100 Prisoners and a Lightbulb

There are 100 prisoners in a jail, each in an isolated cell. In the lobby of the jail is a room with a light and a lightbulb (starts in the off position), which can only been seen from the lobby. Each day, the warden will take 1 random prisoner down to the lobby. The prisoner can flip the switch (either turning the bulb off it was on or on if it was off), or the prisoner can do nothing, or the prisoner can say "each prisoner has been to the lobby at least once." If the prisoner who says this is correct, they all go free. If he is incorrect, they are all killed. The prisoners can get together at the very beginning to discuss a strategy, but can not communicate with one another ever again until the exercise is over. What strategy can the prisoners use to guarantee that they all survive?

1 Million Lockers

There are one million numbered lockers in a row. They all start out closed. On your first pass, you flip every locker, so each one is open. On your second pass, you flip every locker divisible by 2, so you close lockers 2, 4, 6, etc. On the third pass, you flip every locker divisible by 3, so locker 3 becomes closed, locker 6 becomes open, etc. You continue this pattern until the millionth pass, flipping only the millionth locker on the last pass. At the end, is the millionth locker open? What do all the open lockers have in common? Why?

Trains

There are two trains traveling 60 mph towards each other. When they are 120 miles apart, a fly leaves one train heading towards the other at 100 mph. As soon as it reaches the other train, it immediately flies back towards the first train. It continues doing this until the trains collide. How much distance does the fly cover before the trains collide?

Cards in the Dark

You are in a completely dark room with a deck of 52 cards. 12 of the cards are face up, but they are randomly distributed throughout the deck. You need to put the cards into two piles (not necessarily of the same size) so that each pile has the same number of face up cards. There are no markings on the cards that would help you identify the face up cards in any way. You can also flip cards to upside down if you want (changing the number of total face up cards). How do you do it?

25 Horses

You have 25 horses. You need to find out which 3 are the fastest. Only 5 horses can race at a time. You are only given the order in which the horses finish the races, not their actual times. What is the fewest number of races you need to find the 3 fastest?