Trimer Covers in the Triangular Grid — Problem 15

Definitions and mathematical statement

Let $D=\{n\in{\mathbb{Z}}:n\geq 5\}$, equipped with the topology induced by the $2$-adic metric. Write formally \[ a(n)=T_{113}(n,2n-4),\qquad n\in D. \] The supplied record does not define $T_{113}$. In particular, it does not specify the objects counted, the meaning of the subscript $113$, the parameters, or the boundary conditions. Thus this notation cannot be given a complete combinatorial definition from the supplied evidence alone.

Assuming that the intended quantities $a(n)$ are integers, as the source's congruence observations presuppose, the question is whether $a:D\to{\mathbb{Z}}$ is continuous for the induced $2$-adic topologies. Explicitly, ordinary continuity on this domain asks whether \[ \forall n\in D\;\forall r\in{\mathbb{Z}}_{\geq 1}\;\exists s\in{\mathbb{Z}}_{\geq 0}\;\forall m\in D,\qquad m\equiv n\pmod{2^s}\ \Longrightarrow\ a(m)\equiv a(n)\pmod{2^r}. \] Here $s$ may depend on both $n$ and $r$. The source does not specify whether it instead intends uniform continuity, in which $s$ must depend only on $r$, or a continuous extension to the $2$-adic integers. These stronger interpretations are not imposed in the displayed formulation.

The supplied comment reports apparent constancy modulo $2$ and modulo $4$, apparent $2$-periodicity modulo $8$, and apparent $8$-periodicity modulo $16$, with repeating pattern \[ 4,\ 4,\ 4,\ 4,\ 4,\ 12,\ 4,\ 12. \] These are reported observations, not established hypotheses or conclusions. The record does not explicitly identify the starting index of the displayed pattern. It states that $n=4$ breaks the pattern; that index is excluded from the question. The registry labels the problem open, but that status has not been independently verified.

Short English statement

Propp asks whether the sequence $T_{113}(n,2n-4)$, for integers $n\geq 5$, is continuous in the $2$-adic topology. The congruence formulation above uses ordinary continuity. A complete statement still requires the missing definition of $T_{113}$ and confirmation of the intended continuity convention. The reported modular patterns are exploratory observations.

Sources

• Local UnsolvedMath registry, permanent ID $800355$, exact source problem code catalogue entry, title "Trimer Covers in the Triangular Grid --- Problem 15." The supplied record attributes the problem to Propp (2022) and labels it open. Its recorded upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems (2022), source item "Problem 15 (#7)": https://arxiv.org/abs/2206.06472. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• OEIS entry A364416, cited in the supplied problem comment: https://oeis.org/A364416. Its contents have not been independently inspected for this draft.