` `// if odd then neglect the last digit of input in` ` ` `// finding reverse as in case of odd number of` ` ` `// digits middle element occur once` ` ` `if` `(isOdd)` ` ` `n /= 10;` ` ` `// Creates palindrome by just appending reverse` ` ` `// of number to itself` ` ` `while` `(n > 0) {` ` ` `palin = palin * 10 + (n % 10);` ` ` `n /= 10;` ` ` `}` ` ` `return` `palin;` `}` `// Function to generate palindromes` `vector<` `int` `> generatePalindromes(` `int` `N)` `{` ` ` `vector<` `int` `> palindromes;` ` ` `int` `number;` ` ` `// Run two times for odd and even length palindromes` ` ` `for` `(` `int` `j = 0; j < 2; j++) {` ` ` `// Creates palindrome numbers with first half as i.` ` ` `// Value of j decides whether we need an odd length` ` ` `// or even length palindrome.` ` ` `int` `i = 1;` ` ` `while` `((number = createPalindrome(i++, j)) <= N)` ` ` `palindromes.push_back(number);` ` ` `}` ` ` `return` `palindromes;` `}` `// Function to find the minimum` `// number of elements in a sorted` `// array A[i..j] such that their sum is N` `long` `long` `minimumSubsetSize(vector<` `int` `>& A, ` `int` `i, ` `int` `j, ` `int` `N)` `{` ` ` `if` `(!N)` ` ` `return` `0;` ` ` `if` `(i > j || A[i] > N)` ` ` `return` `INT_MAX;` ` ` `if` `(dp[i][N])` ` ` `return` `dp[i][N];` ` ` `dp[i][N] = min(1 + minimumSubsetSize(A, i + 1, j,` ` ` `N - A[i]),` ` ` `minimumSubsetSize(A, i + 1, j, N));` ` ` `return` `dp[i][N];` `}` `// Function to find the minimum` `// number of palindromes that N` `// can be expressed as a sum of` `int` `minimumNoOfPalindromes(` `int` `N)` `{` ` ` `// Getting the list of all palindromes upto N` ` ` `vector<` `int` `> palindromes = generatePalindromes(N);` ` ` `// Sorting the list of palindromes` ` ` `sort(palindromes.begin(), palindromes.end());` ` ` `// Initializing the DP table` ` ` `dp = vector<vector<` `long` `long` `> >(palindromes.size(),` ` ` `vector<` `long` `long` `>(N + 1, 0));` ` ` `// Returning the required value` ` ` `return` `minimumSubsetSize(palindromes, 0,` ` ` `palindromes.size() - 1, N);` `}` `// Driver code` `int` `main()` `{` ` ` `int` `N = 65;` ` ` `cout << minimumNoOfPalindromes(N);` ` ` `return` `0;` `}` |

## Python3

`# Python3 implementation of above approach` `# Declaring the DP table as global variable` `dp ` `=` `[[` `0` `for` `i ` `in` `range` `(` `1000` `)] ` `for` `i ` `in` `range` `(` `1000` `)]` `# A utility for creating palindrome` `def` `createPalindrome(` `input` `, isOdd):` ` ` `n ` `=` `input` ` ` `palin ` `=` `input` ` ` `# checks if number of digits is odd or even` ` ` `# if odd then neglect the last digit of input in` ` ` `# finding reverse as in case of odd number of` ` ` `# digits middle element occur once` ` ` `if` `(isOdd):` ` ` `n ` `/` `/` `=` `10` ` ` `# Creates palindrome by just appending reverse` ` ` `# of number to itself` ` ` `while` `(n > ` `0` `):` ` ` `palin ` `=` `palin ` `*` `10` `+` `(n ` `%` `10` `)` ` ` `n ` `/` `/` `=` `10` ` ` `return` `palin` `# Function to generate palindromes` `def` `generatePalindromes(N):` ` ` `palindromes ` `=` `[]` ` ` `number ` `=` `0` ` ` `# Run two times for odd and even length palindromes` ` ` `for` `j ` `in` `range` `(` `2` `):` ` ` ` ` `# Creates palindrome numbers with first half as i.` ` ` `# Value of j decides whether we need an odd length` ` ` `# or even length palindrome.` ` ` `i ` `=` `1` ` ` `number ` `=` `createPalindrome(i, j)` ` ` `while` `number <` `=` `N:` ` ` `number ` `=` `createPalindrome(i, j)` ` ` `palindromes.append(number)` ` ` `i ` `+` `=` `1` ` ` `return` `palindromes` `# Function to find the minimum` `# number of elements in a sorted` `# array A[i..j] such that their sum is N` `def` `minimumSubsetSize(A, i, j, N):` ` ` `if` `(` `not` `N):` ` ` `return` `0` ` ` `if` `(i > j ` `or` `A[i] > N):` ` ` `return` `10` `*` `*` `9` ` ` `if` `(dp[i][N]):` ` ` `return` `dp[i][N]` ` ` `dp[i][N] ` `=` `min` `(` `1` `+` `minimumSubsetSize(A, i ` `+` `1` `, j, N ` `-` `A[i]),` ` ` `minimumSubsetSize(A, i ` `+` `1` `, j, N))` ` ` `return` `dp[i][N]` `# Function to find the minimum` `# number of palindromes that N` `# can be expressed as a sum of` `def` `minimumNoOfPalindromes(N):` ` ` `# Getting the list of all palindromes upto N` ` ` `palindromes ` `=` `generatePalindromes(N)` ` ` `# Sorting the list of palindromes` ` ` `palindromes ` `=` `sorted` `(palindromes)` ` ` `# Returning the required value` ` ` `return` `minimumSubsetSize(palindromes, ` `0` `, ` `len` `(palindromes) ` `-` `1` `, N)` `# Driver code` `N ` `=` `65` `print` `(minimumNoOfPalindromes(N))` `# This code is contributed by mohit kumar 29` |

**Output:**

3