Three tortoises crawl along the road in a line. “Two tortoises are crawling behind me,” says the first. “One tortoise is crawling behind me, and one tortoise is crawling in front of me,” says the second. “Two tortoises are crawling in front of me, and one tortoise is crawling behind me,” says the third. How can this be?
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!
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).
In a football team (made up of 11 people), a captain and his deputy need to be chosen. How many ways can this be done?