# Questions tagged [polyomino]

The tag has no usage guidance.

28 questions
Filter by
Sorted by
Tagged with
18 views

### Proving T-tetrominoes fit in a chessboard

I'm fairly new to discrete math, and I wasn't sure how to prove the following. Prove that if $n\geq 2$, then every $2^n \times 2^n$ chessboard can be tiled with non overlapping T-tiles. If I draw ...
46 views

### Counting polysticks on the $n$-cube.

Over at Code Golf Stack Exchange, I put up a challenge asking people to count, among other things, the number of ways to take an $n$-cube and color $k$ (connected) edges up to isometries of the $n$-...
28 views

34 views

### dominoes in $4\times4$ square grid

is there an organized way to count the possibilities, or just trial and error. Thanks!
34 views

### Closed Figures Made Up of Unit Squares

This is my first question. I am wondering if there is a formula to find the number of non-congruent figures of area n that are made up of 1 by 1 non-overlapping squares, in which any square must ...
141 views

### 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 ...
101 views

### Minimal covering of a $5 \times 5$ square with $T$ shaped tetrominoes.

The title almost completely describes the problem. So you have a $5 \times5$ square. You have to fit the minimum number of $T$ shaped tetrominoes such that no more $T$ shaped tetrominoes can be fitted ...
125 views

### 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 ...
4k views

### 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 ...
300 views

### 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 ...
24 views

### Using dominating sets to solve a tetrominoe covering problem

So this question is a consequence of The minimum $N \times N$ square covering problem for $1 \times 4$ shaped tetrominoes where Rob Pratt used dominating sets coupled with an integer linear ...
187 views

### 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 ...
33 views

### Why do figures that can be tiled with two different polyominoes tend to be symmetric?

Two polyominoes are compatible when there is a figure that can be tiled by both; a smallest one is called a least common multiple. In looking through tables of these least common multiples of ...
44 views

### How many colors are necessary for a W-polyomino to never cover a color more than once?

A W-polyomino is a polyomino with 2 cells in each row (except possibly the last, which may have one cell), and each row offset once cell to the right. Below are the first few W polyominoes. How many ...
61 views

### Percolation related counting problem

I was trying to look into the following problem, which I intend to use for a lemma for a bigger problem. The question is: For the 2-dimensional integer lattice, what are some good lower and upper ...
64 views

### Finding all rectangles with fault-free tilings of the P-pentomino

I am trying to find all rectangles with fault-free tilings of the P pentomino. (A fault is a vertical or horizontal line inside the rectangle that is not crossed by any tiles; a fault-free tiling is ...
79 views

### Are all polyominoes with even sides tileable by dominoes?

In this paper (Section 8) the author states that it is "trivial" to show that a polyomino with all sides (including the sides of holes in the polyomino) even has a tiling by dominoes. It is indeed ...