Problems

Prove that the number of all arrangements of the largest possible amount of peaceful bishops (figures that move on diagonals and don’t threaten each other) on the \(8\times 8\) chessboard is an exact square.

Giuseppe has a sheet of plywood, measuring \(22 \times 15\). Giuseppe wants to cut out as many rectangular blocks of size \(3 \times 5\) as possible. How should he do it?

Fred and George had two square cakes. Each twin made two straight cuts on his cake from edge to edge. However, one ended up with three pieces, and the other with four. How could this be?

Deep in a forest there is a small town of talking animals. Elephant, Crocodile, Rabbit, Monkey, Bear, Heron and Fox are friends. They each have a landline telephone and each two telephones are connected by a wire. How many wires were required?

Cut a square into three pieces, from which you can construct a triangle with three acute angles and three different sides.

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.

Cut the figure (on the boundaries of cells) into three equal parts (the same in shape and size).

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?

At the disposal of a tile layer there are 10 identical tiles, each of which consists of 4 squares and has the shape of the letter L (all tiles are oriented the same way). Can he make a rectangle with a size of \(5 \times 8\)? (The tiles can be rotated, but you cannot turn them over). For example, the figure shows the wrong solution: the shaded tile is incorrectly oriented.

A rabbit, preparing for the arrival of guests, hung lightbulbs in three corners of his polygonal hole. Winnie the Pooh and Piglet came and noticed that lights did not illuminate all the pots of honey which were in the rabbit hole. When they reached for some honey, two of the lightbulbs broke. The rabbit moved the remaining light bulb into some corner so that the whole hole was lit up. Is this possible? (If yes, draw an example, if not, justify the answer.)