Let us say ABCD is the square with each side = 2 * a. Let O be the center of the square with origin at the center O. We divide the big square into four small squares with one in each quadrant. Each square has a side = a.
Let the mass of the square ABCD be = 4 * M. Then each small square has a mass = M = a². Let us omit one small square in the 3rd quadrant. There remain 3 of them now.
The center of mass of each small square is at its center. So the centers of masses of the three small squares are at :
(- a/2, a/2) , (a/2, - a/2) and (a/2, a/2)
x coordinate of the center of mass:
x = [ M (-a/2) + M (a/2) + M (a/2) ]/ (3 M)
x = a/6
y coordinate of the center of mass
y = [ M (a/2) + M (-a/2) + M (a/2) ]
y = a/6
So CM is at (a/6, a/6) from the center of the big square.