Simple Explicit Formulas for the Frame-Stewart Numbers

@article{Klavar2002SimpleEF,
  title={Simple Explicit Formulas for the Frame-Stewart Numbers},
  author={Sandi Klav{\vz}ar and Uro{\vs} Milutinovi{\'c}},
  journal={Annals of Combinatorics},
  year={2002},
  volume={6},
  pages={157-167}
}
Abstract. Several different approaches to the multi-peg Tower of Hanoi problem are equivalent. One of them is Stewart's recursive formula ¶¶$ S (n, p) = min \{2S (n_1, p) + S (n-n_1, p-1)\mid n_1, n-n_1 \in \mathbb{Z}^+\}. $¶¶In the present paper we significantly simplify the explicit calculation of the Frame-Stewart's numbers S(n, p) and give a short proof of the domain theorem that describes the set of all pairs (n, n1), such that the above minima are achieved at n1. 

Toward a Dynamic Programming Solution for the 4-peg Tower of Hanoi Problem with Configurations

A comparison of different partitioning locations for the Frame-Stewart algorithm indicates that, although certain partitions are optimal for the classic problem, they need to be modified for certain configurations, and that random configurations might require an entirely new algorithm.

Analysis of Recurrence Relations Generalized from the 4-Peg Tower of Hanoi

Recurrence relations generalized from the Tower of Hanoi problem of the form T(n,α,β)=min1≤t≤n{αT(n-t, α,β)+βS(t,3)}, are analyzed, and it is shown that when α and β are natural numbers, the sequence of differences of T (n,β)-T (n-1,β) consists of numbers of the increasing order.

The Explicit Formula of the Presumed Optimal Recurrence Relation for the Star Tower of Hanoi

The explicit formula of the recurrence relation for the Tower of Hanoi on the star graph with four vertices is shown, where the perfect tower of disks on a leaf vertex is transferred to the central vertex.

On a family of triangular arrays of natural numbers and the Tower of Hanoi with four pegs

Several arrangements of the natural numbers N in triangular arrays ( Tj(n, k) ) j∈N, where n ∈ N denotes the row and k ∈ N the column, will be considered. These arrays are defined as follows: The

The Tower of Hanoi - Myths and Maths

This is the first comprehensive monograph on the mathematical theory of the solitaire game The Tower of Hanoi which was invented in the 19th century by the French number theorist douard Lucas and contains a thorough, largely self-contained presentation of the essential mathematical facts with complete proofs.

Hanoi graphs and some classical numbers

Appendix A Hints and Solutions to Exercises

  • Mathematics
  • 2012
Remark A.1. While it is by an easy induction proof that one can show that there is only one mapping F ∈ NN0 0 which fulfills the recurrence (0.4) and we proved the existence by deducing Binet’s

Explorations in 4-peg Tower of Hanoi Ben

It is verified that the presumed-optimal Frame-Stewart algorithm for 4-peg Tower of Hanoi is indeed optimal, and a distributed Tower ofHanoi algorithm is developed, and 2D and 3D representations of the state transition graphs are presented.

The Tower of Hanoi: A Bibliography

In this edition, an attempt has been made to include every relevant document published during the first 100 years of the Tower of Hanoi’s history, from 1883 through 1983.

References

SHOWING 1-10 OF 24 REFERENCES

Generalized multi-peg tower of hanoi problem

  • A. Majumdar
  • Computer Science
    The Journal of the Australian Mathematical Society. Series B. Applied Mathematics
  • 1996
The Dynamic Programming technique has been employed to find the optimality equation satisfied by M(n, p), which is the presumed minimum number of moves required to transfer the tower of n disks from the source peg to the destination peg.

The generalized four-peg tower of hanoi problem

The generalized ρ-peg Tower of Hanoi problem with ρ  4 is considered in this paper. Denoting by M(n, p) the minimum number of (legal) moves required to transfer the tower of n( 1) discs from the

On the Frame-Stewart algorithm for the multi-peg Tower of Hanoi problem

In How Many Steps the k Peg Version of the Towers of Hanoi Game Can Be Solved?

In this we paper we consider the version of the classical Towers of Hanoi games where the game-board contains more than three pegs. For k pegs we give a 2Ckn1/(k-2) lower bound on the number of steps

Error-correcting codes on the towers of Hanoi graphs

The Canterbury Puzzles and other Curious Problem

  • Education
    Nature
  • 1908
THE author of this little book is a well-known expert in the invention and solution of puzzles. Those which he presents to the reader are in the main entirely original; those which are not so are

The average distance on the Sierpiński gasket

SummaryThe canonical distance of points on the Sierpiński gasket is considered and its expectation deduced. The solution is surprising, both for the value and for the method derived from an analysis

Solution to advanced problem 3918

  • Amer. Math. Monthly
  • 1941

Solution to advanced problem 3918

  • Amer. Math. Monthly
  • 1941

The Canterbury puzzles : and other curious problems