Given an integer N, the task is to construct a matrix M of size N x N with numbers in range [1, N^2], such that the sum of diagonal elements for each square submatrix is even.
Input: N = 2
This matrix has 5 square submatrix and 4 of them (, , , ) have a=0 so they satisfy the conditions.
The last square submatrix is the whole matrix M where r=c=a=1. We can see that M(1, 1)+M(2, 2)=1+3=4 and M(1, 2)+M(2, 1)=2+4=6 are both even.
Input: N = 4
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Approach: We know that the sum of two numbers is even when their parity is the same. Let us say the parity of M(i, j) is odd that means the parity of M(i+1, j+1), M(i+1, j-1), M(i-1, j+1), M(i-1, j-1) has to be odd.
Below is the illustration for N = 4 to generate a matrix of size 4×4:
So from the above illustration we have to fill the matrix in the Checkerboard Pattern. We can fill it in two ways:
- All black cells have an odd integer and white cells have an even integer.
- All black cells have an even integer and white cells have an odd integer.
Below is the implementation of the above approach:
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Time Complexity: O(N^2)
Auxiliary Space: O(N^2)
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Improved By : chitranayal