To solve this problem, we aim to find all integer values of $n$ (where $10 < n < 20$) for which an $n \times n$ floor can be completely covered using the same number of $2 \times 2$ and $5 \times 1$ tiles. We cannot cut the tiles and they should not overlap.

First, we calculate the total area of the floor, which is $n \times n$.

### Step 1: Calculating Areas

The area covered by a single $2 \times 2$ tile is:
$$4 \quad \text{(square units)}$$

The area covered by a single $5 \times 1$ tile is:
$$5 \quad \text{(square units)}$$

### Step 2: Setting up Equations

Let $x$ be the number of $2 \times 2$ tiles and $y$ be the number of $5 \times 1$ tiles. According to the problem, $x = y$.

The total area covered by the tiles should equal the area of the floor:
$$4x + 5y = n^2$$
Since $x = y$, we substitute to get:
$$4x + 5x = 9x = n^2$$
Thus,
$$x = \frac{n^2}{9}$$

For $x$ to be an integer, $n^2$ must be divisible by 9. This means that $n$ must be a multiple of 3.

### Step 3: Identifying Values of $n$

We are given $10 < n < 20$. Within this range, the multiples of 3 are:
$$12, 15, 18$$

We will verify that each value of $n$ gives $x$ as an integer:

- For $n = 12$:
$$x = \frac{12^2}{9} = \frac{144}{9} = 16$$

- For $n = 15$:
$$x = \frac{15^2}{9} = \frac{225}{9} = 25$$

- For $n = 18$:
$$x = \frac{18^2}{9} = \frac{324}{9} = 36$$

In each case, $x$ is an integer, confirming that the floor can be completely covered with equal numbers of $2 \times 2$ and $5 \times 1$ tiles.

### Final Answer

Therefore, the possible values of $n$ for which the floor can be laid are:
$$\boxed{12, 15, 18}$$