1. Multi-Peg Towers of Hanoi By Krüz Kalke Summer 2014 1

2. Overview Three-Peg Puzzle Four-Peg Puzzle ▫Example Multi-Peg Puzzle ▫Example My Current Research 2

3. Three-Peg Puzzle The goal is to move the stack of disks from the starting peg to the destination peg Rules: ▫Only one disk can be moved at a time ▫Only the topmost disk on a stack may be moved ▫Larger disks cannot be placed on smaller disks 3

4. Four-Peg Puzzle With more pegs, less moves are required The Frame-Stewart algorithm is the presumed optimal solution: ▫Move k upper disks to temporary peg using four pegs ▫Move bottom disks to destination peg using remaining three pegs ▫Move previous k disks to destination using four pegs 4

5. Four-Peg Puzzle Example 5

6. Multi-Peg Puzzle The previous algorithm works for any number of pegs Each splitting of the stack reduces the number of available pegs by one An open problem is how to decide the optimal k- values at each step 6

7. Multi-Peg Puzzle Example 7

8. My Current Research Analyze algorithm used in Robert Swartz’s Towers of Hanoi applet Verify correctness of his approach Analyze pattern of optimal k-values given by Swartz’s algorithm Attempt to find a closed-form solution that generates k-values given any number of disks and pegs 8

9. Acknowledgments Multipeg Towers of Hanoi Applet by Robert J. Swartz [2014] – www.mathapplets.netwww.mathapplets.net “On the Frame-Stewart Algorithm for the Tower of Hanoi” by Michael Rand [2009] – www2.bc.edu/~grigsbyj/Rand_Final.pdf www2.bc.edu/~grigsbyj/Rand_Final.pdf 9