Multi-Peg Towers of Hanoi By Krüz Kalke Summer 2014 1.
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Transcript of Multi-Peg Towers of Hanoi By Krüz Kalke Summer 2014 1.
1
Multi-Peg Towers of Hanoi
By Krüz KalkeSummer 2014
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Overview
•Three-Peg Puzzle•Four-Peg Puzzle
▫Example•Multi-Peg Puzzle
▫Example•My Current Research
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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
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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
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Four-Peg Puzzle Example
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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
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Multi-Peg Puzzle Example
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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
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Acknowledgments
•Multipeg Towers of Hanoi Applet by Robert J. Swartz [2014] – www.mathapplets.net
•“On the Frame-Stewart Algorithm for the Tower of Hanoi” by Michael Rand [2009] – www2.bc.edu/~grigsbyj/Rand_Final.pdf