In how many ways can we fill a 4 X 4 grid where each cell co

In how many ways can we fill a 4 X 4 grid, where each cell contains either 0 or 1, and the sum of each row and each column is an even number?

Solution

Solution:- Given that a grid contains 4rows and 4columns. First fill 3 rows arbitrarily with 0 or 1.

Given that, sum of each row and each coloumn is even.

There are 2³ ways to do it for each row, for a total of (2³)²=2⁶.

Now, fill the last row to make all coloumn sums even,There is a unique way to do this.

Since n is even, the sum of first three rows is even, so again last row sum is also even.

4X4 grid contains 4 rows and 4 coloumns,thus a magic square of order 4 always contains 4² numbers.

The constant, that is the sum of every row , coloumn and diagonal is called magic constant (M) or magic sum.

Every normal sqaure has a constant dependent on n, calculated by the formula M=n(n²+1) /2

Here, the given grid is a 4X4, which means n=4, so magic sum is 4(4²+1) /2= 4(17) /2=34.