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?
The intelligence agency of the Galactic Empire intercepted the following coded message from the enemy planet Medusa: \(ABCDE+BADC=ACDED\).
It is known that different numbers are represented by different letters, and that the same numbers are represented by the same letters. Two robots attempted to decode this message and each one got a different answer. Is this possible, or should one of the robots be melted down as scrap metal?
Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?
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!
Is it possible to fill a \(5 \times 5\) board with \(1 \times 2\) dominoes?
a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.
b) What can be said about the case of a decagon?
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.
Each cell of a \(2 \times 2\) square can be painted either black or white. How many different patterns can be obtained?
How many ways can Rob fill in one card in the “Sport Forecast” lottery? (In this lottery, you need to predict the outcomes of thirteen sports matches. The result of each match is the victory of one of the teams or a draw, and the scores do not play a role).