Sparse Ruler — open problem in combinatorics

Sparse Ruler

Wichmann's conjecture on optimal rulers. Every optimal ruler of sufficiently large length is a Wichmann ruler W(r, s) (up to reflection, i.e. reversing the segment list). Posed by Wichmann [Wi63].

Current state (tracked automatically)

Formal Conjectures marks the Lean statement wichmann_conjecture as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `wichmann_conjecture`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 solved	Formal Conjectures

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). 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-sparse-ruler,
  title        = {Sparse Ruler},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/sparse-ruler}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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.

The problem

The question

Wichmann's conjecture on optimal rulers. Every optimal ruler of sufficiently large length is a Wichmann ruler $W (r, s)$ (up to reflection, i.e. reversing the segment list). Posed by Wichmann [Wi63]. The Wikipedia article records that no optimal ruler of length $1, 13, 17, 23$ or $58$ is a Wichmann ruler, and that every other optimal length up to $213$ is attained by one; non-Wichmann optimal rulers also occur alongside Wichmann ones at lengths $9, 29, 50$ and $68$ .

A sparse ruler of length $L$ is a sequence of marks $0 = a_{1} < a_{2} < \dots < a_{m} = L$ . A distance $k \in N$ can be measured if there are $i, j \in {1, \dots, m}$ , such that $k = a_{j} - a_{i}$ .

One question concerns the structure of optimal rulers. Wichmann [Wi63] gave a parametric family of sparse rulers and speculated that every sufficiently large optimal ruler is of his type. The Wikipedia article records that no optimal ruler of length $1, 13, 17, 23$ or $58$ is a Wichmann ruler, and that every other optimal length up to $213$ is attained by one; non-Wichmann optimal rulers also occur alongside Wichmann ones at lengths $9, 29, 50$ and $68$ .

The asymptotic growth of the minimal number of marks of an optimal ruler of length $L$ — i.e. the limit of $l (n)^{2} / n$ , conjectured to lie in $[2.434 \dots, 3]$ — is the subject of FormalConjectures.ErdosProblems.«170» (there phrased via $F (N) / N$ ), and is not restated here.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.SparseRuler.

theorem wichmann_conjecture :
    ∃ N : ℕ, ∀ g : List ℕ, IsOptimal g → N < g.sum →
      ∃ r s : ℕ, g = wichmannGaps r s ∨ g = (wichmannGaps r s).reverse

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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

[Wi63] Wichmann, B. "A note on restricted difference bases." Journal of the London Mathematical Society 38 (1963): 465-466.
Wikipedia

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.