6. On one hand, using six straight lines to divide the large square grid into 16 smaller $2 \times 2$ grids, each $2 \times 2$ grid must have at least two cells covered by an L-shape. Therefore, the number of cells covered is no less than 32. Thus, the number of L-shapes is no less than $\left[\frac{32}{3}\right]+1$ $=11$, meaning no fewer than 11 L-shapes.

On the other hand, placing 11 L-shapes as shown in Figure 4 can meet the requirements of the problem.
In conclusion, $k_{\min }=11$.
6. A shape obtained by removing one unit square from a $2 \times 2$ grid is called an "L-shape". In an $8 \times 8$ grid, place $k$ L-shapes, each of which can be rotated, but each L-shape must cover exactly three unit squares of the grid, and any two L-shapes must not overlap. Additionally, no other L-shape can be placed in the grid besides these $k$ L-shapes. Find the minimum value of $k$. (Supplied by Zhang Yuezhu)