Example: On a $18 \times 8$ chessboard, what is the minimum number of L-shaped blocks (each covering exactly 3 small squares) that can be placed without overlapping so that no additional L-shaped block can be placed on the remaining part?
Solution: Suppose $n$ non-overlapping L-shaped blocks are placed on an $8 \times 8$ chessboard. Now, divide the chessboard into 16 $2 \times 2$ squares. Since there are $64-3n$ small squares not covered by the blocks, there must be at least $\left[\frac{64-3n-1}{16}\right]+1$ uncovered squares in one of these 16 squares. If $\left[\frac{64-3n-1}{16}\right]+1 \geqslant 3$, i.e., $n \leqslant 10$, then, this square can still accommodate one more L-shaped block.

On the other hand, as shown in Figure 6-27, after placing 11 non-overlapping L-shaped blocks, the remaining part cannot accommodate another L-shaped block.
Therefore, at least 11 L-shaped blocks must be placed to ensure that the remaining part cannot accommodate another L-shaped block.