Problems

Peter marks several cells on a \(5 \times 5\) board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square (you can only place the corners on the squares). What is the smallest number of cells that Peter should mark so that Richard cannot win?

Twenty-eight dominoes can be laid out in various ways in the form of a rectangle of \(8 \times 7\) cells. In Fig. 1–4 four variants of the arrangement of the figures in the rectangles are shown. Can you arrange the dominoes in the same arrangements as each of these options?

An entire set of dominoes, except for 0-0, was laid out as shown in the figure. Different letters correspond to different numbers, the same – the same. The sum of the points in each line is 24. Try to restore the numbers.

Giuseppe has a sheet of plywood, measuring \(22 \times 15\). Giuseppe wants to cut out as many rectangular blocks of size \(3 \times 5\) as possible. How should he do it?

At the disposal of a tile layer there are 10 identical tiles, each of which consists of 4 squares and has the shape of the letter L (all tiles are oriented the same way). Can he make a rectangle with a size of \(5 \times 8\)? (The tiles can be rotated, but you cannot turn them over). For example, the figure shows the wrong solution: the shaded tile is incorrectly oriented.

What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an \(8\times 8\) square grid, so that no more ’corners’ would fit?