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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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 ...
11
votes
3answers
361 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 ...
13
votes
3answers
835 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 ...
3
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4answers
435 views

Square Room Tiles

The square floor of a square room was covered with colored 1 square unit tiles.The covering tiles were designed to have different (solid) colored tiles that forms rectangular areas but none of these ...
5
votes
1answer
187 views

Cover 3*2.6 rectangle with the least number convex polygons cut from sheets of given size 1.2*2.4

I want to cover a rectangle of given dimensions ($3\times2.6$) with a minimal number of convex polygons cut from rectangular sheets of material of given size ($1.2\times2.4$). These polygons don't ...
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 ...
9
votes
2answers
339 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 ...
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 ...
7
votes
1answer
164 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 ...
5
votes
2answers
324 views

Fill up a tetris field where bordering tiles have different colors

Is it possible to completely fill the infinite tetris field $\mathbb Z^2$ with tetrominoes such that no tetromino borders another one of the same type? Assume that two tetromiones border each other ...
30
votes
2answers
959 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-...
4
votes
2answers
435 views

Puzzle that consists of all possible combinations of pieces containing 5 squares

When I was a child my father gave me a wooden puzzle that consisted of pieces that represented every possible combination of 5 squares. The goal was to arrange the pieces in various rectangular ...
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 ...
13
votes
1answer
668 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 ...
10
votes
1answer
154 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 ...
5
votes
1answer
206 views

Professor Halfbrain and the wonderful rectangles

Professor Halfbrain calls a rectangle wonderful, if it is similar to the rectangle with side lengths $1$ and $2-\sqrt[3]{5}$. The professor claims to have a proof for the following theorem: ...
8
votes
1answer
647 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 ...
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?
7
votes
4answers
286 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 "...
9
votes
2answers
240 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
1answer
379 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?
9
votes
2answers
661 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 $L$?
5
votes
1answer
486 views

Perfectionist Pete's Perfect Pain: Even Tilings

Pete's room has a rectangular floor with dimensions $15 \times 20$. He wants to tile the floor of his room with black and white coloured tiles. Each tile is an unit square (ie. $1 \times 1$ square). ...
4
votes
3answers
535 views

Largest Complete Scrabble Block

Let's define a Complete Scrabble Block as an arrangement of Scrabble tiles such that: The four edges are all flat. All words are words. There is an order in which the tiles can be played in a game of ...
10
votes
2answers
637 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. ...
5
votes
2answers
643 views

Planning an Archipelago

We can place S and Z tetromino shaped islands on an 8x8 grid of water in such a way that no two islands are touching by their edges, because then they would not be two islands. Two islands may touch ...
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 ...
6
votes
1answer
439 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 ...
4
votes
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 ...
5
votes
1answer
267 views

Tiling a 6x6 board with an equal number of horizontal and vertical dominoes

Can you tile a 6x6 chessboard with dominoes, without overlaps or gaps, so that the number of dominoes oriented horizontally is equal to the number of dominoes oriented vertically? Why or why not?
239
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 ...
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 ...
8
votes
2answers
676 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: $2, 4, 6, \ldots, 96, 98, 100, 100, 98, 96, \ldots, 6, 4, 2$.) ...
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 ...
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 ...
8
votes
2answers
665 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 ...
10
votes
3answers
686 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, ...
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 ...
-1
votes
1answer
367 views

Dominoes on a Chess Board [duplicate]

Imagine an 8x8 chessboard has two opposite corners taken out, with 62 squares left over. Is it possible to lay 31 dominoes with the size 2x1 so they cover all of the empty, left over, squares?
2
votes
2answers
326 views

Packing an Efficient 9 Pack

This is sorta like How to ship the new Slurm 7-pack efficiently by Matt Malone but instead of 7, it's the 9 pack! It's basically a 2 by 4 pack with a piece sticking out, like in the above link. What's ...
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 ...
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?