The angle at the top of a crane is \(20^{\circ}\). How will the magnitude of this angle change when looking at the crane with binoculars which triple the size of everything?
Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?
Two play a game on a chessboard \(8 \times 8\). The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?
What is the minimum number of squares that need to be marked on a chessboard, so that:
1) There are no horizontally, vertically, or diagonally adjacent marked squares.
2) Adding any single new marked square breaks rule 1.
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?
Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.
So, the mother exclaimed - “It’s a miracle!", and immediately the mum, dad and the children went to the pet store. “But there are more than fifty bullfinches here, how will we decide?,” the younger brother nearly cried when he saw bullfinches. “Don’t worry,” said the eldest, “there are less than fifty of them”. “The main thing,” said the mother, “is that there is at least one!". “Yes, it’s funny,” Dad summed up, “of your three phrases, only one corresponds to reality.” Can you say how many bullfinches there was in the store, knowing that they bought the child a bullfinch?
In a 10-storey house, 1 person lives on the first floor, 2 on the second floor, 3 on the third, 4 on the fourth, ..., 10 on the tenth. On which floor does the elevator stop most often?
Decipher the quote from "Alice in Wonderland" from the following matrix:
\[\begin{array}{@{}*{26}{c}@{}}
Y&q&o&l&u&e&c&d&a&i&n \\
w&a&r&l&a&w&e&a&t&y&k \\
s&n&t&c&a&e&k&c&e&a&m \\
t&o&d&r&w&e&a&t&a&h&r \\
a&c&n&t&n&e&o&d&t&r&h \\
n&i&d&n&l&g&m&e&x&s&z
\end{array}\]