# Questions tagged [tiling]

A geometric packing puzzle in which a number of shapes have to be assembled into a larger shape, generally without overlaps or gaps.

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### The art of computer programming

EDIT: I know we are not supposed to edit in new requirements after first posting but as far as I understand it this requirement is implicit in all questions here: Explain your answer! At least a ...
104 views

### Any comparison between some variations of T puzzles?

I spent time to experience some variations of the classical T puzzles in here - a kind of dissection/tiling puzzle (Gardner's T, Nob's T, and Asymmetric T). They are 4-piece tangrams. They all give ...
290 views

### A chessboard tiling with corners removed in 3D

A famous problem asks whether an 8x8 chessboard with two opposite corners deleted can be tiled with dominoes, where a domino is a rectangle congruent to two adjacent squares of the board. Now, let C ...
224 views

### Double tiling congruent triangles with little else in common

When you really want to tile more than one layer but triple tiling is just too much of a good thing, surely the happy medium is double tiling. How may a mosaic of more than 900 sections be double ...
137 views

### L-tromino pair!

Amy is playing with different polyominoes. She suddenly thinks of a problem as follows. Choose two positive integers $m,n$. If we can use only L-trominos to tessellate a $m\times n$ rectangle with no ...
119 views

### Building a cube from small bricks such that no lines can be pushed through between the seams

There is a puzzle on fault-free rectangles tiled by dominoes. It is rather known (it is described in Martin Gardner’s “Mathematical puzzles and diversions”, see here) and rather old (it is known at ...
93 views

### Hand tiling - 4x inflated pentominoes with pentominoes and tetrominos

This is a manual tiling puzzle. Difficulty level always hard to estimate, but I would say a few minutes per solution for an accomplished solver, all the way to just about impossible for someone like ...
922 views

### Fillomino Tiling…how many 1's?

Suppose a 'Fillomino tiling', much like a completed Fillomino puzzle, consists of a set of polyominoes covering a region without gaps nor overlaps, with no two n-ominoes of the same size touching ...
153 views

### Sliced sudoku: rearrange the tiles and then solve it

Rearrange the tiles without rotating nor flipping them to form a $9 \times 9$ sudoku. Then solve the sudoku according to the standard rules. Credits: the sudoku is one of the twelve schemas freely ...
624 views

### Filling the plane with two colors

In this puzzle you must tile the plane with colored T-tetraminos. I will start by laying down 3 of them for you like so: Your task will be to tile the entire rest of the plane meeting the following ...
1k views

### Make a hexiamond star by hand

Using some or all of the hexiamonds (pictured), make a star. You may flip pieces. The usual tiling rules apply, no overlaps, no gaps. Use only one or none of each piece. Answer is unique. Target shape ...
637 views

### A cube build with cuboids

You are given 27 pieces of 1x2x4 cuboids. Is it possible to build a 6x6x6 cube using those 27 cuboids?
369 views

### Tile-laying for beginners

Coldport is celebrating having a new town hall by laying a tiled design in the public square in front of the building. The design is also square: $5\times 5$ tiles square, in fact, and features three ...
303 views

An Eisenstein triple is related to 60 degree triangles and a special case of the cosine law. But we need not worry about that except to note that a specific example of an Eisenstein triple is $7^2 = 5^... 3answers 2k views ### Can you put L trominos to fill the figure? In the above picture, there are 24 squares. Can you only use L trominos to fill the figure? If yes, give an example. Otherwise, please explain why. An L tromino is like this: 3answers 308 views ### Four hand tiled squares demonstrating a Pythagorean Quadruple Demonstrating the Pythagorean Quadruple$6\times6 + 6\times6 + 7\times7 = 11\times11$Using the pieces shown in the$11\times11$square: The objective: Arrange the pink pieces (four enneominoes) ... 3answers 515 views ### Tiling a square with rectangles Tile completely this 47 x 47 square with 52 rectangles. Each rectangle must contain precisely one numbered cell, and that number must be the area or perimeter of the rectangle it finds itself in. ... 1answer 304 views ### Surely they can fit? Suppose you have a grid of squares that has even dimensions, with at least one dimension greater than or equal to 4 squares, and from one corner you remove a 1x4 rectangle of those squares for ... 3answers 159 views ### Fit as many overlapping generators as possible Rimworld is a tile-based videogame. One of its constructibles in the wind generator: The wind generator itself occupies a space of 7x2 and can be placed facing the 4 cardinal directions. In order ... 1answer 457 views ### Geometry haberdasher problem - square to equilateral triangle variation Let me remind the haberdasher's problem, proposed in 1907 by the puzzle composer Henry Dudeney. Dissect an equilateral triangle to a square, with only three cuts. I would like to propose the ... 2answers 323 views ### Pentomino solution maximizing straight lines length in rectangle - wood cutter problem Recently in my free time I cut from wood with my scroll saw two pentomino sets. One set made from 10x6 pattern, and then the other set 20x3 pattern. Think of wood cutter difficulties. I would like to ... 1answer 149 views ### Max 4x1 pattern fit within 11x11 area Rimworld is a tile-based videogame. There is a constructible called a sun lamp, which provides light for indoor farming: . As you can see, the area covered by the lamp is 11 * 11, minus 1 for the ... 2answers 201 views ### The Do-It-Yourself Puzzle Can you solve this puzzle? You may (or may not) have noticed that this puzzle doesn't seem to have any pieces. How can it be solved without pieces? The short answer it can't. That's why for this ... 2answers 809 views ### The Pentomino Snake The premise of the puzzle is quite simple. Here's how to set it up. Draw a 5x5 grid of squares. Write the number 1 in the middle. Make a "snake" of numbers up to 25 so that each number is ... 1answer 1k views ### Is there an efficient algorithm for solving tiling puzzles? As an example of the type of problem, consider Stewart Coffin's Cruiser puzzle: Let R be a 48 × 31 rectangle. Let T be a 30°-60°-90° triangle with hypotenuse 34.565 (so legs are 17.2825 and 29.... 2answers 242 views ### Tiling rectangles with a Heptomino plus 2x2 square Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ... 2answers 169 views ### Tiling rectangles with Heptomino plus rectangle #7 Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ... 2answers 250 views ### Tiling rectangles with Heptomino plus rectangle #6 Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ... 2answers 169 views ### Tiling rectangles with Hexomino plus rectangle #3 Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ... 2answers 150 views ### Tiling rectangles with Heptomino plus rectangle #4 Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ... 2answers 104 views ### Tiling rectangles with Hexomino plus rectangle #2 Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ... 1answer 142 views ### Tiling rectangles with Heptomino plus rectangle #3 Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 Next ... 1answer 272 views ### Tiling rectangles with Hexomino plus rectangle #1 Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 Next ... 1answer 186 views ### Tiling a rectangle with an odd number of Y pentomoes Follow-on from Tiling a rectangle with just the Y pentomino Two questions: Find the smallest rectangle that can be tiled with an odd number of Y pentominoes, or prove it impossible Find the smallest ... 3answers 626 views ### Tiling a rectangle with just the Y pentomino Inspired by this question series, which was inspired by this question. They give rise to beautiful pictures (at least in the eye of the beholder mathematician) and some nice generalizable solutions. ... 1answer 144 views ### Tile a square with five rectangles with 10 distinct edges The baby brother of: Cutting a square into seven rectangles Tile a square with five rectangles. Select the lengths of the edges of the rectangles from the set$1$through$10$, with no length ... 2answers 256 views ### 20 right isosceles triangles into a square Similar: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Five graded difficulty isosceles right triangle into square tilings Two difficult "Seventeen right isosceles ... 2answers 150 views ### Two difficult “Seventeen right isosceles triangles into a square” tilings Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Five graded difficulty isosceles right triangle into square tilings V.hard problem, 20 right isosceles triangles ... 2answers 203 views ### Five graded difficulty isosceles right triangle into square tilings Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Two difficult "Seventeen right isosceles triangles into a square" tilings V.hard problem, 20 right ... 2answers 289 views ### Unlucky tiling: Arrange thirteen right isosceles triangles into a square Link to next puzzle in this series:Five graded difficulty isosceles right triangle into square tilings Two difficult "Seventeen right isosceles triangles into a square" tilings V.hard ... 3answers 325 views ### Tiling rectangles with X pentomino plus rectangles Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ... 2answers 259 views ### Tiling rectangles with V pentomino plus rectangles Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ... 3answers 210 views ### Tiling rectangles with U pentomino plus rectangles Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ... 2answers 255 views ### Tiling rectangles with T pentomino plus rectangles Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ... 6answers 541 views ### Tiling rectangles with N pentomino plus rectangles Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with T pentomino plus rectangles ... 3answers 619 views ### Tiling rectangles with F pentomino plus rectangles Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with N pentomino plus rectangles Tiling rectangles with T pentomino plus rectangles ... 2answers 562 views ### Tiling rectangles with W pentomino plus rectangles Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ... 3answers 576 views ### The$1 question: Tiling a triangle with trapezoids (the hard way)

Take a triangular grid consisting of 64 equilateral triangular cells in the shape of a larger triangle, and remove a single triangle at one of the tips. Can you tile this shape with 21 trapezoidal ...
We all know the classic puzzle of trying to tile a mutilated chessboard with $2\times1$ dominoes. Let's imagine that, instead of dominoes, we have $n^2$ L-shaped tetrominoes and we want to tile a \$2n\...