Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Geometry P-moser-worm

Moser's Worm

Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?

Current state (tracked automatically)

  • Best known bounds for constant 13a: 0.239 ≤ C ≤ 0.2617993878. Upper bound ([PW2021], 30° circular sector of radius 1 (“Wetzel’s sector”)); lower bound ([K2026], Theorem 1.1), not yet independently verified.
  • The best independently verified lower bound is 0.232239 ([KPS2013], Min–max estimate for convex hull area of certain forced configurations).
All tracked facts and sources
FactValueSource
Formal Conjectures: research statements in the file2 open, 4 solvedFormal Conjectures
Best lower bound with an explicit value (constant 13a)0.239*
[K2026] — Theorem 1.1.
Tao's optimization constants
Best lower bound with an explicit value, independently verified (constant 13a)0.232239
[KPS2013] — Min–max estimate for convex hull area of certain forced configurations
Tao's optimization constants
Best upper bound with an explicit value (constant 13a)0.2617993878
[PW2021] — 30° circular sector of radius 1 (“Wetzel’s sector”).
Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12), Tao's optimization constants (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

Try it with your chatbot Submit a claim Follow (RSS)
Cite 
@misc{cairn-moser-worm,
  title        = {Moser's Worm},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/moser-worm}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Fmoser-worm.json)](https://cairn-commons.com/problems/moser-worm)
Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

or , ,

ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

mosers_worm_problem. Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?

convex_mosers_worm_problem. Convex Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a convex shape that can cover every unit-length curve?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.MoserWorm (2 statements).

theorem mosers_worm_problem :
    IsGLB {v | ∃ X ∈ WormCovers, volume X = v} mosers_worm_problem.answer
theorem convex_mosers_worm_problem :
    convexWormCoverConstant = convex_mosers_worm_problem.answer

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.