On the Frame-Stewart algorithm for the multi-peg Tower of Hanoi problem
نویسندگان
چکیده
It is proved that seven different approaches to the multi-peg Tower of Hanoi problem are all equivalent. Among them the classical approaches of Stewart and Frame from 1941 can be found.
منابع مشابه
Explorations in 4-peg Tower of Hanoi
Finding an optimal solution to the 4-peg version of the classic Tower of Hanoi problem has been an open problem since the 19th century, despite the existence of a presumed-optimal solution. We verify that the presumed-optimal Frame-Stewart algorithm for 4-peg Tower of Hanoi is indeed optimal, for up to 20 discs. We also develop a distributed Tower of Hanoi algorithm, and present 2D and 3D repre...
متن کاملOn the Frame--Stewart Conjecture about the Towers of Hanoi
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...
متن کاملToward a Dynamic Programming Solution for the 4-peg Tower of Hanoi Problem with Configurations
The Frame-Stewart algorithm for the 4-peg variant of the Tower of Hanoi, introduced in 1941, partitions disks into intermediate towers before moving the remaining disks to their destination. Algorithms that partition the disks have not been proven to be optimal, although they have been verified for up to 30 disks. This paper presents a dynamic programming approach to this algorithm, using tabli...
متن کاملOn generalized Frame-Stewart numbers
For the multi-peg Tower of Hanoi problem with k ≥ 4 pegs, so far the best solution is obtained by the Stewart’s algorithm [15] based on the the following recurrence relation: Sk(n) = min 1≤t≤n � 2 · Sk(n− t) + Sk−1(t) � , S3(n) = 2 − 1. In this paper, we generalize this recurrence relation to Gk(n) = min 1≤t≤n � pk · Gk(n− t) + qk · Gk−1(t) � , G3(n) = p3 · G3(n− 1) + q3, for two sequences of a...
متن کاملCombinatorics of topmost discs of multi-peg Tower of Hanoi problem
Combinatorial properties of the multi-peg Tower of Hanoi problem on n discs and p pegs are studied. Top-maps are introduced as maps which reflect topmost discs of regular states. We study these maps from several points of view. We also count the number of edges in graphs of the multi-peg Tower of Hanoi problem and in this way obtain some combinatorial identities.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Applied Mathematics
دوره 120 شماره
صفحات -
تاریخ انتشار 2002