Akira Iino | Searcharxiv

arXiv2026

Planar lattices and equilateral odd-gons

For a planar integral lattice $L$, let $ν(L)$ denote the square-free part of the integer $D(L)^2$, where $D(L)$ stands for the area of a fundamental parallelogram of $L$. For each odd integer $n$ with $3 \leq n<29$, a planar lattice $L$ contains an equilateral $n$-gon if and only if $L$ is similar to an integral lattice $L'$ such that $ν(L')\equiv 3 \pmod 4$ and the largest prime factor $p$ of $ν(L')$ satisfies $p \leq n$. Moreover, such $L$ contains a convex equilateral $n$-gon, which answers a problem posed by Maehara.

math.CO↗

arXiv2026

Covering a Polyomino-Shaped Stain with Non-Overlapping Identical Stickers

You find a stain on the wall and decide to cover it with non-overlapping stickers of a single identical shape (rotation and reflection are allowed). Is it possible to find a sticker shape that fails to cover the stain? In this paper, we consider this problem under polyomino constraints and complete the classification of always-coverable stain shapes (polyominoes). We provide proofs for the maximal always-coverable polyominoes and construct concrete counterexamples for the minimal not always-coverable ones, demonstrating that such cases exist even among hole-free polyominoes. This classification consequently yields an algorithm to determine the always-coverability of any given stain. We also show that the problem of determining whether a given sticker can cover a given stain is $\NP$-complete, even though exact cover is not demanded. This result extends to the 1D case where the connectivity requirement is removed. As an illustration of the problem complexity, for a specific hexomino (6-cell) stain, the smallest sticker found in our search that avoids covering it has, although not proven minimum, a bounding box of $325 \times 325$.

cs.DS↗