# 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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### 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 ...
2answers
65 views

### 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 ...
2answers
95 views

### 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 ...
0answers
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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 ...
2answers
153 views

### 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: ...
1answer
247 views

### 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. ...
0answers
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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 ...
1answer
117 views

### 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 ...
1answer
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0answers
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### Is there a devious starting position to this tile puzzle?

I recently asked a question on the puzzling site, where I placed three colored T-tetraminos on the plane and asked for a tiling of the plane with T-tetraminos that fulfilled the following three ...
0answers
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1answer
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### Can a $10\times 10$ square be entirely covered by 25 $T$-shape bricks?

Let $ABCD$ be a square in which length of a side is $10$ meters. Suppose that we have $T$-shape brick which consists of $4$ smaller squares in which a side of each smaller square has length of $1$ ...
1answer
228 views

### Find a great strategy to a pentomino type game

I have a game. Given an $8\times 8$ square and a set, which contains the pentominoes and four $1\times 1$ squares. Players alternately pick one item from the set. Then players (starting with the ...
1answer
218 views

### 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 ...
2answers
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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 ...
0answers
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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 ...
1answer
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### Polyominoes with the most reflex exterior angles

What is the polyomino with the largest number $n$ of reflex (i.e., $270^\circ$) exterior angles that can fit in a $L \times L$ grid? How does $n$ scale with $L$?
2answers
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### Generalization of pentomino-rectangle tiling

It is very well known that there are $12$ pentominos and they can tile $6 \times 10$, $5 \times 12$, $4 \times 15$ and $3 \times 20$ rectangles. Now, let's define a function for simplify this. t(n)=\...
2answers
871 views

### Placing tetrominos in square, maximum size

I am currently coding an algorithm which places a list of Tetrominos (tetris pieces) in the smallest square possible. My question is : is there a mathematical way to know the maximum size (upper ...
1answer
498 views

### Tiling squares with L-Trominoes

Is there a simple proof that any square besides a 3x3 square with area divisible by 3 is tileable with L-trominos?