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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24 views

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
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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 ...
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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 ...
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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 $$F_n^2+F_{n+1}^2=F_{2n+1}\;.$$...
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1answer
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 ...
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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: ...
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1answer
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 ...
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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 ...
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1answer
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.
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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 ...
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1answer
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 ...
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2answers
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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 ...
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1answer
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 ...
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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 ...
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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 ...
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1answer
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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. ...
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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. ...
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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-...
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1answer
28 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 ...
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1answer
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Does COVID-19 fit into the Caspar-Klug (Quasi-Equivalence) Theory for virus architecture?

The following is compiled largely from my "Applications of Group Theory to Virology" module I took at The University of York as an undergraduate back in 2012. The icosahedral group $I$ with identity $...
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1answer
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When two tiles meet, the edges that come together must be of the same colour. [closed]

Each 1x1 tile has a red side opposite a yellow side,and a blue side opposite a green side.An 8x8 chessboard is formed from 64 of these tiles,which may be turned around or turned over.When two tiles ...
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1answer
112 views

Different bricks making a cube

We want to build an $n \times n \times n$ cube using bricks that have integer sides and are all different. As a function of $n$, what is the maximum number of bricks we can use? For $n=1$ or $2$ it ...
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1answer
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Inductive proof showing a tiling of trominoes on a $2i \times 3j$ board with no squares missing.

Question: How to use mathematical induction to show how to tile a $2i \times 3j$ board with $L$ trominoes. The board must be tiled with no squares missing. Where I am at: What I don't understand: ...
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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 ...
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2answers
131 views

Find a generating function involving a tiling problem

Let $h_n$ be the number of ways to tile a $1 \times n$ rectangle with $1 \times 1$ tiles that are red or blue and $1 \times 2$ tiles that are green, yellow, or white. Find a closed formula for $$H(x)=\...
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1answer
37 views

Tiling a $2n-1 \times 2n-1$ board by $L$ triominos, $Z$ tetrominoes and box tetrominoes [closed]

A $2n-1 \times 2n-1$ board is going to be tiled by L triominoes, Z tetrominoes and Box tetrominoes. Prove that at least 4n − 1 L triominoes must be used. Any ideas how can I solve this? Any help will ...
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3answers
50 views

Placing $t$ horizontal dominoes in a $2 \times n$ table with some restrictions

Given positive integers $n$ and $t$ find the number of ways to place $t$ horizontal dominoes in a $2\times n$ table so that no two dominoes form a $2\times 2$ square and no $2\times 3$ rectangle ...
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Combinatorial interpretation for sum of powers (specially power=$3$) of Fibonacci numbers

Consider the following sum: $$\sum_{k=0}^{n}F_k^2=F_nF_{n+1}$$ Where $F_n$ is the nth Fibonacci number. Using a little of algebra it's easy to show that the sum telescopes to $F_nF_{n+1}$. Now ...
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1answer
34 views

dominoes in $4\times4$ square grid

is there an organized way to count the possibilities, or just trial and error. Thanks!
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1answer
69 views

Number of ways to packing $2 \times 1$ dominos into a $3 \times 6$ grid

I tried counting the different ways but couldn't figure out a consistent method, and would appreciate any insights.
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Penrose tiling may have reflection symmetry.

I tried to draw PT (Penrose tiling) starting by a small wide rhomb and going upward in hierarchy. See the following picture where I started from the small light green rhomb and proceeding one ...
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1answer
64 views

How to skew-stack tetrahedral-octahedral honeycombs?

In 1D, the densest packing of 0-sphere in a line is by apeirogon, placing their centre on the apeirogon's vertices. In 2D, the densest packing of 1-sphere in a plane is by triangular tiling, which ...
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2answers
106 views

recurrence initial conditions

I'm working on a homework assignment involving recursion and I'm having trouble finding an easy way to determine the initial conditions. Heres the problem: We want to tile ann×1 strip with tiles of ...
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1answer
43 views

Tiling a $2\times 6$ grid

Find the number of ways to tile a $2 \times 6$ grid board with the following 2 types of dominoes - (i) A $2×1$ black dominoes (ii) A $2×2$ red dominoes Here's how i though of this - It is trivial ...
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Tiling a N×M rectangle with 1×1, 1x2 and 2×1 tiles

I have read articles explaining the solution to the ways to tile a 2xM rectangle, but I'm not able to think the recursive ecuations for the general case. I tried to solve the problem initially by ...
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2answers
162 views

is it possible to divide $9\times 11$ rectangle into one $1\times 3$ rectangle and N tetrominoes?

There is a rectangle of size $9\times 11$. Is it possible to divide it using 1 tromino and $N$ tetrominoes? I have tried a lot and it seems that this is not possible, thus I want to prove it is not ...
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1answer
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Domino tilings in a specific figure, math olympiad problem

There was a reddit post a month ago in learnmath about this question: "Prove that the number of possible domino tilings in this figure is a square number" The last paragraph was the wrong try by ...
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Tiling Matrix Question

How many ways can you tile a $2\times2n$ board completely with $1\times2$ and $2\times1$ tiles? Your answer should be an explicit formula, found by deriving a matrix equation, diagonalizing the ...
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1answer
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What is the difference between an $m$-archimedean and a $k$-uniform/isogonal tiling?

According to Wikipedia and one of its sources (archived), the terms are defined as follows: An $m$-archimedean tiling is a tiling with $m$ distinct vertex figures A $k$-uniform tiling is a tiling ...
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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 ...
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What are functions on tuples called for which $f(a,b)=f(f(a),f(b))$?

Suppose $f$ is a function defined for all tuples on a set $S$, with the result also in $S$. (Something like $f: \bigcup \{S, S\times S, S \times S \times S, \cdots \} \rightarrow S$.) I will use $f(...
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2answers
269 views

Algorithm for filling a rectangle with tiles of known relative size

I checked existing questions but couldn't find the case I wanted. I have a set of tiles of known relative size. The tiles could be squares, circles, or rectangles. They all have the same aspect ...
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2answers
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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 ...
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7answers
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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 ...
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1answer
100 views

Properly drawing a Penrose tiling using the pentagrid method

As part of my work, I create tools for artists to make various types of patterns for artistic purposes. I am trying to make a tool to create a Penrose tiling and I would like to use the pentagrid ...
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2answers
64 views

Structure of inflation/deflation on Penrose Tilings?

Thinking about P2/P3 type penrose tilings (kites/darts or rhombs - for this question they should be equivalent?) we know we can "inflate"/"deflate" any tiling of the plane to get another. We also know ...
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54 views

How to calculate the quantity of different size of tiles to fit rectangle area

Let's say I have 3 type of tiles in option. (Width x Height) Tile A = 30x30 cm Tile B = 30x40 cm Tile C = 60x60 cm I want to fit rectangle area of 70x100 cm How to calculate the quantity of each ...
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39 views

List of pseudo-Catalan solids

Convex solids can have all sorts of symmetries: the platonic solids are vertex and face-transitive, meaning there is a subgroup of the rotations of 3-dimensional space which can bring any vertex onto ...
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1answer
97 views

Colorings of an $m \times n$ board with constraints

Partly inspired by this question: I'm interested in the number $A_{m, n}$ of $2$-colorings of an $m \times n$ grid, say red and blue, such that no two (vertically or horizontally) adjacent squares are ...

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