Can you pay 20 p in seven coins?
Can you pay 20 p in seven coins using 1p and 5p coins only?
Can you pay 25 p in eight coins using 1p and 5p coins only?
(a) Show that it is impossible to find five odd numbers which all add to 100.
(b) Alice wrote several odd numbers on a piece of paper. The Hatter did not see the numbers, but says that if he knew how many numbers Alice wrote down, than he would say with certainty if the sum of the numbers is even or odd. How can he do it?
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 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?
Prove that the equation \(x^2 + 4034 = y^2\) does not have solutions in integer numbers.