Trimer Covers in the Triangular Grid — Problem 4

Definitions and mathematical statement

Let $D$ denote the parameter domain intended in the cited source, let $B_{a,b}$ denote its $(a,b)$-benzel for $(a,b)\in D$, and let $I(a,b)$ denote its Conway--Lagarias invariant, with the sign convention used there. Let $\mathcal T_{013}(B_{a,b})$ denote the set of tilings called type $013$ in the source, described in the supplied statement as tilings by bones and left stones.

The intended question is whether \[ \forall (a,b)\in D,\qquad I(a,b)\leq 0\ \Longrightarrow\ \mathcal T_{013}(B_{a,b})\neq\varnothing. \] The equality case $I(a,b)=0$ is explicitly included.

The supplied statement also asserts the obstruction \[ I(a,b)>0\ \Longrightarrow\ \mathcal T_{013}(B_{a,b})=\varnothing, \] and reports positive entries in an accompanying table whenever the invariant is nonpositive. Those observations are supplied motivation, not a proof of the proposed implication; the table itself is not supplied.

Unresolved qualifications. The supplied record does not specify $D$, the geometric definition and boundary conventions of the benzel, the permitted tile shapes and orientations, the precise type $013$ convention, or the definition and sign normalization of the invariant. The notation above therefore expresses the intended implication conditionally on those source definitions; it is not a self-contained geometric formulation.

Short English statement

Propp asks whether every admissible $(a,b)$-benzel with nonpositive Conway--Lagarias invariant admits a type $013$ tiling, described as a tiling by bones and left stones. The supplied registry labels the question open. Essential geometric definitions and parameter restrictions are missing from the supplied excerpt.

Sources

• Local UnsolvedMath registry, permanent ID $800350$, source code catalogue entry, title "Trimer Covers in the Triangular Grid --- Problem 4." The supplied record attributes the problem to Propp (2022) and labels it open; this status is supplied metadata, not an independently verified literature assessment.
• Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems (2022), source item "Problem 4 (#2)." Supplied reference: https://arxiv.org/abs/2206.06472. The provenance records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance details, not a claim of independent consultation.