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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.

On a plane there is a square, and invisible ink is dotted at a point P. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does P lie $($ if P lies on the line, then he says that P lies on the line $)$.

What is the smallest number of such questions you need to ask to find out if the point P is inside the square?

During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men $($ at least 80% $)$ – with a girl who was at the same time more beautiful and more intelligent. Could this happen? $($ There was an equal number of boys and girls at the ball.$)$

A square piece of paper is cut into 6 pieces, each of which is a convex polygon. 5 of the pieces are lost, leaving only one piece in the form of a regular octagon $($see the drawing$)$. Is it possible to reconstruct the original square using just this information?

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?

Is it possible to cut a square into four parts so that each part touches each of the other three $($ie has common parts of a border$)$?

Decipher the following rebus $\\$

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All the digits indicated by the letter “E” are even $($ not necessarily equal $)$; all the numbers indicated by the letter O are odd $($ also not necessarily equal $)$.

In Neverland, there are magic laws of nature, one of which reads: “A magic carpet will fly only when it has a rectangular shape.” Frosty the Snowman had a magic carpet measuring $9 \times 12$. One day, the Grinch crept up and cut off a small rug of size $1 \times 8$ from this carpet. Frosty was very upset and wanted to cut off another $1 \times 4$ piece to make a rectangle of $8 \times 12$, but the Wise Owl suggested that he act differently. Instead he cut the carpet into three parts, of which a square magic carpet with a size of $10 \times 10$ could be sown with magic threads. Can you guess how the Wise Owl restructured the ruined carpet?

What is the maximum number of pieces that a round pancake can be divided into with three straight cuts?

Is it possible to bake a cake that can be divided by one straight cut into 4 pieces?

How can you divide a pancake with three straight sections into 4, 5, 6, 7 parts?

Two people had two square cakes. Each person made 2 straight cuts from edge to edge on their cake. After doing this, one person ended up with three pieces, and the other with four. How could this be?

Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?

Four lamps need to be hung over a square ice-rink so that they fully illuminate it. What is the minimum height needed at which to hang the lamps if each lamp illuminates a circle of radius equal to the height at which it hangs?

Several pieces of carpet are laid along a corridor. Pieces cover the entire corridor from end to end without omissions and even overlap one another, so that over some parts of the floor lie several layers of carpet. Prove that you can remove a few pieces, perhaps by taking them out from under others and leaving the rest exactly in the same places they used to be, so that the corridor will still be completely covered and the total length of the pieces left will be less than twice the length corridor.

In a $10 \times 10$ square, all of the cells of the upper left $5 \times 5$ square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square $($on the boundaries of the cells$)$ so that in every polygon there would be three times as many white cells than black cells? $($Polygons do not have to be equal in shape or size.$)$

Every day, James bakes a square cake size $3\times3$. Jack immediately cuts out for himself four square pieces of size $1\times1$ with sides parallel to the sides of the cake $($not necessarily along the $3\times3$ grid lines$)$. After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?

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 $BE.$