# Maximum number of ones in a N*N matrix with given constraints in C++

Given the task is to find the maximum number of ones in a binary matrix possible with the following constraints.

Two integers N and X are given where X<=N. The size of the binary matrix should be N*N and every sub-matrix of size X*X should contain at least one zero.

Let’s now understand what we have to do using an example −

Input − N=4, X=2

Output − 12

Explanation − The resultant matrix will be −

1 1 1 1
1 0 0 1
1 0 0 1
1 1 1 1

Input − N=7, X=3

Output − 45

## Approach used in the below program as follows

• To obtain the maximum number of ones, first we will have to find the minimum number of zeros required in the given matrix.

By observing a common pattern in all the matrices, it can be seen that the number of zeroes required = (N / X)2

So the maximum number of ones = Total elements in matrix – number of zeros\

• In function MaxOne() create a variable Z of type int and store in it the minimum number of zeros required which is equal to (N / X)2

• Then initialize another variable total = N*N of type int to store the total size of the matrix.

• Then finally initialize int ans = total – Z to store the final answer and return ans.

## Example

Live Demo

#include <bits/stdc++.h>
using namespace std;
int MaxOne(int N, int X){
// Minimum number of zeroes that are needed
int Z = (N / X);
Z = Z * Z;
/* Totol elements in matrix = square of the size of the matrices*/
int total =N * N;
//Final answer
int ans = total - Z;
return ans;
}
int main(){
int N = 4;
int X = 2;
cout << MaxOne(N, X);
return 0;
}

## Output

If we run the above code we will get the following output −

12

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