What is the largest number of horses that can be placed on an \(8\times8\) chessboard so that no horse touches more than seven of the others?
Chess board fields are numbered in rows from top to bottom by the numbers from 1 to 64. 6 rooks are randomly assigned to the board, which do not capture each other (one of the possible arrangements is shown in the figure). Find the mathematical expectation of the sum of the numbers of fields occupied by the rooks.
A tennis tournament takes place in a sports club. The rules of this tournament are as follows. The loser of the tennis match is eliminated (there are no draws in tennis). The pair of players for the next match is determined by a coin toss. The first match is judged by an external judge, and every other match must be judged by a member of the club who did not participate in the match and did not judge earlier. Could it be that there is no one to judge the next match?
30 teams are taking part in a football championship. Prove that at any moment in the contest there will be two teams who have played the same number of matches up to that moment, assuming every team plays every other team exactly once by the end of the tournament.
A White Rook pursues a black bishop on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play to take the bishop? White makes the first move.
Fill the free cells of the “hexagon” (see the figure) with integers from 1 to 19 so that in all vertical and diagonal rows the sum of the numbers, in the same row, is the same.
Six chess players participated in a tournament. Each two participants of the tournament played one game against each other. How many games were played? How many games did each participant play? How many points did the chess players collect all together?
Is it possible to fill a \(5 \times 5\) table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?
Find out the principles by which the numbers are depicted in the tables (shown in the figures below) and insert the missing number into the first table, and remove the extra number from the second table.
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?