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 \(n\).
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?
What is the smallest number of cells that can be chosen on a \(15\times15\) board so that a mouse positioned on any cell on the board touches at least two marked cells? (The mouse also touches the cell on which it stands.)
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 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.
The White Rook pursues a black horse on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play in order to take the horse? White makes the first move.
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?
Two play a game on a chessboard \(8 \times 8\). The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?
What is the minimum number of squares that need to be marked on a chessboard, so that:
1) There are no horizontally, vertically, or diagonally adjacent marked squares.
2) Adding any single new marked square breaks rule 1.