1. For what smallest $k$ can $k$ cells be marked on an $8 \times 9$ board such that for any placement of a four-cell figure on the board, it can be rotated and flipped.
Answer: 16.

Solution. It is not hard to notice that in any $2 \times 4$ rectangle, there are at least two marked cells. Since from an $8 \times 9$ board, 8 non-overlapping $2 \times 4$ rectangles can be cut out, there must be at least 16 marked cells in it. An example with 16 marked cells is shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_9304f7a3bc6f36cb1f9ag-14.jpg?height=294&width=340&top_left_y=664&top_left_x=915)