In the \(4 \times 4\) square, the cells in the left half are painted black, and the rest – in white. In one go, it is allowed to repaint all cells inside any rectangle in the opposite colour. How, in three goes, can one repaint the cells to get the board to look like a chessboard?
On a chessboard, \(n\) white and \(n\) black rooks are arranged so that the rooks of different colours cannot capture one another. Find the greatest possible value of \(n\).
Several football teams are taking part in a football tournament, where each team plays every other team exactly once. Prove that at any point in the tournament there will be two teams who have played exactly the same number of matches up to that point.
A \(3\times 3\) square is filled with the numbers \(-1, 0, +1\). Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.
Some whole numbers are placed into a \(10\times 10\) table, so that the difference between any two neighbouring, horizontally or vertically adjacent, squares is no greater than 5. Prove that there will always be two identical numbers in the table.
Is it possible to place the numbers \(-1, 0, 1\) in a \(6\times 6\) square such that the sums of each row, column, and diagonal are unique?
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?
If a class of 30 children is seated in the auditorium of a cinema there will always be at least one row containing no fewer than two classmates. If we do the same with a class of 26 children then at least three rows will be empty. How many rows are there in the cinema?
Two grandmasters in turn put rooks on a chessboard (one turn – one rook) so that they cannot capture each other. The person who cannot put a rook on the chessboard loses. Who will win with the game – the first or second grandmaster?
In a tournament by the Olympic system (the loser is eliminated), 50 boxers participate. What is the minimum number of matches needed to be conducted in order to identify the winner?