Is it possible to bake a cake that can be divided by one straight cut into 4 pieces?
What is the maximum number of pieces that a round pancake can be divided into with three straight cuts?
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?
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)?
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.
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?
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.)
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?
A game takes place on a squared \(9 \times 9\) 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
1) the first player can secure a win of no less than \(B\), no matter how the second player played;
2) the second player can always make it so that the first player will receive no more than \(B\), no matter how he plays.