At the tea party the Hatter, who loves everything being odd, decided to divide 25 cakes between himself, the March Hare, Alice, and the Dormouse in such a way that everybody receives an odd number of cakes. Show that he would never be able to do it.
Alice went to a shop to buy flowers for her sister. She bought 6 roses £1 each, 4 lilies 82p each, and 4 freesias 76p each. At the till she was asked to pay £12.25, which she asked to recalculate straight away pointing out that the amount was not correct. Alice did not calculate the full amount, but how did she know that they made a mistake?
The Hatter says that he knows four numbers such that their sum and their product are odd. Is he right? Can you expain why?
The four numbers 1, 1, 1, 2 are written on a piece of paper. Alice and the Hatter are playing a game. They add in turns 1 to any two numbers to make the new four numbers. The winner is the one to make all the four numbers equal. If Alice goes first, who will win, if any?
The March Hare and the Dormouse also decided to play a game. They made two piles of matches on the table. The first pile contains 7 matches, and the second one 8. The March Hare set the rules: the players divide a pile into two piles in turns, i.e. the first player divides one of the piles into two, then the second player divides one of the piles on the table into two, then the first player divides one of the piles into two and so on. The loser is the one who cannot not find a pile to divide. The March Hare starts the game. Can the March Hare play in such a way that he always wins?
The Hatter is obsessed with odd numbers. He is very determined to represent 1 as \[1 = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d},\] where \(a\), \(b\), \(c\), and \(d\) are all odd.
Alice is very sceptical about it. Do you think you can help Alice to persuade the Hatter that it is impossible?
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?