1 The game “Towers of Hanoi” consists of three rods, on the first of them a

the first of them a pyramid of n disks is

installed, the radius of which decreases from the lower disk to the upper one. It is required to transfer disks to the third core, using the second core as an auxiliary one and following the following rules.
a) In one operation, you can transfer only one disk.
b) You can not put a larger disc diameter on a smaller disc diameter.

Problems

1. Transfer 5 disks
2. Prove that N disks can be moved.

begin
Solve(n-1, a, c, b);
Writeln(‘transfer', a, ‘to rod',b);
Solve(n-1,

c, b, a);
end;
end;
begin
Solve(4, '1','2','3');
end.

the plane. Prove that it is possible to color the

plane in two colors so that the two areas that have a common part of the border have a different color. Areas that have only one common vertex may be of the same color.

2. Prove that a square can be cut into any number of squares, starting with 6.

3. Prove that for every N > 2 exists N–gon with three acute angles.

4. Prove that the square 2N х 2N, from which one cell was cut can be cut into “corners” of three cells.

any n people there is a different person from them

who is familiar with each of them. Prove that in this company there is a person who knows everyone.

From n to (n +1)

2. Among the participants of the conference, everyone has at least one friend. Prove that the participants can be distributed in two rooms so that each participant has a friend in the other room.

We can reduce the problem for example with a tree