Abstract
In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs $H_p^n$ in a more general setting. Here $p$ stands for the number of pegs and $n$ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length $p-1$ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all $n\geqslant p(p-2)$, of optimal paths where $p-1$ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.
Highlights
The general task of the Tower of Hanoi (TH) puzzle reads as follows
In contrast to the widespread interest in the Frame-Stewart Conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs Hpn in a more general setting
Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results
Summary
The general task of the Tower of Hanoi (TH) puzzle reads as follows. There are p 3 pegs, n 1 discs and two regular distributions of the discs, i.e. no disc lies on a smaller one, on the pegs. For p = 4 we need n = 4 discs for an example where the bound in Theorem 1.1 is sharp, namely a task whose bitcode is larger than 3: let s = 0233 and t = 3001, the shortest paths using 1, 2 or 3 LDMs are unique, respectively, and all have length 6; the LDM code is 111 ∼= 7. It is natural to ask whether for all p there are shortest paths with p − 1 LDMs, i.e. tasks with bitcodes greater than or equal to 2p−2, and if so, for which numbers of discs the electronic journal of combinatorics 21(4) (2014), #P4.38 n; as we have seen, for p = 3 this is so for all n 2. For p > 3 we will first summarize some theoretical results and approach these questions by computer experiments
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