Pith Number

pith:MY4V5WNA

pith:2026:MY4V5WNAB4PNGDRS4JRYKRUI3V

not attested not anchored not stored refs resolved

The Weighted Tower of Hanoi: Algebraic Structure, Phase Transitions, and Integer Sequences

Andreas M. Hinza, El-Mehdi Mehiri

The weighted Tower of Hanoi admits explicit closed forms for minimal transfer costs from matrix formulations of one-LDM and two-LDM strategies.

arxiv:2605.16956 v1 · 2026-05-16 · math.CO · cs.DM

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{MY4V5WNAB4PNGDRS4JRYKRUI3V}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

Starting from a general optimality recurrence with one-LDM and two-LDM strategies, the authors derive complete matrix formulations and obtain explicit closed forms for the minimal transfer cost, with the one-LDM dynamics governed by a linear operator whose spectral decomposition connects to the Jacobsthal and Lichtenberg sequences.

C2weakest assumption

The assumption that the optimal solution is always achieved by one of the two competing strategies (one largest-disc move or two largest-disc moves) in the recurrence relation, without other hybrid or more complex move sequences becoming cheaper for arbitrary nonnegative symmetric weights.

C3one line summary

The authors give matrix-based closed forms for minimal costs in weighted Tower of Hanoi problems, connect them to Jacobsthal and other integer sequences, and identify a phase transition in optimal move strategies when some moves are forbidden.

References

11 extracted · 11 resolved · 0 Pith anchors

[1] Aumann, S., G¨ otz, K. A. M., Hinz, A. M., & Petr, C. (2014). The number of moves of the largest disc in shortest paths on Hanoi graphs.The Electronic Journal of Combinatorics, 21(4), Article P4.38.ht 2014 · doi:10.37236/4252

[2] Wildberger and Dean Rubine 2025 · doi:10.1080/00029890.2025.2491975

[3] Hinz, A. M. (1992). Shortest paths between regular states of the Tower of Hanoi.Information Sciences, 63(1–2), 173–181.https://doi.org/10.1016/0020-0255(92)90067-I 18 1992 · doi:10.1016/0020-0255(92)90067-i

[4] Hinz, A. M. (2017). The Lichtenberg sequence.The Fibonacci Quarterly, 55(1), 2–12.https: //doi.org/10.1080/00150517.2017.12427786 2017 · doi:10.1080/00150517.2017.12427786

[5] M., Klavˇ zar, S., & Petr, C 2018 · doi:10.1007/978-3-319-73779-9

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:32.808286Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

66395ed9a00f1ed30e32e263854688dd418b9e5f81d2954bb5dbfeb1a15955c8

Aliases

arxiv: 2605.16956 · arxiv_version: 2605.16956v1 · doi: 10.48550/arxiv.2605.16956 · pith_short_12: MY4V5WNAB4PN · pith_short_16: MY4V5WNAB4PNGDRS · pith_short_8: MY4V5WNA

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MY4V5WNAB4PNGDRS4JRYKRUI3V \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 66395ed9a00f1ed30e32e263854688dd418b9e5f81d2954bb5dbfeb1a15955c8

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "db1df0b4a888e967d2177ea51a279ac6d27536041dea678ebe6c83af2dd7ab89",
    "cross_cats_sorted": [
      "cs.DM"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-16T12:16:44Z",
    "title_canon_sha256": "281a73f3630040c8d8d37e71bf33f2b6762fc57e0a5cd7e4d1cc5aaa9520c72e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16956",
    "kind": "arxiv",
    "version": 1
  }
}