Five Coins
You play a game with 5 coins. On turn 1 you flip all of them.
On turn 2, you pick up all the ones that came up tails
(leaving all the heads alone) and flip them again. You
continue to do this until all the coins are heads.
For example:
Turn 1: H T T H T
Turn 2:  H T  H ( means leave this coin alone)
Turn 3:   T  
Turn 4:   T  
Turn 5:   H  
Done in 5 turns.

As a reduced fraction, what is the expected number
of turns you'll need to finish the game?

As a reduced fraction, what is the probability
you will finish the game in 3 turns or less?

Can you find a general formula? If you have N coins to start, find
the expected number of turns and the
probability you will finish in k turns or less?
Larry Baum also suggests trying to solve for the general case:
With N coins, probability p of getting heads on each flip, find the
expected number of turns and the
probability you will finish in k turns.
Source: Denis Borris, citing Larry Baum, citing The Price is Right.
Solution
Mail to Ken