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A game takes place on a squared 9 × 9 piece of checkered paper. Two players play in turns. The first player puts crosses in empty cells, its partner puts noughts. When all the cells are filled, the number of rows and columns in which there are more crosses than zeros is counted, and is denoted by the number K, and the number of rows and columns in which there are more zeros than crosses is denoted by the number H $($ 18 rows in total $)$. The difference B = K – H is considered the winnings of the player who goes first. Find a value of B such that

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1$)$ the first player can secure a win of no less than B, no matter how the second player played;

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2$)$ the second player can always make it so that the first player will receive no more than B, no matter how he plays.

17 squares are marked on an 8×8 chessboard. In chess a knight can move horizontally or vertically, one space then two or two spaces then one – eg: two down and one across, or one down and two across. Prove that it is always possible to pick two of these squares so that a knight would need no less than three moves to get from one to the other.

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.

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?

In the isosceles triangle ABC, the angle B is equal to 30$^{\circ}$, and AB = BC = 6. The height CD of the triangle ABC and the height DE of the triangle BDC are drawn. Find the length BE.

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?

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

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.

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?