Some fun questions I thought you might like:
1. If I take a square and split it into 4 similar smaller squares, and remove one of them, how can I split the remaining shape into 4 equal shapes? Can you do the same with an equilateral triangle? (You can show me your answers in Paint.)
2. A gamber plays at a casino (which had unlimited funds). He starts with £x. Each turn, he flips a coin. If it lands heads up, he wins £1, but if it lands tails up, he loses £1. With a fair coin, the game would be expected to last forever. However, the casino (naturally!) is using a loaded coin, that has a probability 'p' of landing heading (p<0.5). How long can we expect the game to last for now, before the gambler runs out of money (in terms of p and x)?
3. There are 100 doors, arranged in a circle, all of which start locked. A man walks around, unlocking every door, and then walks around again, locking every other door. He then goes around again, locking or unlocking every third door. The man continues doing this, until he has locked/unlocked every 100th door (i.e. the last). Which doors are now unlocked?
Each question earns (a little) rep, but more importantly, you get the satisfaction of doing maths!
3) i answered this a while ago in a 'questions you might get asked at a cambridge interview' thread. now give me rep.
2) E(x) = p - (1-p) = 2p-1
n is number of goes, we want nE(x) = x = n(2p-1)
n = x/(2p-1)
1) which triangle is removed in the second case? and edge or the middle one?
3. All the square numbers because of the repitition of factors, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100
Ah that's why I couldn't do question one and it seemed ridiculous. I figured once you'd cut the square in 4 the original lines would remain.