NDLI: On the Frame--Stewart Conjecture about the Towers of Hanoi

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Shen, Jian Chen, Xiao
Copyright Year	2004
Abstract	The multipeg Towers of Hanoi problem consists of k pegs mounted on a board together with n disks of different sizes. Initially these disks are placed on one peg in the order of their size, with the largest at the bottom. The rules of the problem allow disks to be moved one at a time from one peg to another as long as a disk is never placed on top of a smaller disk. The goal of the problem is to transfer all the disks to another peg with the minimum number of moves, denoted by H(n,k). An easy recursive argument shows that H(n,3)= 2n -1. However, the problem of computing the exact value of H(n,k) for $k \ge 4$ has been open since 1939, and in particular, the special case of H(n,4) has been open since 1907.In 1941, Frame and Stewart each gave an algorithm to solve the Towers of Hanoi problem based on an unproved assumption. The Frame--Stewart number, denoted by FS(n,k), is the number of moves needed to solve the Towers of Hanoi problem using the "presumed optimal" Frame--Stewart algorithm. Since then, proving the Frame--Stewart conjecture FS(n,k) =H(n,k) has become a notorious open problem. In this paper, we prove that FS(n,k) and H(n,k) both have the same order of magnitude of $2^{(1\pm o(1))(n(k-2)!)^{1/(k-2)}}$. This provides the strongest evidence so far to support the Frame--Stewart conjecture.
Starting Page	584
Ending Page	589
Page Count	6
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539703431019
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2004-01-01
Access Restriction	Subscribed
Subject Keyword	Towers of Hanoi problem Frame--Stewart conjecture Frame--Stewart algorithm Factorials, binomial coefficients, combinatorial functions Combinatorics optimal algorithm Asymptotic enumeration
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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