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

TL;DR

This work studies Troublesome Sticker: the question of whether a stain $Q$ (a polyomino) is always flatly coverable by non-overlapping copies of a single sticker $P$ under translations, rotations, and reflections. It delivers a complete classification: no polyomino with $|Q|\ge 7$ is always-coverable, while a finite family $\mathcal{J}$ consists of always-coverable shapes (with pentomino Y and T explicitly proven), and a finite $\mathcal{I}$ contains not always-coverable shapes; an $O(1)$-time decision procedure decides the property for any $Q$. The paper also establishes $\mathrm{NP}$-hardness of Flat Cover in 2D via a gadget-based reduction from 3-Precoloring Extension and $\mathrm{NP}$-completeness in 1D via a reduction from X3C using a Golomb-ruler-based template, with correctness arguments tying covers to exact colorings or set covers. Together, these results yield both a practical decision tool for always-coverable stains and strong complexity barriers for the general covering problem, plus open questions about broader domains and constrained variants.

Abstract

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 $325 \times 325$.

Figures (41)

• Figure 1: Pentomino I ($1 \times 5$) shaped stain
• Figure 2: The smallest known polyomino ($33 \times 33$) that never covers pentomino I
• Figure 3: Not always-coverable polyominoes $\mathcal{I}$ and always-coverable polyominoes $\mathcal{J}$
• Figure 4: Pentomino "Y"
• Figure 5: Pentomino "T"

Theorems & Definitions (29)

• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Theorem 3.4
• Corollary 3.5
• Remark 3.6
• proof
• Lemma 3.7
• proof
• Lemma 3.8