Three wise men ride on a train. Suddenly the train drives into a tunnel, and after the lights come on, each of the men sees that the faces of his colleagues are stained with soot that has flown through the car window. All three begin to laugh at their stained companions, but suddenly the most intelligent man guesses that his face is also stained. How did he do it?
Is it possible to arrange 6 long round pencils so that each of them touches all the other ones?
With the help of scissors, cut a hole in a notebook through which an elephant could climb!
In each cell of a \(25 \times 25\) square table, one of the numbers 1, 2, 3, ..., 25 is written. In cells, that are symmetric relative to the main diagonal, equal numbers are written. There are no two equal numbers in any row and in any column. Prove that the numbers on the main diagonal are pairwise distinct.
There are three groups of stones: in the first – 10, in the second – 15, in the third – 20. During one turn, you are allowed to split any pile into two smaller ones; the one who cannot make a move loses.
Numbers from 1 to 20 are written in a row. Players take turns placing pluses and minuses between these numbers. After all of the gaps are filled, the result is calculated. If it is even, then the first player wins, if it is odd, then the second player wins. Who won?
Two players take turns to put rooks on a chessboard so that the rooks cannot capture each other. The player who cannot make a move loses.
On a board there are written 10 units and 10 deuces. During a game, one is allowed to erase any two numbers and, if they are the same, write a deuce, and if they are different then they can write a one. If the last digit left on the board is a unit, then the first player won, if it is a deuce then the second player wins.
The numbers 25 and 36 are written on a blackboard. Consider the game with two players where: in one turn, a player is allowed to write another natural number on the board. This number must be the difference between any two of the numbers already written, such that this number does not already appear on the blackboard. The loser is the player who cannot make a move.
Consider a chessboard of size (number of rows \(\times\) number of columns): a) \(9\times 10\); b) \(10\times 12\); c) \(9\times 11\). Two people are playing a game where: in one turn a player is allowed to cross out any row or column as long as there it contains at least one square that is not crossed out. The loser is the player who cannot make a move. Which player will win?