A mathematical chess problem is a mathematical problem which is formulated using a chessboard and chess pieces. These problems belong to recreational mathematics. The most well-known problems of this kind are the eight queens puzzle and the knight's tour problem, which have connection to graph theory and combinatorics. Many famous mathematicians studied mathematical chess problems, such as, Thabit, Euler, Legendre and Gauss. Besides finding a solution to a particular problem, mathematicians are usually interested in counting the total number of possible solutions, finding solutions with certain properties, as well as generalization of the problems to N×N or M×N boards.
1. The chess knight stands in the lower left corner of the board. Can he through a) 4; b) 5; c) 2013 moves to return to the original field?
2. Is it possible to place the pieces in the cells of an 8×8 chessboard (in each cell there is no more than one piece) so that there are equal numbers of pieces in all verticals, but not equally in any two horizontals?
3. On a 4x4 chessboard there is a piece - a “flying rook”, which moves in the same way as an ordinary rook, but cannot move to a square adjacent to the previous one in one move. Can she go around the entire board in 16 moves, standing on each square once, and return to the original square?
4. What is the largest number of a) rooks; b) can kings be placed on a chessboard so that they do not beat each other?
5. Each cell of a 7x7x7 triangular board contains a beetle. At one point, each beetle crawled onto the cell next to it. a) Prove that at least one cell was free. b) What is the smallest number of cells that could be free? c) Come up with such a “crawling” of beetles so that as many cells as possible are empty.
6. Arrange 32 knights on a chessboard so that each of them beats exactly two others.
7. Is it possible to cut a chessboard into dominoes such that no two dominoes form a 2x2 square?
8. Is it possible to cut a chessboard into 15 vertical and 17 horizontal dominoes?
9. Prove that the number of ways to place 8 queens on a chessboard is even.
10. Given a cube 4x4x4. Arrange 16 rooks in it so that they do not attack each other.
eleven.
11. On a 5x5 chessboard, the maximum number of knights was placed so that they did not beat each other. Prove that this arrangement is the only one.
12. What is the largest number of squares that can be marked on a chessboard so that from each of them it is possible to move from each of them to any other marked square in exactly two moves of a chess knight?
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