Mr Tan, an interior designer, tiled a square floor completely with square tiles of side 1 cm as shown in Figure 1. He used a total of 73 grey square tiles which formed a shape of an X as shown in Figure 2. Find the perimeter of the square floor.

1. Determine the relationship between the number of grey tiles and the side length of the square.

The grey tiles form an 'X' shape, which consists of the two main diagonals of the square floor. Let the number of tiles along one side of the square be $n$. The number of tiles on one diagonal is $n$. The two diagonals intersect at exactly one tile in the center. Therefore, the total number of grey tiles is $n + n - 1 = 2n - 1$.

2. Calculate the number of tiles along one side of the square.

Given that the total number of grey tiles is $73$, we have:
$2n - 1 = 73$
$2n = 74$
$n = 37$
So, there are $37$ tiles along one side of the square floor.

3. Calculate the perimeter of the square floor.

Since each tile has a side length of $1$ cm, the side length of the square floor is $37 \times 1 = 37$ cm.
The perimeter of the square floor is $4 \times 37 = 148$ cm.