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Find K numbers with sum equal to N and sum of their squares maximized

  • Last Updated : 22 Mar, 2021
Geek Week

Given two integers N and K, the task is to find K numbers(A1, A2, …, AK) such that i=1KAi is equal to N and i=1KAi2 is maximum.
Examples: 
 

Input: N = 3, K = 2 
Output: 1 2 
Explanation: The two numbers are 1 and 2 as their sum is equal to N and 12 + 22 is maximum.
Input: N = 10, K = 3 
Output: 1 8 1 
 

 

Approach: The idea is to take number 1, K – 1 times and number N – K + 1 once. The sum of these numbers is equal to N and sum of squares of these numbers is always maximum. For any two non-negative numbers a and b, (a2 + b2) is always less than 1 + (a + b – 1)2.
Below is the implementation of the above approach:
 

C++




// C++ program to find K numbers
// with sum equal to N and the
// sum of their squares maximized
#include <bits/stdc++.h>
using namespace std;
 
// Function that prints the
// required K numbers
void printKNumbers(int N, int K)
{
    // Print 1, K-1 times
    for (int i = 0; i < K - 1; i++)
        cout << 1 << " ";
 
    // Print (N-K+1)
    cout << (N - K + 1);
}
 
// Driver Code
int main()
{
    int N = 10, K = 3;
 
    printKNumbers(N, K);
 
    return 0;
}

Java




// Java program to find K numbers
// with sum equal to N and the
// sum of their squares maximized
class GFG{
 
// Function that prints the
// required K numbers
static void printKNumbers(int N, int K)
{
    // Print 1, K-1 times
    for(int i = 0; i < K - 1; i++)
        System.out.print(1 + " ");
 
    // Print (N - K + 1)
    System.out.print(N - K + 1);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 10, K = 3;
    printKNumbers(N, K);
}
}
 
// This code is contributed by Amit Katiyar

Python3




# Python3 program to find K numbers
# with a sum equal to N and the
# sum of their squares maximized
 
# Function that prints the
# required K numbers
def printKNumbers(N, K):
     
    # Print 1, K-1 times
    for i in range(K - 1):
        print(1, end = ' ')
     
    # Print (N-K+1)
    print(N - K + 1)
     
# Driver code
if __name__=='__main__':
     
    (N, K) = (10, 3)
     
    printKNumbers(N, K)
     
# This code is contributed by rutvik_56

C#




// C# program to find K numbers
// with sum equal to N and the
// sum of their squares maximized
using System;
     
class GFG{
 
// Function that prints the
// required K numbers
static void printKNumbers(int N, int K)
{
     
    // Print 1, K-1 times
    for(int i = 0; i < K - 1; i++)
        Console.Write(1 + " ");
 
    // Print (N - K + 1)
    Console.Write(N - K + 1);
}
 
// Driver code
public static void Main(String[] args)
{
    int N = 10, K = 3;
     
    printKNumbers(N, K);
}
}
 
// This code is contributed by shivanisinghss2110

Javascript




<script>
 
    // Javascript program to find K numbers
    // with sum equal to N and the
    // sum of their squares maximized
     
    // Function that prints the
    // required K numbers
    function printKNumbers(N, K)
    {
        // Print 1, K-1 times
        for (let i = 0; i < K - 1; i++)
            document.write(1 + " ");
 
        // Print (N-K+1)
        document.write(N - K + 1);
    }
 
     let N = 10, K = 3;
   
    printKNumbers(N, K);
     
</script>
Output: 
1 1 8

 

Time Complexity: O(K) 
Auxiliary Space: O(1)
 

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