# 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.

## 14 Citations

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

- Computer ScienceArXiv
- 2013

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

- MathematicsIEICE Trans. Inf. Syst.
- 2011

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

- Computer ScienceIEICE Trans. Inf. Syst.
- 2019

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

- Mathematics
- 2021

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

- Computer Science
- 2013

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.

### 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

- Computer Science
- 2004

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

- Computer Science
- 2005

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.

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