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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