1. It is impossible to measure (or calculate) the length of an island’s coastline.
2. If there are 23 or more people in a room there is a better than even chance that two have the same birthday.
3. Imagine carrying a large object down a corridor and you have to negotiate a right angle turn. We still don’t know what the biggest object, capable of making that turn, is.
4. The Hairy Ball theorem says that if your ball is hairy you cannot comb all the hair flat. It follows from this argument that at any one moment in the world there must be a place with no wind.
5. Get a list of prime numbers. There’s a method of then finding another prime number not on that list. Therefore there are infinitely many prime numbers.
6. You can draw any pattern and you will only need 4 colours to colour it in so that no two areas the same colour touch.
7. The set of decimals between 0 and 1 has more numbers in it than the set of all the whole numbers.
8. If you divide the universe up into cubes, each cube having a side equal to the Planck length (the smallest possible length) and wrote a digit in each cube you wouldn’t have enough space to write down the largest number ever used in Mathematics.
9. It is rumoured that Pythagoras killed one of his students for proving that the square root of 2 is irrational.
…and my personal favourite:
1+2+3+4+5+6+…. = -1/12
think about that! How the hell can it come out negative??