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

Qi Junyi

TL;DR

The paper revisits the multi-peg Tower of Hanoi problem by combining Sierpiński-style self-similarity, invariant-based proofs, and computational verification to scrutinize Frame–Stewart strategies. It shows that the commonly taught balanced split $k=\lfloor n/2\rfloor$ is optimal only up to eight discs and then increasingly suboptimal, with empirical ratios $\rho(n)$ climbing to at least $3.9$ by $n=20$; this is demonstrated via a memoised solver, along with Bousch’s four-peg optimality framework and related invariants. The work provides reproducible data, three publication-ready TikZ figures, and a detailed pipeline that links recursive theory to practical heuristic robustness, ultimately reframing open problems as questions of heuristic robustness rather than premature theorems. It also highlights connections to graph-theoretic structures, Gray codes, and applications in logistics and computational search, illustrating how a classical puzzle can inform modern algorithmic thinking and pedagogy.

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.

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

TL;DR

The paper revisits the multi-peg Tower of Hanoi problem by combining Sierpiński-style self-similarity, invariant-based proofs, and computational verification to scrutinize Frame–Stewart strategies. It shows that the commonly taught balanced split is optimal only up to eight discs and then increasingly suboptimal, with empirical ratios climbing to at least by ; this is demonstrated via a memoised solver, along with Bousch’s four-peg optimality framework and related invariants. The work provides reproducible data, three publication-ready TikZ figures, and a detailed pipeline that links recursive theory to practical heuristic robustness, ultimately reframing open problems as questions of heuristic robustness rather than premature theorems. It also highlights connections to graph-theoretic structures, Gray codes, and applications in logistics and computational search, illustrating how a classical puzzle can inform modern algorithmic thinking and pedagogy.

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.

Paper Structure

This paper contains 23 sections, 3 theorems, 5 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and historiography
  2. Notation and conventions
  3. Recursive warm-up and Sierpiński geometry
  4. Structural invariants and uniqueness
  5. Induction in slow motion
  6. Lower bounds via parity, Gray codes, and energy
  7. Asymptotics for multiple pegs
  8. Catalogue of invariants
  9. From Frame--Stewart to Bousch
  10. Balanced Frame--Stewart bounds
  11. Sensitivity of split choices
  12. Computational data
  13. Extended numerical trends
  14. Experimental pipeline and validation
  15. Graph-theoretic insight
  16. ...and 8 more sections

Key Result

Lemma 1

Once the largest disc in a legal four-peg solution has been moved, the remaining $n-1$ discs evolve as two independent Tower-of-Hanoi subproblems, each respecting the standard rules on a disjoint subset of pegs.

Figures (3)

  • Figure 1: Sierpiński arrow depiction of the three-peg recursion.
  • Figure 2: Macro-level state diagram for the four-peg recursion.
  • Figure 3: Base-$2$ logarithmic growth of $T_p(n)$ for three to five pegs.

Theorems & Definitions (6)

  • Lemma 1: Subtower independence
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof