# Questions tagged [tiling]

Use this tag for questions about (not necessarily periodic) tilings of metric spaces, their combinatorial, topological and dynamical properties, as well as basic definitions and concepts.

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### Basic question with tiling tiles in a rectangle

Find the number of ways of tiling a 2 by 10 rectangle with 1 by 2 tiles which can be placed in any orientation. I’m just not sure how to approach it begin counting it
33 views

### Tiling a chessboard with triomino (L-shaped tile)

Can someone please help me to prove the following statement? Prove that it is impossible to tile an 8 × 8 chessboard missing two opposite corners using right triominoes. I suppose that the proof ...
17 views

### How to calculate the number of unique patterns in a 3x3 grid using 4 different elements

I have a 3x3 grid (9 cells) I would like to populate with 4 different elements (e.g., colors or shapes or numbers, etc.) in four of the cells. How many different unique patterns can I generate with ...
44 views

### Using 'tiling' proof technique to create combinatoric proofs about number relationships. [closed]

The number of ways of tiling a $1\times n$ rectangle with $1\times 1$ and $1\times 2$ tiles is $F_{n+1}$. (a) Use a tiling argument to give a combinatorial proof that $$F_n^2+F_{n+1}^2=F_{2n+1}\;.$$...
25 views

### Tiles, dominos, black and white, even and odd [closed]

A cell is good if the board without this cell can be tiled by dominoes ($1\times 2$ tiles). What is the number of good cells? (In other words, you want to delete one cell in such a way that the rest ...
66 views

### Tiling a rectangle with both rational and irrational sided squares

We define a 'tiling of rectangle with squares' as the process of dividing the rectangle into finitely many squares so that they do not overlap and cover up the whole rectangle. Here is my question: ...
40 views

### Why are these triangles & their circumcircles collinear upon inversion?

Consider an arbitrary logarithmic spiral of growth rate $q$ per angle $\theta$ and flair coefficient $b=\ln q/\theta$. Plot the spiral $z=e^{(b+i)\theta}$ and mark off the points that are equally ...
57 views

### What is the smallest diameter of a set of $n$ points in the plane which are all at least 2 meters apart from each other?

This question is similar to https://en.wikipedia.org/wiki/Circle_packing_in_a_circle except I am looking for the smallest diameter, i.e: I want the smallest maximum distance between the centers of the ...
60 views

### Tiling a 4xN board with L shaped 3x2 tiles [closed]

I need to find the number of the possible combinations of tilings of a 4xN board, with L shaped tiles that are 3x2. The tiles can be rotated freely. Any solution? Thanks.
32 views

### How to derive relations between the sides and angles of equilateral hyperbolic triangles

I hope everyone in this community is staying safe, well and isolated. In this unprecedented situation I am starting to learn about some non-Euclidean geometry and explore down a fractal. In the ...
40 views

### Is Cairo pentagonal tiling belong to pentagonal tilings type 8?

I am interested in Cairo pentagonal tiling. In following link of wikipedia: https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling It claims "The Cairo pentagonal tiling has two lower symmetry forms ...
46 views

### Tiling a $2\times3n$ floor with trominoes

Consider a floor of size $2\times3n$ to be tiled with trominoes of which there are two kinds: three blocks connected vertically, and 3 blocks connected in an 'L' shape. If $x_n$:= the number of ...
24 views

### In how many ways $A_n$ can we cover a $2 \times n$ rectangle with $1 \times 2$ and $2 \times 2$ polyominoes?

This is my answer: (if rotations are allowed) Let An be the number of ways to completely cover a 2 times n checkerboard with 1x2 and 2x2 dominoes There 3 conditions: The upper right corner can be ...
48 views

### How to proof that distinct “recursive geometrical constructions” converge to the same object?

It seems like a topological isomorphism problem ... I'm starting with an "intuitive isomorphism recognition" of 3 objects, which are defined by geometric construction. I'm looking for a formalization ...
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### In how many ways can a $3 \times 10$ rectangle be tiled with tiles of size $1 \times 2$? [duplicate]

In how many ways can a $3 \times 10$ rectangle be tiled with tiles of size $1 \times 2$? I know the Fibonacci Sequence can solve $2 \times n$ rectangles and $2 \times 1$ rectangles, but I'm not sure ...
38 views

### Covering a n x n square grid with minimum squares [duplicate]

Given an N x N square grid what is the minimum number of overlapping squares you need to draw such that every line of the grid is covered by the side of a square. The question can be rephrased as ...
20 views

### Inductive definition for number of ways to tile a $2$ by $n$ grid?

Inductive definition for number of way to tile a 2xn grid? I have attempted to formulate an inductive definition, thinking that every extra column we add, adds an extra n-1 ways to tile the grid. ...
1k views

### Convex polygons that do not tile the plane individually, but together they do

I am looking for two convex polygons $P,Q \subset \Bbb R^2$ such that $P$ does not tile the plane, $Q$ does not tile the plane, but if we allowed to use $P,Q$ together, then we can tile the plane. ...
29 views

### What is the complete classification of the uniform tesselations?

A uniform tesselation is a set of regular polygons in the plane that meet edge to edge, such that any two vertices are related by a symmetry of the whole figure. Regular polygons include self-...
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 ...