9. (10 points) Place $n$ identical square pieces of paper on a table without overlapping, forming a multi-layer rectangular (including square) shape with at least two layers, and each upper square piece must have two vertices each on the midpoint of a side of a lower square piece. The figure below shows all different placement methods when $n=6$. How many different placement methods are there when $n=9$?
【Analysis】Through analysis, we can know:
When the number of layers is 2: $8+1$ arrangement has 7 kinds; $7+2$ arrangement has 5 kinds; $6+3$ arrangement has 3 kinds; $5+4$ arrangement has 1 kind; when the number of layers is 3:
$6+2+1$ arrangement has 4 kinds; $5+3+1$ arrangement has 4 kinds; $4+3+2$ arrangement has 1 kind;
When the number of layers is 4, it cannot be satisfied, so there are a total of $7+5+3+1+4+4+1=25$ different placement methods. According to this, the answer can be given.
【Solution】Solution:
When the number of layers is 2:
(1) $8+1: 7$ kinds
(2) $7+2: 5$ kinds
(3) $6+3: 3$ kinds
(4) $5+4: 1$ kind

When the number of layers is 3:
(1) $6+2+1: 4$ kinds
(2) $5+3+1: 4$ kinds
(3) $4+3+2$ : 1 kind

When the number of layers is 4, it cannot be satisfied.
Therefore, there are a total of:
$$
7+5+3+1+4+4+1=25 \text { (kinds) }
$$

Answer: When $n=9$, there are 25 different placement methods.