6. There is a paper rectangle $3 \times 100$, divided into 300 cells $1 \times 1$. What is the maximum number of pairs consisting of one corner and one $2 \times 2$ square that can be cut out along the grid lines? (A corner is obtained from a $2 \times 2$ square by removing one of its corner cells).
Answer: 33.

Sketch of the solution. The square in the middle row occupies two cells, and the corner - at least one, so the pair occupies at least three cells in the middle row. If there are no fewer than 34 pairs, then they occupy at least $34 \times 3=102$ cells in the middle row, while there are only 100 there.

Example: we cut out 33 squares $3 \times 3$. From each $3 \times 3$ square, we cut out a $2 \times 2$ square and a corner.

Criteria. Answer only: 0 points.