6

Part of 1999 Tournament Of Towns

Problems(3)

TOT 1999 Spring AJ6 64 moves of a rook in a 8x8 chessboard

Source:

A rook is allowed to move one cell either horizontally or vertically. After

64

moves the rook visited all cells of the

8 \times 8

chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction cannot be equal.
(A Shapovalov, R Sadykov)

combinatoricsChessboardRook

n rectangular holes in rectangle paper

Source: Tournament Of Towns Spring 1999 Junior A Level p6

Inside a rectangular piece of paper

n

rectangular holes with sides parallel to the sides of the paper have been cut out. Into what minimal number of rectangular pieces (without holes) is it always possible to cut this piece of paper?
(A Shapovalov)

geometryrectangle

TOT 1999 Autumn AS6 rook on a large chessboard, rectangle combo

Source:

On a large chessboard

2n

1 \times 1

squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles.
(A Shapovalov)

combinatorial geometrycombinatoricsrectangleChessboard