Trimer Covers in the Triangular Grid — Problem 7

Definitions and mathematical statement

Let $D$ denote the set of admissible parameter pairs for benzels in the cited source. For $(a,b)\in D$, write $B_{a,b}$ for the $(a,b)$-benzel, $I(B_{a,b})$ for its Conway--Lagarias invariant with the source's sign convention, and $\mathcal T_{103}(B_{a,b})$ for the set of its tilings of type $103$, using the source's tile and placement conventions. These symbols are placeholders for source definitions: the supplied record does not specify $D$, the geometry of a benzel, the definition or normalization of the invariant, or the precise meaning of type $103$.

The question attributed to Propp (2022), Problem 7 (#3), is whether \[ \forall (a,b)\in D,\qquad I(B_{a,b})\geq 0 \ \Longrightarrow\ \mathcal T_{103}(B_{a,b})\neq\varnothing. \] The equality case $I(B_{a,b})=0$ is explicitly included.

The supplied background states that a strictly negative Conway--Lagarias invariant prevents a benzel from being tiled by bones and right-pointing stones, forcing the corresponding table entries to vanish. It reports observed positive entries when the invariant is nonnegative, while explicitly withholding a general justification. The table and the observations preceding Problem 4 are not supplied. The context associates these tile restrictions with the question about type $103$, but the exact type encoding and allowed tile orientations are absent. Consequently, the displayed implication records the intended question subject to unresolved source definitions; it is not a fully specified geometric formulation.

Short English statement

Propp asks whether every admissible benzel with nonnegative Conway--Lagarias invariant has a type $103$ tiling, including when the invariant is zero. The supplied record reports supporting table observations, but omits the definitions needed to specify the regions, invariant, and permitted tilings precisely.

Sources

• Local UnsolvedMath registry: permanent ID $800351$, exact source problem code catalogue entry, title Trimer Covers in the Triangular Grid --- Problem 7. The supplied record attributes the problem to Propp (2022) and labels it open; this is supplied registry status, not an independently verified literature assessment.
• Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems (2022), source item Problem 7 (#3). Supplied source URL: https://arxiv.org/abs/2206.06472. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access or verification.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The excerpt refers to observations preceding Problem 4 and to a table, neither of which is included in the supplied record.