Answer to POW #2:

In general, the least number of moves required to move n rings is (2^n)-1. Therefore, to move 9 rings requires (2^9)-1=511 moves.

The formula (2^n)-1 can be found by inspection, by examining the number of moves necessary to move 1 ring, 2 rings, 3 rings, 4 rings, and so on until the pattern is clear. Once you've "guessed" the formula, if you're hankering for some rigor you can prove the formula by induction.

Source: This problem is an easier version of a problem from Spivak's calculus book.