Suppose $S_{i}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a similarity with contraction factor $c_{i} \in(0,1)$ for $1 \leqslant i \leqslant k$. Let $X$ be the unique non-empty compact invariant set for the $S_{i}$ 's. State a formula for the Hausdorff dimension of $X$, under an assumption on the $S_{i}$ 's you should state. Hence compute the Hausdorff dimension of the subset $X$ of the square $[0,1]^{2}$ defined by dividing the square into a $5 \times 5$ array of squares, removing the open middle square $(2 / 5,3 / 5)^{2}$, then removing the middle $1 / 25$ th of each of the remaining 24 squares, and so on.