Trimer Covers in the Triangular Grid — Problem 17

Definitions and mathematical statement

Let $D=\{n\in{\mathbb{Z}}:n\geq 5\}$, equipped with the topology induced by the $2$-adic metric. For nonzero $a\in{\mathbb{Z}}$, let $v_2(a)$ be the largest integer $r\geq 0$ such that $2^r\mid a$; put $v_2(0)=+\infty$ and $d_2(a,b)=2^{-v_2(a-b)}$.

The supplied record does not define $T_{113}(a,b;3)$, its underlying objects, or its counting conventions. Subject to supplying that definition and establishing that the displayed values are integers, set \[ f(n)=T_{113}(n,2n-4;3),\qquad n\in D. \] The question is whether $f$, with the $2$-adic topology on its values, is continuous. Explicitly, is the following assertion true? \[ \forall n\in D\;\forall k\in{\mathbb{Z}}_{\geq 1}\;\exists \ell\in{\mathbb{Z}}_{\geq 0}\;\forall m\in D,\qquad m\equiv n\pmod{2^\ell} \ \Longrightarrow\ T_{113}(m,2m-4;3)\equiv T_{113}(n,2n-4;3)\pmod{2^k}. \] Here $\ell$ may depend on both $n$ and $k$. This expresses continuity on the stated domain; the record does not specify a stronger uniform continuity or extension requirement.

The source comment reports apparent constancy modulo $2$, $4$, and $8$, and apparent $8$-periodicity modulo $16$, with repeating pattern \[ 4,\ 4,\ 12,\ 12,\ 4,\ 12,\ 12,\ 4. \] These are supplied observations, not established assertions in this draft. The starting index of the displayed pattern is not explicitly specified. The comment says that $n=4$ breaks the pattern; that index is excluded from $D$.

Short English statement

Propp asks whether the values $T_{113}(n,2n-4;3)$, for integers $n\geq 5$, vary continuously in the $2$-adic topology. At each fixed index, sufficiently close indices must give values congruent modulo any prescribed power of $2$. The supplied record omits the definition of $T_{113}$, so this formulation remains conditional on recovering that notation.

Sources

• Local UnsolvedMath registry, permanent ID $800357$, source code catalogue entry, title "Trimer Covers in the Triangular Grid --- Problem 17," in the AMR Open Problem Lists collection. The supplied registry attributes the problem to Propp (2022) and labels it open; that status is supplied metadata, not independently verified here.
• Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems (2022), source item "Problem 17 (#9)." 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 access or verification for this draft.
• OEIS entry A364418, cited in the supplied source comment: https://oeis.org/A364418. Its contents were not supplied or independently checked for this draft.