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?
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.
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 2026-09-28}
} Also: CITATION.cff · Atom feed of results
- Claims
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- Verified
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- Disputed
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- Refuted
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- On the literature board
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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). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem mosers_worm_problem :
IsGLB {v | ∃ X ∈ WormCovers, volume X = v} answer(sorry)theorem convex_mosers_worm_problem :
IsGLB {v | ∃ X ∈ WormCovers, Convex ℝ X ∧ volume X = v} answer(sorry)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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.