Method of Coloring Roman A. Kvasov
Problem #1 Somebody cut out top-right and bottom-left squares from the 8 × 8 board. Is it possible to fill the remained 62-square figure with rectangles 1× 2 ? Problem #2 Is it possible to visit all squares of the board 9 × 9 exactly once and come back to the initial square with a knight? Problem #3 A mouse eats a piece of cheese 3 × 3 × 3 divided into 27 little cubes. After the mouse eats one little cube completely it starts to eat another adjacent little cube. Is it possible for the mouse to eat all little cubes except for the one in the center? Problem #4 In every square of the board 5 × 5 there is a bug. At some moment of time every bug climbed out horizontally or vertically to an adjacent square. Is it true that after this at least one square becomes empty? Problem #5 In every square of board 9 × 9 there is a bug. At some moment of time every bug climbed out diagonally to an adjacent square. It is true that after this at least 9 squares become empty? Problem #6 On the graph paper three crazy grasshoppers sit in three vertices of a square 1× 1 . Grasshoppers can only jump one over the other but have to land on the same line and form the same distance they had before. Is it possible for one grasshopper to get into the 4th vertex of an initial square sometime? Problem #7 Six crazy grasshoppers sit in six vertices of a regular hexagon. Grasshoppers can only jump one over the other but have to land on the same line and form the same distance they had before. Is it possible for one grasshopper to get into the center of the hexagon? Problem #8 Is it possible to fill the board 10 × 10 with rectangles 1× 4 ? Problem #9 Is it possible to fill the board 6 × 6 with rectangles 1× 3 and one 3-square angle?