Painting Method

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?

Problem #10 The bottom of a rectangular box is filled with figures of two types: squares 2 × 2 and rectangles 1× 4 . Somebody stole one 2 × 2 square and substituted it for 1× 4 rectangle. Is it still possible to fill the bottom of the box?

Problem #11 Find the maximum number of boxes 1× 1× 4 that a 6 × 6 × 6 cube can contain without intersections.

Problem #12 Is it possible to fill the board 10 × 10 with 4-square figures of “T”-shape?

Problem #13 Numbers 2, 7, 9, 10, 3, and 12 are written in the vertices of a hexagon. You can add or subtract the same number from two consecutive numbers at a time. Is it possible to obtain 5, 11, 6, 15, 8 and 14 this way?

Problem #14 On the graph paper one chose 2008 squares. Prove that from these 2008 squares you can choose at least 502 that do not have common points.

Problem #15 On the graph paper one chose n squares. Prove that you can choose from these n squares at least n 4 squares that do not have common points.

Problem #16 2 From a square 1993 × 1993 drawn on the graph paper somebody cut out ( 399 − 1) squares 3 × 3 . Is it possible to cut out one more such square?

Problem #17 Prove that if vertices of the convex polygon are in the nodes of the graph paper and there are no other nodes inside or on the sides of the polygon, then the polygon is a quadrilateral or a triangle.

Problem #18 A convex n-gon is divided into triangles with nonintersecting diagonals. Prove that if there are an odd number of triangles at every vertex, then n must be a multiple of 3.

Problem #19 Is it possible to fill the board 5 × 7 with 3-square angles with multiple layers?

Problem #20 An equilateral triangle is divided into n 2 equal equilateral triangles. To some of them numbers 1, 2,..., m are assigned in such a way that triangles with consequent numbers are adjacent. Prove that m ≤ n 2 − n + 1 .