Problems

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).

Cut the board shown in the figure into four congruent parts so that each of them contains three shaded cells. Where the shaded cells are placed in each part need not be the same.

A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?

In a \(10 \times 10\) square, all of the cells of the upper left \(5 \times 5\) square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)

Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?

Is it possible to cut a square into four parts so that each part touches each of the other three (ie has common parts of a border)?