Example 2 Use $1 \times 1, 2 \times 2, 3 \times 3$ tiles to cover a $23 \times 23$ floor (no overlapping, no gaps), how many $1 \times 1$ tiles are needed at least?
Solve: In a $23 \times 23$ square floor (divided into $1 \times 1$ small squares), the small squares in columns $1, 4, 7, \cdots, 19, 22$ are colored black, and the rest of the columns are colored white. If $1 \times 1$ tiles are not used, then each $2 \times 2$ tile either covers 2 white squares and 2 black squares or covers 4 white squares, and each $3 \times 3$ tile covers 3 black squares and 6 white squares. Therefore, regardless of how many $2 \times 2$ and $3 \times 3$ tiles are used, the number of white squares covered is always even. However, there are $15 \times 23$ white squares, and $15 \times 23$ is an odd number, which is a contradiction. Hence, it is impossible to cover the $23 \times 23$ square without using $1 \times 1$ tiles.

Furthermore, Figure 6-18 shows that $2 \times 2$ and $3 \times 3$ tiles can cover a $12 \times 11$ rectangle, and Figure 6-19 shows that using just 1 $1 \times 1$ tile can cover the $23 \times 23$ square.
In conclusion, at least 1 $1 \times 1$ tile is needed.