A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?
1. **Initial Assumption and Coloring:**
   - Suppose we do not need any $1 \times 1$ tiles.
   - Color the $23 \times 23$ square in a checkerboard pattern, starting with the first row as black. This means alternate rows are black and white.
   - In this pattern, there will be $12$ black rows and $11$ white rows, each containing $23$ squares.
   - Therefore, the number of black squares is $12 \times 23 = 276$ and the number of white squares is $11 \times 23 = 253$.
   - The difference between the number of black and white squares is $276 - 253 = 23$.

2. **Properties of $2 \times 2$ and $3 \times 3$ Tiles:**
   - **Observation 1:** A $2 \times 2$ tile covers $2$ black and $2$ white squares, so the difference in the number of black and white squares within any $2 \times 2$ tile is $0$.
   - **Observation 2:** A $3 \times 3$ tile covers either $6$ black and $3$ white squares or $3$ black and $6$ white squares. Thus, the difference in the number of black and white squares within any $3 \times 3$ tile is $\pm 3$.

3. **Divisibility Argument:**
   - The difference in the number of black and white squares in the entire $23 \times 23$ square is $23$, which is not divisible by $3$.
   - Since the difference contributed by $2 \times 2$ tiles is $0$ and by $3 \times 3$ tiles is $\pm 3$, the total difference must be a multiple of $3$ if only $2 \times 2$ and $3 \times 3$ tiles are used.
   - However, $23$ is not a multiple of $3$, leading to a contradiction.

4. **Conclusion:**
   - Therefore, it is impossible to tile the $23 \times 23$ square using only $2 \times 2$ and $3 \times 3$ tiles without using at least one $1 \times 1$ tile.

5. **Example Construction:**
   - Place one $1 \times 1$ tile in the center of the $23 \times 23$ square.
   - This leaves a $22 \times 22$ area to be tiled.
   - The $22 \times 22$ area can be divided into four $11 \times 11$ squares.
   - Each $11 \times 11$ square can be tiled using $2 \times 2$ and $3 \times 3$ tiles.

The final answer is $\boxed{1}$