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   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.

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