10.5. What is the maximum number of 2 by 2 squares that can be placed on a 7 by 7 grid of squares so that any two placed squares share no more than one common cell? The 2 by 2 squares are placed along the grid lines such that each covers exactly 4 cells. The squares do not extend beyond the boundaries of the board.
Answer: 18 squares.

Solution: First, let's provide an example of laying out 18 squares of 2 by 2: 9 of them cover the left bottom square of size 6 by 6 of the board, and the other 9 cover the right top square of size 6 by 6 of the board.

We will prove that it is impossible to lay out 19 squares correctly. Note that if a cell of the board, adjacent to the edge, is covered by two squares, then they intersect in at least two cells, and if a cell of the board, not adjacent to the edge, is covered by three squares, then two of them intersect in at least two cells. Consequently, the edge cells of the board can be covered by squares no more than once, and the internal cells no more than twice. Therefore, the squares can contain no more than \(24 \cdot 1 + 25 \cdot 2 = 74\) cells in total, and the total number of squares cannot exceed \(74 / 4 = 18.5\) pieces.

Grading criteria: Any correct example: 2 points. Proof that the number of squares is no more than 18: 5 points.