The Tower of Hanoi: Optimality Proofs, Multi-Peg Bounds, and Computational Frontiers

TL;DR

This paper revisits the Tower of Hanoi problem, providing modern proofs for classical results, analyzing multi-peg bounds, and highlighting the limitations of common heuristics through updated data and visualizations.

Contribution

It offers modern invariant-based proofs for three-peg and four-peg cases, updates optimality data for multiple pegs, and reframes open problems as heuristic robustness tests.

Findings

Balanced split k = floor(n/2) is optimal for n <= 8

Optimal cost exceeds the balanced split at n=9 and grows significantly by n=20

Heuristic methods should be tested for robustness rather than assumed optimal

Abstract

The Tower of Hanoi continues to provide a surprisingly rich meeting point for recursive reasoning, combinatorial geometry, and computational verification. Motivated by the editorial standards of the Bulletin of the Australian Mathematical Society, we revisit the classical three-peg problem through Sierpinski-style self-similarity, bring Stockmeyer's uniqueness argument into a modern invariant-based framework, and then pivot to four pegs via the Frame-Stewart strategy and Bousch's optimality proof. The heart of this note is a cautionary data-and-proof cycle: the balanced split k = floor(n/2) is indeed optimal for n <= 8, but our corrected tables show that it already exceeds the optimal cost by 20% at n = 9, crosses the 1.5 mark at n = 13, and comes close to quadrupling the optimum by n = 20. We complement this diagnosis with a subtower-independence lemma, a reproducible table for n <= 15, three publication-ready TikZ figures (recursion arrow, four-peg state diagram, and multi-peg growth curves), and a bibliography exceeding thirty sources that foreground Bulletin and Gazette contributions. The concluding section reframes the open problems as robustness tests for heuristics rather than premature theorems.