Transcript Multi-Peg Towers of Hanoi

Multi-Peg Towers of Hanoi
By Krüz Kalke
Summer 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 kvalues 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
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