Classic Mathematical puzzles

A duck is in the center of a circular pond, with a bear waiting on the shore to kill it. If the duck can reach the shore without immediately being pounced upon it will fly away to safety, but it can't get traction to fly from water. Trying to outwait the bear is not an option: The bear is hungry, but the duck is hungrier.

The duck paddles at 1 meter per second. If the bear runs at less than 3.14159 (π) meters per second, the duck will escape just by paddling away from the bear's starting position. But the duck can do better than that. What is the fastest-running bear from whom the duck can escape?

The warden has decided to shut down his prison, but hasn't decided whether to release all 100 prisoners, or to just execute them all. He decides to stage a contest. He writes the 100 prisoner names (along with their prisoner ID numbers, 1 to 100) on 100 slips of paper, shuffles them thoroughly, and places them in 100 numbered drawers. Each prisoner will be given a chance to open drawers one-by-one until he either finds his name or has opened fifty drawers. If all prisoners locate their own slip within fifty drawers then all hundred of them are set free, but if even one prisoner fails, then all 100 will be executed.

The prisoners are allowed to plan a strategy, but once the drawer-opening begins no further communication is permitted. The prisoners will not be able to observe which drawers their fellows open; the drawers and their contents will all be restored to their initial state between prisoners.

For any single prisoner it is easy to see that his chance of success is 50/100. If the prisoners' chances are independent then their combined chance will be (50/100)^100, which is less than one chance in a trillion billion billion. Can they improve on this?

The Sultan will show you one thousand of his daughters, one-by-one in an order determined by random chance, tell you how many dinars the maiden's dowry is, and ask if you want to marry her. Your decision is irrevocable: if you say 'Yes' you won't even meet the other daughters; if you say 'No' you won't be able to pick that maiden later. In other words, if you reject the first 999, you're stuck with daughter #1000.

There's one catch. The Sultan wants to know if you're clever enough to pick the girl with the largest dowry. If you pick any of the other 999 girls, you'll be executed instead of married.

How do you play, and what is your chance of success? (For a variation, how about if the Sultan lets you live and marry either of the two highest-dowried daughters?)

A spectator chooses five cards from an ordinary deck of fifty-two. The magician's assistant removes one of the five cards, and arranges the other four into some order. No other signalling is permitted, just the choice among 4! = 24 possibilities for ordering the four cards.

The magician comes into the room, inspects the four cards in order, and announces what the fifth card was. How does he do it.