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.

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Can we form a rectangle with integral lengths using an odd number of copies of this domino?

Question: This figure is made up of 6 unit cells. Can we form a rectangle with integral lengths using an odd number of copies of this domino? Rotating and flipping of the figure is allowed. This ...
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How many Katamino solutions are there on a $5 \times 12$ board?

Katamino is the puzzle of placing twelve polygonal pieces so as to form a $5\times 12$ rectangular array. The pieces consist of all possible arrangements of five connected $1\times 1$ squares. For ...
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All the solutions to tile an 8 by 8 square with tetrominoes

I want to fully tile an 8x8 board using the 19 fixed Tetrominoes, allowing repeats of Tetrominoes. For example, this is a valid solution: Here are the 19 fixed Tetrominoes: I am looking for a list ...
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Coloring a polyomino tiling so that no two pieces with the same color have a common point

How many colors are enough to color all polyomino tilings so that no two adjacent or touching polyominoes have the same color? In the following example 6 colors are required (each region has a common ...
134 views

How many domino tilings of a $2n\times 2n$ board are uniquely 3-colourable?

The closed-form formula for the number of domino tilings of a $2n\times 2n$ board is known to be $$\prod_{j=1}^{n}\prod_{k=1}^{n}\Big{(}4\cos^2\frac{\pi j}{2n+1} + 4\cos^2\frac{\pi k}{2n+1}\Big{)}.$$ (...
81 views

Largest number of corner pieces in an $m \times n$ grid?

The other day the following combinatorics problem popped into my head: Given an $m \times n$ grid, how many corner pieces can fit in it without overlapping? A corner piece is defined as such: ...
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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 $O$ ...
166 views

A game from fault lines in domino tilings

It is known that, for any tiling of a $6\times6$ rectangle with dominoes, there must exist a fault line, or a line cutting the square without cutting any domino. (There is a nice elementary proof of ...
1 vote
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Recurrence relation convex polyominoes of perimeter $2n+8$

This question concerns Exercise 4.24 from generatingfunctionology. Find a three term recurrence relation, whose coefficients are polynomials in $n$, that is satisfied by  f(n) = (2n+11)4^n - 4(2n+1) ...
1 vote
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Name of polyomino using shapes other than a square [closed]

Is there a name to describe polyominoes that use shapes other than a square? Wolfram alpha could not answer this.
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Tile homotopy and T-tetromino packing of rectangles

From my old question (Which rectangles can be tiled with L-trominos, when only two orientations are allowed?), I learned a very interesting way to deal with tiling problems. I was wondering about T-...
86 views

Which pentacube oddities can be solved or improved?

A few years ago I studied pentacube oddities. A pentacube is a polycube with 5 cells, and an oddity is an arrangement of an odd number of copies of a polyform that has binary symmetry (or stronger). ...
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Which rectangles can be tiled with triangles $T_n$, when only two orientations are allowed?

This question is a generalization of another question asked here: Which rectangles can be tiled with L-trominos, when only two orientations are allowed? A triangle $T_n$ is a polyomino with columns on ...
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A recursive relation for the number of ways to tile a 2 x n grid with 2x1, 1x2, 1x1 and 2x2 dominos

I'm trying to solve this problem: In how many ways can you cover a 2xn grid with 1x1, 1x2, 2x1, 2x2 dominos? And here is my attempt: Let a(n) be the number of ways we can cover the grid. Then if we ...
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1 vote
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Area of math dealing with sliding tiles?

I am not sure if this may be better suited for Computer Science, but I have recently gotten interested in problems having to do with different shapes of tiles sliding around in a grid. For example, ...
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Tiling of a grid using triominoes

I want to tile a $12 \times 12$ grid using L-shaped triominoes. There must be no overlaps or missing spaces, and I know that it is possible to do so. Now, I want to know about a new condition: each ...
461 views

Which rectangles can be tiled with L-trominos, when only two orientations are allowed?

This is a question that I got after reading this: https://www.cut-the-knot.org/Curriculum/Games/LminoRect.shtml. (This link already gave me the same result as theorem 1.1 of the article https://www....
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Are maximum-density fixed polyomino packings always isohedral?

Consider, for a polyomino $P$ made from $n$ unit squares joined at the edges, the arrangements of non-overlapping translations of $P$. Sometimes we can cover the infinite plane with such translations ...
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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:                                          ...
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1 vote
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Prove that any polyomino of size n > 1 and perimeter p can be built by adding 1 square to some other polyomino of size n-1 and perimeter p or p-2.

It is well known that any polyomino of size n > 1 can be built by adding a square to a smaller one. Can the same thing be proven if we add the criterion that said smaller polyomino has a perimeter ...
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Is there a polyhedron all of whose faces are distinct polyominoes?

The title is pretty much the complete question. An example of this would (I believe) have to be a nonconvex polycube. If there is one, I am also interested in the smallest example. The analogous ...
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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 ...
93 views

Can a scaled $L$-tromino be cut into two congruent polyominoes?

The $L$-tromino can trivially be cut into two congruent trapezoidal pieces: It can also be trivially cut into three squares, and into four other $L$-trominoes of half the side length. I am curious ...
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1 vote
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Can the following figures be divided into 6 equal parts?

This question asks whether a figure can be divided into $2$ and $3$ equal parts, but not $6$. It is in turn based off of an earlier puzzling.SE question. One natural approach is to consider the case ...
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How to find winning Strategy for 4 celled animals of Harary's generalized tic tac toe

A polyomino is a structure made of unit squares joined along their sides. A single square is called a monomino. Two make a domino. Three join in two different ways to make two trominoes. Let's call ...
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Finding the largest polyominoes that can fit in a rectangular space

I am making a program that generates $3$ random polyominoes of size $x \le n$ (referring to the number of squares in the shape). Each polyomino fits within a space: $k \times k$ ($k=6$ in my case). I ...
261 views

Determine whether a polyomino has a hole

Suppose I know the $(x,y)$ coordinates of the corners of all the unit squares making up a connected polyomino. Is there a simple/elegant way to determine if the polyomino has a hole in it, merely by ...
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1 vote
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Are there polyominoes which tile half-plane but tile no strip with any width?

In Golomb's hierarchy: If a polyomino tiles strip then tiles half-plane. (Ok, it's trivial.) But what is with other direction? Is there an example which tiles half-plane but doesn't tile any strip?
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What is the minimum size of a region that can be tiled by every polyomino on up to $4$ cells?

I am interested in regions that can be tiled by all $k$-ominoes for each $k\le n$. If we take $n=3$, it is obvious that the $2\times 3$ rectangle is the minimal region that can be tiled by the ...
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Can the $X$ pentomino and the $2\times 2$ square mutually tile any nontrivial cofinite region?

It is easy to see that there is no nonempty finite region in the square grid that can be tiled by both the $X$ pentomino and the $2\times 2$ square: if we look at any cell of maximal $y$-coordinate, a ...
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Can every polyomino of even size be tiled by $L$-trominoes when scaled up by a factor of $3$?

The $L$-tromino does not tile a $3\times 3$ square. However, it can tile $3\times3$ squares glued together in various ways:                                                                     I am ...
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How many fixed polyominoes does it take to force an aperiodic tiling of the plane?

A longstanding open problem asks whether there is a single connected tile that only tiles the plane aperiodically. As far as I know, this is still open even in the case of polyominoes: we do not know ...
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Smallest non-space-filling polycube?

The title nearly says it all: what is the fewest number of cubes that can be fused face-to-face into a polyhedron that does not fill space? The smallest that seemed like a sure non-tiler to me was 9: ...
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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. ...
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Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb

Drake Thomas and I have proposed a sequence A343909 to the On-Line Encyclopedia of Integer Sequences (OEIS), which counts "generalized polyforms": generalizations of free polyominoes (Tetris ...
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Are the only polyominoes that can tile triangles right trominoes?

A triangle $T(n)$ is a polyomino with columns on the same base with lengths $1, 2, 3, \cdots, n$. In this slide deck (original PPT, images may be corrupt), Friedman looks at tilings of these triangles ...
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