Trimer Covers in the Triangular Grid — Problem 1

Definitions and mathematical statement

Write \[ a_k=\frac{k(3k-1)}{2},\qquad b_k=\frac{k(3k+1)}{2}. \] The source asks for an exact formula for the tiling count \[ T_{003}(a_k,b_k) =T_{003}\!\left(\frac{k(3k-1)}{2},\frac{k(3k+1)}{2}\right). \] The surrounding text places this question in the context of no-stones tilings of benzels. However, the supplied record does not define the benzel region, the permitted tiles, the restrictions encoded by the subscript $003$, or the admissible values of $k$. Consequently, the displayed expression preserves the source's target, but a precise counting problem and its parameter domain require those missing definitions; no range or universal quantifier for $k$ is imposed here.

The source also identifies as worthwhile any method for determining these tiling counts that is more efficient than brute-force enumeration, for example a recurrence relation, even without an exact formula. The record supplies no computational model or precise efficiency criterion.

Short English statement

Propp asks for an exact formula for the displayed family of tiling counts. A method more efficient than brute-force enumeration, such as a recurrence, would also be of interest. The supplied excerpt leaves the tiling model and the range of $k$ undefined, so the statement requires completion from its source.

Sources

• Local UnsolvedMath registry, permanent ID $800349$, source problem code catalogue entry, entitled Trimer Covers in the Triangular Grid --- Problem 1. The supplied record attributes the problem to Propp, with proposed year 2022, and labels it open; this status is supplied metadata, not an independently verified assessment.
• Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems (2022), Problem 1 (#1). Supplied source URL: https://arxiv.org/abs/2206.06472. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of access or verification for this draft.
• The source excerpt refers to [KiPr] for further discussion of no-stones tilings of benzels. No bibliographic identification or URL for [KiPr] is supplied.