Number of edges of the cuboid \(= x = 12\) Number of faces \(= y = 6\) \[ \therefore x + y = 12 + 6 = 18 \] The nearest perfect square greater than 18 is \(25\) \[ \therefore x + y + a = 25 \Rightarrow 18 + a = 25 \Rightarrow a = 25 - 18 = 7 \] \(\therefore\) The minimum value of \(a\) required to make \((x + y + a)\) a perfect square is 7.