Problem 1272

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 $\boxed{1}$