A board \(7\times7\) is coloured in chessboard fashion in such a way that all the corners are black. The Queen orders the Hatter to colour the board white but sets the rule: in one go it is allowed to repaint only two adjacent cells into opposite colours. The Hatter tries to explain that this is impossible. Can you help the Hatter to present his arguments?
Prove that the equation \(x^2 + 4034 = y^2\) does not have solutions in integer numbers.
Find a solution of the equation \(x^2 + 2017 = y^2\) in integer numbers.
A snail climbs a 10-meter high tree. In day time the snail manages to climb 4 meters, but slips down 3 meters during the night time. How long would it take the snail to reach the top of the tree if it started the journey on a Monday morning?
A girl and a boy are sitting on a long playground bench. Twenty other children approach them, and one by one sit down in-between two already sitting children until everybody is sitting comfortably on the bench. We call a boy “brave” if he sat between two girls, and we call a girl “brave” if she sat between two boys. How many boys and girls are “brave” if the boys and the girls who sit on the bench alternate?
A natural number \(n\) can be exchanged to number \(ab\), if \(a+b=n\) and \(a\) and \(b\) are natural numbers. Is it possible to receive 2017 from 22 after such manipulations?
Alice marked several points on a line. Then she put more points – one point between each two adjacent points. Show that the total number of points on the line is always odd.
The Dormouse brought a \(4\times 10\) chocolate bar to share at the tea party. She needs to break the bar by the lines into single pieces (without any lines on it). In one turn she can cut one piece into two along the lines. What is the least number of cuts she needs to make to break the bar into single pieces?
Tweedledum and Tweedledee travel from their home to the castle of the White Queen. Having only one bicycle among them, they take turns in riding the bike. While one of them is cycling, the other one walks (walking means walking and not running). Nevertheless, they manage to arrive to the castle nearly twice as fast in comparison to if they both were walking all the way to the castle. How did they manage it?
The Hatter plays a computer game. There is a number on the screen, which every minute increases by 102. The initial number is 123. The Hatter can change the order of the digits of the number on the screen at any moment. His aim is to keep the number of the digits on the screen below four. Can he do it?