Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [polyomino]

A polyomino is an edge-connected union of grid-aligned squares in the plane; in some contexts, they may be viewed as subsets of Z^2. This tag is for questions about the properties of polyominoes, including questions about how they tile different shapes, how they may be dissected, and assembly puzzles with a given set of polyominoes.

93 questions
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Can any number of squares sum to a square?

Suppose $$a^2 = \sum_{i=1}^k b_i^2$$ where $a, b_i \in \mathbb{Z}$, $a>0, b_i > 0$ (and $b_i$ are not necessarily distinct). Can any positive integer be the value of $k$? The reason I am interested in this: in a irreptile tiling where the…
Herman Tulleken
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Are tetromino packings always 4-colorable?

Suppose that I have a $2n \times 2n$ grid which has had $n^2$ tetrominoes packed into it, as in the following illustration: Can I always assign the numbers 1,2,3,4 to the squares of the grid so that (1) each tetromino contains each number exactly…
Gregory J. Puleo
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What is the smallest polyomino that can't surround a $1\times 1$ hole?

Given a polyomino $P$, we can ask if it is possible for disjoint copies of $P$ to surround a single cell in the square grid - i.e., for the complement of their union to have a connected component of size $1$. We can further refine this into…
RavenclawPrefect
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Are there polyominoes that can't tile the plane, but scaled copies can?

I'm wondering where there is a finite set $\mathcal{T}$ of polyominoes that are pairwise similar that can tile the plane, but a single element from the set cannot. (All orientations are allowed.) To show what I mean, here is a tiling by two similar…
Herman Tulleken
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21
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1 answer

How densely can the :...: polyomino fill the plane?

This is a follow-up to the question How good can a "near-miss" polyomino packing be?. Let $P$ be the heptomino shown below: I am interested in the packing density of $P$ on the square grid. (Unlike all polyominoes on $6$ or fewer cells, $P$ does…
RavenclawPrefect
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21
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Smallest region that can contain all free $n$-ominoes.

A nine-cell region is the smallest subset of the plane that can contain all twelve free pentominoes, as illustrated below. (A free polyomino is one that can be rotated and flipped.) A twelve-cell region is the smallest subset of the plane the can…
Peter Kagey
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Does the $Z$ pentomino tile a 3D box?

Some polyominoes are rectifiable, meaning they can tile some rectangle in the plane. For instance, the following tiling shows the $Y$-pentomino is rectifiable:                                                                             On the other…
RavenclawPrefect
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16
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Union of two disjoint congruent polygons is centrally symmetric. Must the polygons differ by a 180 degree rotation?

Let $P$ be a polygon with $180^\circ$ rotational symmetry. Let $O$ be the center of $P$ and suppose $P$ is dissected into congruent polygons $A$ and $B$. Must the $180^\circ$ rotation around $O$ send $A$ to $B$? This conundrum appears to have…
greenturtle3141
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Is every "even" polyomino with one hole tileable by dominoes?

In Conformal Invariance of Domino Tiling the author defines an even polyomino as a polyomino with all corners (of all borders, inside and outside) "black" if the polyomino is colored with the checkerboard coloring. A corner is black if the interior…
Herman Tulleken
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14
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Random domino tilings: Is this distribution well-defined, and how can it be sampled from?

I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means. My first instinct was to do something like "the center of a random tiling of a large…
RavenclawPrefect
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14
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Are there polyominoes that tile L-shapes and not rectangles? (Except L-shapes)

An L-shape is a polyomino with 6 vertices (5 convex, 1 concave). I am investigating polyominoes that can tile some L-shape. Two non-square or three square rectangles can be put together to make an L-shape. So any polyomino that can tile a…
Herman Tulleken
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14
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3 answers

How to classify polyominoes by shape

I am trying to find a robust way to classify and distinguish polyominoes. I would like to write a simple algorithm that could identify similar free polyominoes (under translation, rotation, reflection or glide reflection), given a set. Since I am…
Tropilio
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Finding all 15-ominoes that tile the plane and have distinct internal adjacencies

Problem Description: This problem oddly came up in Minecraft with some friends. Not sure what the best terms are; but that's partly why I'm here. So a polyomino is built up from squares. This problem is to find 15-ominoes (pentdecominoes?) that…
WhiteStoneJazz
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Tiling of a $9\times 7$ rectangle

Can a rectangle $9\times 7$ be tiled by "L-blocks" (an L-block consists of $3$ unit squares)? Although the problem seems to be easy, coloring didn't help me. The general theory is interesting, but I'm looking for an elementary and relatively simple…
Ievgen
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Is there only one way to dissect a $p\times p$ square into $p$ congruent $p$-ominoes, for prime $p$?

The posted is posted at MO, it is here. The problem is as this. Suppose there is a $p\times p$ square grid, and we tile it by a set of $p$ congruent $p$-ominoes (a $p$-omino being a connected piece of $p$ squares). We allow rotation and reflection.…
JetfiRex
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