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

• Difficulty Level : Medium
• Last Updated : 08 Oct, 2021

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

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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 ``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 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 >(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`

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