7.5. What is the maximum number of three-cell corners that can be cut out from a grid rectangle of size: a) $5 \times 10$ cells; b) $5 \times 9$ cells?
Answer: a) 16; b) 15. Solution. a) It is clear that no more than 16 corners can be cut out, since

![](https://cdn.mathpix.com/cropped/2024_05_06_0e4929195c2239774641g-2.jpg?height=331&width=334&top_left_y=2013&top_left_x=193)
otherwise the rectangle must contain no fewer than \(17 \cdot 3=51>50\) cells. The left image shows an example of cutting one \(5 \times 5\) square into 8 corners and one (shaded) cell. The adjacent \(5 \times 5\) square

![](https://cdn.mathpix.com/cropped/2024_05_06_0e4929195c2239774641g-2.jpg?height=331&width=571&top_left_y=2013&top_left_x=1279)
is cut in the same way. b) The right image shows an example of cutting into 15 corners (each \(2 \times 3\) rectangle here is clearly cut into two corners).