What is $n=?$

The side length of the larger square is equal to the sum of the lengths of one side and two diagonals of the smaller squares, i.e., $(2\sqrt{2} + 1)\sqrt{3}$ , since the side length of the smaller squares is $\sqrt{3}$ . So as there are 9 small squares, we are looking for $n$ such that

$((2\sqrt{2} + 1)\sqrt{3})^{2} - 9 \times 3 = \sqrt{n} \Longrightarrow (9 + 4\sqrt{2}) \times 3 - 9 \times 3 = \sqrt{n} \Longrightarrow 12\sqrt{2} = \sqrt{n} \Longrightarrow \sqrt{144}\sqrt{2} = \sqrt{n} \Longrightarrow \boxed{n = 288}$ .