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.
Questions tagged [polyomino]
72 questions
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votes
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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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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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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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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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15
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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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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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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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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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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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If a subset of the square grid can be tiled by $1\times n$ rectangles for every $n$, can it be tiled by infinite rays?
Suppose that we have some set $S$ of grid-aligned squares in the plane; equivalently, we can think of our set as $S\subset \mathbb{Z}^2$. Suppose that for every positive integer $n$, $S$ can be tiled by disjoint copies of a $1\times n$ rectangle…
RavenclawPrefect
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10
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The minimum $N \times N$ square covering problem for $1 \times 4$ shaped tetrominoes
OK, so me and my friend are working on a problem in which you do the opposite to trying to stuff as many of the I shape tetrominoes in a square as possible. Trying to find the smallest number of I shape tetrominoes that you have to place in the…
Niks Topics
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Can all convex $3n$-iamonds be tiled by $3$-iamonds?
Background
A polyiamond is a plane figure constructed by joining together equilateral triangles of the same size along their edges.
The number of convex polyiamonds is given by A096004.
Based on enumerating examples up to $n=8$, it looks plausible…
Peter Kagey
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How good can a "near-miss" polyomino packing be?
Given a polyomino $P$ with $n$ cells, we can ask about its maximal packing density $\delta_P$ in the plane (perhaps the limsup if we are concerned about convergence issues, though I don't think this actually comes up).
If $\delta_P=1$, then $P$…
RavenclawPrefect
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