# Questions tagged [tiling]

For challenges that involve partitioning a space (usually the plane) into small tiles without gaps (usually using a finite set of proto-tiles). See also [set-partitions].

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### Can this polyomino tile the toroidal grid?

Inspired by certain puzzles on Flow Free: Warps. Background We all know that L-triominos can't tile the 3x3 board, and P-pentominos can't tile the 5x5 board. But the situation changes if we allow the ...
355 views

### Identify the smallest possible tile in the matrix

Challenge Given a matrix of digits (0-9), find the smallest (in terms of area) rectangular matrix of digits where one or more copies of itself, possibly rotated, can tile the original matrix. ...
2k views

### Seamless conversion from square to hexagon

For many games played on a grid, hexagons are the Clearly Superior Choice™. Unfortunately, many free game art sites only have seamless tile sets for square maps. On a past project, I used some of ...
2k views

### Draw the GKMS aperiodic tile

Chaim Goodman-Strauss, Craig Kaplan, Joseph Myers and David Smith found the following simple (both objectively and subjectively) polygon that tiles the plane, but only aperiodically: Indeed they ...
2k views

### Extending OEIS: Counting Diamond Tilings

I promise, this will be my last challenge about diamong tilings (for a while, anyway). On the bright side, this challenge doesn't have anything to do with ASCII art, and is not a code golf either, so ...
661 views

### Complete the landscape

Carcassonne is a tile-based game, where the objective is to construct Roads, Cities and Monasteries, in order to score points. The game works by players taking turns to draw and place tiles to ...
3k views

### Triangular domino tiling of an almost regular hexagon

Background An almost regular hexagon is a hexagon where all of its internal angles are 120 degrees, and pairs of the opposite sides are parallel and have equal lengths (i.e. a zonogon). The ...
2k views

### Piet (Mondrian)'s Puzzle

For more information, watch this video, and go to A276523 for a related sequence. The Mondrian Puzzle (for an integer n) is the following: Fit non-congruent ...
537 views

### Simplest Tiling of the Floor

You should write a program or function which receives a string describing the floor as input and outputs or returns the area of the simplest meta-tiling which could create the given pattern of the ...
339 views

### AoCG2021 Day 25: Stitching maps together

Part of Advent of Code Golf 2021 event. See the linked meta post for details. Related to AoC2020 Day 20, Part 1. (This day is a dreaded one for many of you, I know :P) Obligatory final "but you'...
2k views

### Game of Life, but on a 4-8-8 tiling

Background The 4-8-8 tiling looks like this: For the purpose of this challenge, we take the orientation of the tiling as exactly shown above. In plain English words, we take the tiling so that it can ...
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### Is it a checkered tiling?

Background A checkered tiling of a rectangular grid is a tiling using some polyominoes, where each region can be colored either black or white so that no two polyominoes sharing an edge has the same ...
532 views

### Counting maximal domino placements

Background A maximal domino placement (MDP) on a rectangular grid is a non-overlapping placement of zero or more dominoes, so that no more dominoes can be added without overlapping some existing ...
292 views

### The number of tilings of a grid

Setup: A block is any rectangular array of squares, specified by its dimensions $(w,h)$. A grid is any finite ordered list of blocks. For example, $\lambda = ((3,2),(3,1),(1,2))$ defines a grid. ...
307 views

### Counting polydominoes

Background A polyomino of size $n$ is a contiguous shape made from joining $n$ unit squares side by side. A domino is a size-2 polyomino. A polydomino of size $2n$ is defined as a polyomino of ...
382 views

### Domino Recurrence Generator

Challenge We once had a challenge to count domino tilings of m by n grid, and we all know that, for any fixed number of rows, the number of domino tilings by columns forms a linear recurrence. Then ...
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### Concentric rings on a snub square tiling

This challenge takes place on the snub square tiling. Start by choosing any triangle, and color it $c_1$. Next, find all tiles which touch this triangle at any vertex, and color them $c_2$. Next, ...
686 views

### Tiling a staircase with staircases

Background A staircase polyomino is a polyomino made of unit squares whose shape resembles a staircase. More formally, a staircase polyomino of size $n$ is defined as follows: A staircase polyomino ...
939 views

### How should I tile my kitchen?

I recently ordered some new and colorful tiles to replace my boring old white tiling for my kitchen. However, when the tiles arrived, they were all in a weird shape! Therefore, I need a program to ...
629 views

### Maximal saturated domino covering of a rectangle

Inspired by this OEIS entry. Background A saturated domino covering is a placement of dominoes over an area such that the dominoes are completely inside the area, the dominoes entirely cover the ...
4k views

### Is this a robbery?

Backstory You own a tiny jewellery shop in the suburbs of the city. The suburbs are too much overpopulated, so your shop has a thickness of only one character to fit in the busy streets. Recently, ...
988 views

### Generate valid Fibonacci tilings

Background The Fibonacci tiling is a tiling of the (1D) line using two segments: a short one, S, and a long one, L (their length ratio is the golden ratio, but that's not relevant to this challenge). ...
1k views

### Number of domino tilings

Write a program or function that given positive n and m calculates the number of valid distinct domino tilings you can fit in a n by m rectangle. This is sequence A099390 in the Online Encyclopedia of ...
349 views

### ASCII Exact Cover with Rectangles

Challenge Given a rectangular area arrange a group of rectangles such that they cover the rectangular area entirely. Input An integer denoting the height. An integer denoting the width. The ...
1k views

### Integers, Assemble!

Your task is to assemble the integers from 1 to N (given as input) into a rectangle of width ...
552 views

### Number of tilings on a triangular board with triangular tiles

Background Consider the shape $T(n)$ consisting of a triangular array of $\frac{n(n+1)}{2}$ unit regular hexagons: John Conway proved that $n = 12k + 0,2,9,11$ if and only if $T(n)$ can be ...
579 views

### Test a polyomino against Conway criterion

Background Conway criterion is a method to test if a given polygon can tile (i.e. cover without overlapping) an infinite plane. It states that a polygon can tile the plane if the following conditions ...
2k views

### Random ASCII Art of the Day #5: Diamond Tilings

Mash Up Time! This is instalment #5 of both my Random Golf of the Day and Optimizer's ASCII Art of the Day series. Your submission(s) in this challenge will count towards both leaderboards (which you ...
2k views

### Arranging arbitrary rectangles to fill a space

Can these rectangles fill a rectangular space? Given a bunch of rectangles, you are asked whether or not they can be arranged to fill a rectangular space. Specs Given a bunch of arbitrary ...
697 views

### Is my kids' alphabet mat properly grouped by colors?

My kids have an alphabet mat to play with, something like this: After months with the tiles of the mat randomly placed, I got tired and placed all the tiles of the mat grouped by sections according ...
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### Finite tilings in one dimension

The purpose of this challenge is to determine if a collection of one-dimensonal pieces can be tiled to form a finite continuous chunk. A piece is a non-empty, finite sequence of zeros and ones that ...
230 views

### Arranging arbitrary shapes to fill a rectangular space

A while ago, I posted a challenge asking to determine whether or not it's possible to arrange arbitrary rectangles to fill a rectangular space, here. That got answers, so clearly it was too easy. (...
609 views

### Mondrian Puzzle Sequence

Partition an n X n square into multiple non-congruent integer-sided rectangles. a(n) is the least possible difference between ...
542 views

### Let's Tessellate!

Introduction From Wikipedia: A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. A fairly well known ...
1k views

### ASCII Jigsaw Puzzle

This is a 3x3 ASCII jigsaw puzzle: ...
539 views

### Make a ASCII Hexagon Ring Tiling

Using ASCII print a section of a hexagon ring tiling. Here's a small section: ...
596 views

### Rotate a diamond tiling

Any regular hexagon can be tiled with diamonds, for instance like so (stolen from this question): ...
322 views

### Tiling by substitution

EDIT: The incorrect A rhomb substitution has been fixed. Apologies to anoyone who had started working on a solution. Consider the following substitutions, where the substituted rhomb(us) is scaled up ...
2k views

### Fibonacci domino tiling

There's classic combinatorial result that the number of ways to tile a 2*n strip by 1*2 dominoes is the nth Fibonacci number. ...
2k views

### Tiling a 2^N by 2^N Grid with L-Shaped Trominoes

When students are first taught about the proof technique of mathematical induction, a common example is the problem of tiling a 2N×2N grid with L-shaped trominoes, leaving one predetermined grid ...
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### Fill the Screen with Wang Tiles

It has been proven that the following 13 square Wang tiles always tile the plane aperiodically. This means that when the squares are arranged in a grid with all neighboring sides the same color, a ...
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### ASCII connected hexagons

Overview Given a number of hexagons, arrange them into a connected shape within the confines of a 50 by 50 ASCII art image. The shape you choose can be arbitrary - whatever you find most amenable to ...
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### 64 bit ASCII weaving

Input Two integers: A non-negative integer W in the range 0 to 2^64-1, specifying the weave. A positive integer S in the range 1 to 255, specifying the side length. These can be taken in whichever ...
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### Me Want Honeycomb

Write the shortest program that prints this ASCII art section of a hexagonal tiling or honeycomb: ...
1k views

### Scale up a Diamond Tiling

Any regular hexagon can be tiled with diamonds, for instance like so: ______ /_/_/\_\ /_/\_\/\_\ /\_\/_/\/_/\ \/_/\_\/_/\/ \_\/_/\_\/ \_\_\/_/ We'll ...
706 views

### Supersonic domino tilings

Task Write a program that reads three integers m, n either from STDIN or as command-line arguments, prints all possible tilings of a rectangle of dimensions m × n by 2 × 1 and 1 × 2 dominos and ...
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### Tile the plane with this modified circle

Take a unit circle centered on the origin. In any two neighboring quadrants, mirror the curve of the circle across the lines connecting the circle's x and y intercepts. With the resulting shape, you ...
1k views

### Tiling, given vertex configuration

Task The task is to tile polygons, given a vertex configuration. Scoring Your score is equal to the "complexity level" your submission reaches. Complexity levels are cumulative, meaning that to ...