# 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 = 11
Output: 1
11 is itself a palindrome.

Input: N = 65
Output: 3
65 can be expressed as a sum of three palindromes (55, 9, 1).

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 ` `using` `namespace` `std; ` ` `  `// Declaring the DP table as global variable ` `vector > 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 revers ` `    ``// 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 >(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 revers ` `    ``# 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
```

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