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.
103 questions
0 votes
0 answers
63 views
How to find all polyominoes that are intersections of the integer lattice with an open disk?
Given a positive integer $n$, how to find all $n$-ominoes that are intersections of the integer lattice with an open disk? Denote $I=\left\{ t\in\mathbb{R}\colon0\le t<1\right\} $ and $\mathbb{R}_{...
- 1,551
-1 votes
1 answer
81 views
What is the minimum number of line segments required to split a rectangle into the 12 free pentominoes?
Let $R_{m,n}$ be an $m$x$n$ rectangle where $m$ is the width and $n$ is the length, and $R_{m,n}$ is subdivided into a grid of unit squares. A line segment is a horizontal or vertical segment drawn ...
- 115
0 votes
1 answer
53 views
What's the minimum number of sets in which you can tile all asymmetric n-ominoes into a non-trivial symmetric shape?
For clarification, s sets of all asymmetric n-ominoes means s copies of each of the k asymmetric n-ominoes. Non-trivial means that for $n = 5$, you can’t take the F, L, P, N, Y pentominoes and do this ...
- 115
4 votes
1 answer
84 views
What is the smallest rectangle tileable by copies of each n-omino that can tile a rectangle?
If the wording of the question was a little confusing, here is the main idea: Some $n$-ominoes cannot tile rectangles. We are excluding those. Excluding those $n$-ominoes not tiling any rectangle, ...
- 115
1 vote
1 answer
54 views
How many distinct free polyforms can be made using n triangles and n squares attached at edges?
For $n = 1$ and $n = 2$ the amounts are 1 and 15 respectively. For $n ≥ 3$, it's harder to do by hand, and is also hard to build a script due to some complexities, such as this shape -ST TSS T-- which ...
- 115
3 votes
1 answer
44 views
What's the minimum number of sets of n-ominoes needed for it to able to be tiled into a rectangle?
For any $n$, what's the minimum number of sets needed for all free $n$-ominoes to be able to be tiled in a rectangle? For $n = 1$ and $n = 2$ the answer is 1 as there is only one monomino and domino. ...
- 115
6 votes
1 answer
117 views
Are all non-rectangular order-N polyominoes reachable from each other by valid move sequences?
Are all non-rectangular polyominoes of order $N > 3$ reachable from each other by finite sequences of valid moves of this type? To perform a single move, choose some $1\times K$ subblock (a ...
- 1,551
5 votes
2 answers
291 views
Snake polyominoes placing
I came up with this idea. Let's draw an $n\times n$ grid of squares and then try to place as many "snakes" polyominoes as possible. The objective is to place $k$ polyominoes of resp. lengths ...
- 89
0 votes
0 answers
101 views
Counting problem for V-tromino pattern
Can someone give some insight on how to write down the sum function $S(m)$, given this pattern/table: Notation wise, $n$ shows the size of the square size of the $n \cdot n$ grid, $m$ indicates the ...
- 91
-1 votes
1 answer
59 views
Prove/disprove that it is possible to tile a $4\times (4k+2)$ boards by using T-shaped tetrominoes [closed]
The tiling allows for any rotation of T-shaped tetrominoes, as long as all gaps are completely filled and there is no overlapping. I believe the answer is no (that a tiling isn't possible), but I ...
- 1
2 votes
1 answer
255 views
Do regions which can be tiled with trominoes using either 1 or 2 L-trominoes necessarily have a hole?
It seems that the original question is not clear enough. I don't agree with this, but ok, here is a more formal writing. Let P be a polyomino of area 3n. Say that P is of type (k,n-k) if it is ...
- 49
1 vote
0 answers
95 views
Find the 'closest' polycube that tiles the space
Does the polycube of size 81 composed of a $5 \times 5 \times 5$ cube with its edges removed tile the space? If not, what is the minimal change (i.e the addition of extra cubes) we need to make so ...
- 121
1 vote
0 answers
75 views
Lattice polyomino tiling without rotations and reflections of a tile
Consider a lattice $\mathbb{Z}^2$, picture it as an infinite board of cells. If we are given some connected shape(a set of cells), what are the necessary and sufficient conditions for it to be ...
- 103
4 votes
3 answers
166 views
smallest polyomino that has n squares with 1 neighbour, n squares with 2 neighbours, n squares with 3 neighbours, and n squares with 4 neighbours
I have no idea what to call this problem, so I'm gonna call it the neighbouring problem sense I couldn't find any other math problem with the same name when I searched it up. but first I need to ...
- 145
1 vote
0 answers
76 views
How to prove an estimation about polyominoes?
Let's call a polyomino consisting of at least $n$ cells a cluster. We say that a cluster is solid if it cannot be split into two or more clusters. How can I find the maximum possible number of cells ...
- 129