in reply to Recursion: The Towers of Hanoi problem
Another related fun 'n' interesting problem is to start with the disks placed randomly onto the three pegs and then try to sort them all onto one peg. In this version, you are allowed to place any disk onto any other disk (no size restrictions). In fact, a question on a recent algorithms Ph.D qualifying exam here at UIUC was to show that disks can be sorted in O(n log n) moves, and to also show that in fact, &Omega(n log n) moves are required in general. I won't give away how to sort the disks in O(n log n) time, but here's a hint: adapt quicksort.
If we aren't allowed to put larger disks on top of smaller disks (as in the classic example above), you need an exponential number of moves. If T(n) is the number of moves needed to sort n disks, your recursion gives the recurrence T(n) = 2T(n1). You can verify that T(n) = 2^n.
Towers of Hanoi problems are good examples in that they are easy to visualize, and their relationship to sorting with stacks is obvious.
Visualizing (was Re^2: Visiting the Towers of Hanoi.) by merlyn (Sage) on Oct 19, 2004 at 03:50 UTC 
Towers of Hanoi problems are good examples in that they are easy to visualize, and their relationship to sorting with stacks is obvious.
However, if you are selecting problems for a syllabus, please be aware that
"easy to visualize" can actually give an unfair disadvantage to some of your students. I've commented on my own affliction with regard to using GUIs and learning to program.
I completely understand the algorithm for Towers of Hanoi, and why it works, and have translated it into a few languages. But I understood it only by playing with physical disks or marks (and erasing) on a piece of paper. I could never have constructed it by "visualization", since my brain doesn't work that way.
So, beware when you pretend you know what skills are "universal" about thinking... you may have wrong thinking about that. {grin}
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