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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.

234
votes
1answer
24k 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 ...
44
votes
3answers
2k 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 ...
37
votes
6answers
3k views

One rectangle, indivisible

Here is a rectangle made out of 2x1 dominoes: It can be divided along a line into two smaller rectangles: What is the smallest (in area) rectangle of (edit) multiple 2x1 dominoes that cannot be ...
29
votes
2answers
936 views

Rebuilding the Rio 2016 Olympics logo

This puzzle belongs to the puzzle series: hyper-modern art It is also an entry to the 13th fortnightly challenge You two friends are still venturing through the endless halls of the gallery of hyper-...
20
votes
4answers
3k views

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 ...
20
votes
4answers
3k views

Mosaic with tetris blocks

Create the pattern shown in the picture below using the set of standard tetris blocks. This is a rectangular arrangement of 6×5 squares where the first and third squares have been removed from the ...
20
votes
9answers
2k views

Occupy a field with tetrominos

You have a square 8x8 board, and tetrominoes of all possible shapes. Each individual square on a tetromino is the same size as one square on the board (i.e. any given tetromino will cover exactly 4 ...
19
votes
4answers
3k views

Mutilated chessboard

Remove the square in the top-left corner of a $2015 \times2015$ chessboard. Can the remaining mutilated chessboard be tiled with $1\times4$ and $4\times1$ rectangles?
18
votes
9answers
3k views

Tiling a Hexagon with Diamonds

A regular hexagon is divided into a triangular grid, and completely tiled with diamonds (two triangles glued together). Diamonds can be placed in one of three orientations. Prove that, no matter how ...
18
votes
4answers
3k views

The Rectangle Puzzle

A solution to the Rectangle Puzzle of size n is an arrangement of n rectangles into a larger rectangle, such that no smaller rectangle is formed by outlining 2 or more of the placed pieces. For ...
14
votes
5answers
2k views

Find a heptagon with mirror symmetry that can tile a flat plane

A seven-sided flat shape of fixed size in which all angles are equal and all sides of the same length, called a regular heptagon, cannot tile a flat plane. The only regular shapes that can are the ...
14
votes
2answers
345 views

Hexominoes into 7 simultaneous congruent shapes

I came up with this puzzle 16 years ago, it was on Ed Pegg's Mathpuzzle site but nobody solved it AFAIK. The 35 hexominoes (which look like this): are to be arranged, in groups of five, into seven ...
14
votes
2answers
501 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 ...
13
votes
3answers
532 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 ...
13
votes
2answers
2k views

Placing 2x1 dominoes on a chessboard with two corners removed

Suppose you have a checkerboard with two opposite corner squares removed, like this: Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
13
votes
1answer
665 views

Symmetrical hexiamond figure problem

Objective is to make as much symmetrical figures as plausible using one of each L (red/pink), E(green/lime), H(blue) hexiomond. You can rotate, translate, flip them as you please. My work so far.. 2 ...
12
votes
2answers
1k views

How to ship the new Slurm 7-pack efficiently

The six-pack is a thing of the past. Beverages of the future will use the seven-pack format. But how will the mighty spacemen of the future manage to ship the Slurm seven-pack efficiently in ...
12
votes
2answers
779 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 ...
12
votes
2answers
251 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 ...
11
votes
3answers
348 views

Tiling a hexagonal chessboard with “tribones”

A tribone is a tile made of three hexagons in a line. A hexagonal chessboard is a hexagonal grid of 91 cells in the shape of a larger hexagon. When 30 tribones are placed on a hexagonal chessboard ...
11
votes
2answers
341 views

Combinatorial Agriculture

You have just acquired a $64$ acre farm, in the shape of a square, and divided into an eight by eight array of one acre subplots. You have $21$ crops to plant. Each crop requires its own $3$ acre plot ...
11
votes
2answers
247 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 ...
10
votes
1answer
961 views

Hexomino Puzzle

First, draw out a 10x10 grid. Take the shape below and see how many you can fit in the 10x10 grid. It should take up 6 grid squares. Also, it can be rotated. 1) How many can you fit in the grid? 2) ...
10
votes
2answers
611 views

Occupy a field with your choice of tetromino

You have an 8x8 board, and you must partially cover it with a single kind of tetromino in such a way that it is impossible to place any additional tetrominoes of that kind in the empty spaces. ...
10
votes
3answers
670 views

The Erasmus isosceles triangle

Professor Erasmus has constructed a special isosceles triangle that he modestly calls the "Professor-Erasmus-triangle". The professor claims that he can cut his triangle into three smaller triangles, ...
10
votes
1answer
153 views

Combinatorial Agriculture, part 2

Your 64 acre square farm has been doing well and you decide to expand with an orchard. The orchard is also in the form of an 8-by-8 square of subplots, each of which can hold one tree. You have 3 each ...
9
votes
2answers
658 views

Tiling by trapezoids

An equilateral triangle with sidelength $L$ can be tiled by trapezoids with sidelengths $2,1,1,1$. What are the possible values for $L$?
9
votes
3answers
524 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. ...
9
votes
2answers
338 views

Cover the terrace with “slashed” tiles

Peter has a rectangular terrace with integer sides. He plans to tile this terrace, with the ultra-modern 'slashed tiles". Each tile is a $2\times 1$ rectangle, with a diagonal, called its "slash". A ...
9
votes
2answers
239 views

Jigsaw Logic: Opheodrys Vernalis

Challenge: Given unlimited copies of each of the above type of puzzle piece, create a $9\times 9$ square which features a picture of snake. Rules: There must be only one snake, with a head at one ...
8
votes
2answers
671 views

Tiling a diamond-shaped grid with tetrominoes

You have a grid like this: (The entire grid isn't shown as it would be too large, but the number of squares in each row are as follows: $2, 4, 6, \ldots, 96, 98, 100, 100, 98, 96, \ldots, 6, 4, 2$.) ...
8
votes
2answers
662 views

Dominoes on a chessboard

Mary has a box with special $2\times1$ dominoes. Each dominoe has two red corners and two blue corners, and these dominoes come in two different types: The first type has the lower left and the upper ...
8
votes
2answers
323 views

Covering a chessboard with L-tetrominoes

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\...
8
votes
1answer
376 views

Cutting a 7-by-9 rectangle

Is it possible to dissect a $7\times9$ rectangle into $21$ pieces that are $L$-shaped and that consist of three little squares?
8
votes
2answers
220 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 ...
8
votes
1answer
644 views

Jepetto's punctured chessboard [duplicate]

Jepetto the toymaker was thinking about a new toy to add to his tiling product line. His new design involved a punctured chessboard: an ordinary $8 \times 8$ chessboard, except with a single square ...
8
votes
3answers
199 views

Hand tiling puzzle

Here's a set of polyominoes (sizes 4,5,6,7,8,9,10) that you can print and cut out. You can use the two smallest to make a 3x3. Nice easy one to get you started. Add a piece to make a 3x5. Add another ...
7
votes
3answers
4k 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 ...
7
votes
6answers
449 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 ...
7
votes
4answers
285 views

Tiling an Odd Polygon with Dominoes

There is a polygon whose edge lengths are all odd integers. Prove that this polygon's interior cannot be tiled by dominoes whose dimensions are $1\times 2$. An example of such a polygon is a "...
7
votes
3answers
475 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 ...
7
votes
1answer
276 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 ...
7
votes
2answers
171 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 ...
7
votes
1answer
156 views

Reassembling the Marquetry II: The Coffee Table Strikes Back

After your last fateful encounter with a workbench, you decide to high-tail it out of the woodworker's home. You've gained a new sense of respect for marquetry from your last visit, finding the task ...
7
votes
2answers
250 views

Dissect a square into 3:2 non-congruent integer-sided rectangles

(Similar to the recent 3:1 rectangle question) Tile a square completely with rectangles which have aspect ratio 3:2, integral side lengths and all different sizes. In other words selected from 2x3, ...
6
votes
3answers
273 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 ...
6
votes
1answer
422 views

2 of each Tetris Puzzle

In Is this Tetris puzzle solvable? we established that it is not possible to form a rectangle with an uneven number of each Tetris piece. But if we had a solution for 2 of each piece, we would now ...
6
votes
2answers
238 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 ...
6
votes
1answer
264 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 ...
6
votes
1answer
169 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 ...