# 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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### 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 ...
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 ...
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 ...
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-...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
459 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. ...
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 ...
353 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 ...
507 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 ...
536 views

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### 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?
222 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 ...
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 ...
201 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 ...
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 ...
453 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 ...
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 "...
482 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 ...
288 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 ...
175 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 ...
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 ...
266 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, ...
278 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 ...
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 ...