Existence of benzel tilings with nonnegative Conway–Lagarias invariant (Propp's Problem 7)

Let $T_{103} (a,b)$ denote the number of tilings of the $(a,b)$ -benzel using right-pointing stones and all three orientations of bones, with left-pointing stones forbidden.

Let $CL(a,b)$ be the Conway–Lagarias invariant of the $(a,b)$ -benzel.

Since a $103$ -tiling contains no left-pointing stones, a necessary condition for such a tiling to exist is

CL(a,b)\ge 0.

Problem (Propp's Problem 7). Is this condition also sufficient? In other words, is

CL(a,b)\ge 0 \quad\Longrightarrow\quad T_{103}(a,b)>0

for every valid $(a,b)$ -benzel?

References

James Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems. https://www.samuelfhopkins.com/OPAC/files/proceedings/propp.pdf

James Propp, Benzel Tilings (problem-status page). https://faculty.uml.edu/jpropp/benzels.html

Jesse Kim and James Propp, A Pentagonal Number Theorem for Tribone Tilings, Electronic Journal of Combinatorics 30(3) (2023), P3.26. https://arxiv.org/abs/2206.04223

Colin Defant, Leigh Foster, Rupert Li, James Propp, and Benjamin Young, Tilings of Benzels via Generalized Compression, SIAM Journal on Discrete Mathematics 39(1) (2025), 146–162. https://arxiv.org/abs/2403.07663

Alex Chengyu Li, Finite-Defect Path Models for Peripheral Benzel Tilings: Propp's Problem 6 and the First Defect Diagonal (2026). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7344859

Posted by Akshat Sharma Posted 3 weeks ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed solved

A reader-submitted manuscript claims to prove the conjecture for every valid benzel, but no independent source has verified the argument.

James Propp posed the question of whether nonnegative Conway–Lagarias invariant guarantees a $103$ -tiling, namely $CL(a,b)\ge 0\Longrightarrow T_{103}(a,b)>0$ . The published record still lists this problem as open.

Known results

Kim and Propp (2022) reported computational evidence for $T_{103}(a,b)>0$ when $CL(a,b)>0$ or $CL(a,b)=0$ , without proving sufficiency.
Defant, Foster, Li, Propp, and Young (2025) left Problem 7 open and reported further computations consistent with the conjecture.

Community submission (unverified), September 8, 2026

A submitted manuscript claims a constructive proof of the remaining case $a+b\equiv 0\pmod{3}$ with $CL(a,b)>0$ , using base cases, first-layer extensions, and layer lifting. It therefore claims the full implication, but the proof is unverified.

Current status (as of September 2026): the implication remains unverified; a community submission claims to prove it for all valid $(a,b)$ , while the published record still treats Problem 7 as open.

Sources

ProofI prove Propp's Problem 7 for benzel tilings: every valid

(a,b)

-benzel with nonnegative Conway--Lagarias invariant admits a tiling using right-pointing stones and all three orientations of bones. Equivalently,

CL(a,b)\ge 0

implies

T_{103}(a,b)>0

. The proof is constructive and covers the previously unresolved positive-invariant class.See full solutionHide full solution

A complete proof is provided in the attached manuscript.

The argument is constructive. The cases $a+b\equiv 1 \pmod 3$ and $a+b\equiv 2 \pmod 3$ are already known, and the case where the Conway--Lagarias invariant is zero is also known. Thus it remains to treat the case $a+b\equiv 0 \pmod 3$ with strictly positive Conway--Lagarias invariant.

For every admissible pair $(a,b)$ in this remaining region, we give an explicit recursive construction of a valid $103$ -tiling. The construction is established through a sequence of lemmas covering the base cases, the first-layer extension, and the general layer-lifting step.

Together these results prove that every valid $(a,b)$ -benzel with nonnegative Conway--Lagarias invariant admits a $103$ -tiling, resolving Propp's Problem 7.

propp_problem_7_solution.pdf507,657 bytesOpen