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# 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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### A rectangular room has a floor tiled with tiles of two shapes: 1×4 and 2×2

A rectangular room has a floor tiled with tiles of two shapes: 1×4 and 2×2. The tiles completely cover the floor of the room, and no tile has been damaged, or cut in half. One day, a heavy object is ...
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:
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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 288 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 3k views ### Tiling with T-tetrominos in gravity The goal is to tile all the white squares using T-tetrominos when there is gravity pulling the tetrominos downwards like regular tetris. The black squares are void and the ground is just below the ... 1answer 25k views ### Is this Tetris puzzle solvable? As a birthday present last year, I received some fridge magnets. They didn't come as a puzzle, so I don't know if they have a solution, but I made a puzzle out of them anyway. The magnets are ... 3answers 487 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. ... 3answers 843 views ### Near-fill with 3x1 long triominos, how to do a different void square than the center square? It's rather easy to fill a$7 \times 7$board with 16 long triominos, leaving the center square void: see the picture below. But if I want to move the void square in another position, where else could ... 1answer 296 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 147 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 330 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 266 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 89 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 179 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 690 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 ... 3answers 5k views ### Tiling a Chessboard with tetrominos Is it possible to tile a$10\times10$chessboard with (non-overlapping) T-tetrominos? If so, how? If not, prove it's impossible. Bonus: Which Tetris pieces can used to tile a 10$\times$10 ... 2answers 149 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 ... 3answers 502 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 527 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 ... 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 210 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 154 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 240 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 136 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 245 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 ... 1answer 242 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 348 views ### Hard tiling puzzle Your goal is to make two squares of the same size from a set of rectangles. Each of the rectangles has an aspect ratio of 1:2. Select two sets of rectangles from the list: ... 1answer 132 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 ... 2answers 94 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 ... 6answers 469 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 ... 1answer 180 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 ... 2answers 234 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 ... 3answers 1k views ### Fitting rectangles into square (optimal/perfect rectangle packing) I gave the puzzle you can see on the image below to a friend of mine for christmas last year. I thought it would be fun to dump it out in front of him so he would not know the solution. Unfortunately ... 3answers 553 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. ... 2answers 143 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 198 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 231 views ### Challenging 15 rectangle tiling problem This will test you, a computer will definitely help. Just one set of$1:2$aspect ratio rectangles this time, but$15$of them. Short side only is listed. The challenge is to arrange them into a ... 3answers 196 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 231 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 ... 3answers 294 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 ... 1answer 139 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 273 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 542 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 ...
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### Slightly bigger 1:2 hand tiling, getting a little harder

Pretty sure an expert hand tiler would be able to find this. Please post a spoiler only if you hand tile it... Tile a $60\times60$ square with $1:2$ rectangles of sizes ...
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### Two similar hand tiling puzzles

Make a square from each of these lists of aspect ratio $1:2$ rectangles 1, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17 ...
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### Trickier hand tiling

This should be a little trickier. Using $20$ rectangles of aspect ratio $1:2$, make two squares. Use sizes 2,3,7,12,16 in both squares. And split this list among ...