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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9 votes
2 answers
489 views

Tiling a 5-by-5 bathroom with L-shaped triomino tiles

The following puzzle appeared in the Cambridge, UK February 2024 MathsJam Shout: Tiling with Triominoes Imagine tiling a 5-by-5 bathroom with L-shaped triomino tiles. Clearly, not every square can be ...
Will Octagon Gibson's user avatar
18 votes
1 answer
1k views

Which heptomino is it obvious can't tile the plane?

A polyomino is a collection of equal-sized squares joined edge-to-edge in the plane (think Tetris pieces, but with an arbitrary number of squares instead of just four). A heptomino is a polyomino ...
Lieutenant Zipp's user avatar
23 votes
3 answers
2k views

Tiling a square with right-angled triangles

Tile a square with twenty congruent right-angled triangles. For each triangle, one leg is of length 1 and the other leg is of length 2.
Will Octagon Gibson's user avatar
3 votes
0 answers
84 views

Another hand tiling puzzle - 8 convex shapes from 7 polyiamonds

Arrange the 7 polyiamonds in the image into 8 different convex shapes. One after the other, not simultaneously... Rotating and flipping allowed. No gaps or overlaps. Should be a fairly easy puzzle.
theonetruepath's user avatar
3 votes
0 answers
125 views

Hand tiling polyiamond puzzle. Non-trivial

Group these 30 polyiamonds into five sets of six, then use each set of six to make five different convex shapes. There are some repeated polyiamonds, no set of six may contain a duplicate. The convex ...
theonetruepath's user avatar
2 votes
0 answers
118 views

Make three different convex shapes with four polyiamonds, by hand. Five times

To while away the endless holidays. Start with the four polyiamonds in row one. Arrange them without gaps or overlaps (flipping/rotating allowed) in a convex shape. Repeat for two more convex shapes, ...
theonetruepath's user avatar
1 vote
0 answers
227 views

Covering a Square Floor with Square Rugs [closed]

You are given a finite collection of axis-aligned square rugs. (You do not choose the collection of rugs that you receive and the rugs are not necessarily all the same size.) Your objective is to move ...
Basset Hound Video's user avatar
11 votes
2 answers
364 views

Fit small squares in a large square

Fit the given square pieces in a larger square! Warmup In a 10x10 square, fit 5 1x1 squares, 8 2x2 squares, and 7 3x3 squares. Solution Puzzles In an 11x11 square, fit 3 1x1 squares, 12 2x2 squares, ...
Pontus von Brömssen's user avatar
7 votes
3 answers
690 views

Making a 3 x 8 grid with tetrominoes

I'm wondering if it is possible to make a 3x8 grid using 6 tetrominoes: 2 Is, 2 Ls, a Z skew, and a square tetromino. I believe it may be impossible but would like to know why.
John Williams's user avatar
1 vote
7 answers
529 views

Dissect this figure into four pieces which can be reassembled to form a square

How can you cut this figure into four (not necessarily identical) pieces which can be reassembled to form a square? Rotating and flipping the pieces is allowed. Hole(s) in the final square are allowed....
Will Octagon Gibson's user avatar
4 votes
1 answer
298 views

Xmas-colored tiles

If you subdivide a 2x2 tile into 4 unit squares and then color each unit square either red or green, then there are $2^4=16$ ways you can do this as shown below: Can you use all 16 tiles (rotations ...
Will Octagon Gibson's user avatar
7 votes
3 answers
469 views

Ziggy - Can you make a square from 49 polyomino pieces

This puzzle is variant of the puzzle Ziggy - Make a square from 8 polyomino pieces by jaap-scherphuis. This puzzle is based on the fact that $1+2+3+\cdots+49=35^2$. It consists of 49 zigzag polyomino ...
Will Octagon Gibson's user avatar
4 votes
2 answers
208 views

Polyomino equation part 2

There is a particular shape (holes are permitted) that you can build with four copies of the piece on the left, or with 5 copies of the piece on the right. You may rotate and flip the pieces, but ...
Will Octagon Gibson's user avatar
3 votes
2 answers
707 views

The Diamond of Columbus

There are up to flipping / rotation 12 distinct pentominoes. Can you fit them without overlap in the white area?
loopy walt's user avatar
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7 votes
1 answer
396 views

Polyomino equation

There is a particular shape (holes are permitted) that you can build with three copies of the piece on the left, or with 5 copies of the piece on the right. You may rotate and flip the pieces, but ...
Will Octagon Gibson's user avatar
8 votes
4 answers
1k views

How much area can Alice tile?

Alice is tiling a very large area using square tiles. She can tile at most ten tiles each time before getting some rest. When she rests, Bob the trouble maker will sneak in to remove one connected ...
Eric's user avatar
  • 6,458
7 votes
1 answer
411 views

Closed path on a dodecahedron

Your task is to draw lines between edges on a regular pentagon such that if you tile a dodecahedron with 12 identical copies of that pentagon you get a single closed line which does not intersect ...
Herbert Kociemba's user avatar
46 votes
4 answers
2k views

A colorful dodecahedron

Divide a "base" edge of a regular pentagon into three equal parts. Then draw two lines from the base to the center of the other edges such that the lines do not intersect. This splits the ...
Herbert Kociemba's user avatar
2 votes
1 answer
354 views

Tile 1x2 dominoes in a 2x10 space

How many ways are there to tile unmarked 1x2 dominoes in a 2x10 space? Bonus: What if the dominoes were identical and had pips on their front (face-up), so they could be distinguished by 180 degree ...
qwr's user avatar
  • 693
11 votes
2 answers
443 views

Tiling a dodecahedron

The surface of a dodecahedron is tiled with 6 of the shown tiles, each tile covering two faces of the dodecahedron. In how many essentially different ways this can be done? Two tiled dodecahedrons are ...
Herbert Kociemba's user avatar
5 votes
1 answer
2k views

How many ways are there to solve the Mensa cube puzzle?

The Mensa cube is a puzzle in which a solid cube has been partitioned into $N=11$ rigid parts. The goal of the puzzle is to re-assemble the cube from its parts and place it back in its rigid box. See ...
fromscratch's user avatar
8 votes
2 answers
690 views

Flipped Einsteins in the Einstein Tiling

The single-tile aperiodic tiling by Goodman-Strauss, Kaplan, Myers and Smith has been all the rage recently: In this tiling a minority of tiles, coloured purple above, are flipped with respect to the ...
Parcly Taxel's user avatar
  • 7,359
6 votes
2 answers
351 views

Boxeslayers to the rescue

This is about layering boxes, not about slaying them. We have 1,830 2×5 boxes to stack safely as 10 alternating contiguous layer patterns of 183 boxes each. Layers have identical silhouettes that fit ...
humn's user avatar
  • 21.9k
5 votes
3 answers
1k views

How to fully tile an 8 by 8 square with Z-tetrominoes?

Is it possible to fully tile an 8 by 8 square grid using only Z-tetrominoes? (I don't know the answer...)
Daniel's user avatar
  • 69
0 votes
2 answers
262 views

Popping pentominoes

I started with the following configuration: I then popped a few spots in the shape of the same pentomino to arrive at this configuration: The shapes did not overlap and I was allowed to rotate and ...
Dmitry Kamenetsky's user avatar
6 votes
0 answers
220 views

Set of magic polyominoes that can tile a square

Let's first look at this square grid of numbers. The 9 squares in yellow is what we are looking at and the green numbers are the sums for the digits within the rows and columns. The red squares are ...
Maff's user avatar
  • 621
1 vote
0 answers
153 views

Arranging shapes into a similar shape

The goals if possible. Goal 1. In the image there are 12 shapes each containing 15 cells. Take any 3 shapes from the set and arrange them into any new shape, 2 example shapes that you could use are ...
Maff's user avatar
  • 621
8 votes
2 answers
330 views

Polyominos packing into a square

Rules of the game: Take a square grid nxn. Populate the grid with polyonimoes of area size 1 to area of 9. Polyominoes can be any shape. There must be one each of every size polyomino in every row ...
Maff's user avatar
  • 621
7 votes
2 answers
399 views

Smallest square that can pack thin digits

If we draw the digits 0 to 9, segmented into squares, across a rectangle of 2x5 (except the 1) they use up 81 total squares. Is it possible to pack them all into a 9x9 grid. What is the smallest n by ...
Maff's user avatar
  • 621
9 votes
3 answers
1k views

Tiling a chessboard

Say I have an eleven by eleven chessboard, so there's 121 squares total. I remove the centermost piece so there's 120 pieces. I want to tile the chessboard with 1x4 or 4x1 pieces in a way that none of ...
Joey's user avatar
  • 203
3 votes
1 answer
328 views

Seven octahedral nets to cover an octahedron

After solving Cover a single cube with FIVE identical cube nets I had the idea for this puzzle, which may be regarded as a natural generalisation to triangular grids. Find two different nets, A and B, ...
Parcly Taxel's user avatar
  • 7,359
16 votes
1 answer
497 views

Cover a single cube with FIVE identical cube nets

Start with five identical cubes: Your challenge: Cut and unwrap all five cubes into five identical cube nets. Show how to re-fold these five cube nets to form the surface of a single larger cube, ...
plasticinsect's user avatar
3 votes
2 answers
867 views

Covering a room with 34 carpets

There are 34 rectangular integers less than 100 which are the product of two primes. Can a room be covered precisely (no overlaps, etc.) with the 34 rectangles that have as areas these products?
Bernardo Recamán Santos's user avatar
10 votes
3 answers
2k views

Covering a large room with 55 carpets

How many different rectangular rooms, if any, is it possible to cover with all the 55 carpets of different dimensions 1 x 1, 1 x 2, 1 x 3,..., 1 x 10, 2 x 2, 2 x 3, ..., 8 x 9, 8 x 10, 9 x 9, 9 x 10, ...
Bernardo Recamán Santos's user avatar
3 votes
0 answers
222 views

Cover a 15×8 board with the pentominoes [closed]

Cover the entire area of the 15×8 square shape with two complete sets of 12 different types of pentominoes. The pentominos can be rotated and flipped. Two pentominoes of the same type must not be ...
Bill's user avatar
  • 31
13 votes
4 answers
850 views

What approach can I use to solve this wooden packing puzzle?

I am looking for an approach to solve a wooden packing puzzle which my three-year-old got as a present. We enthusiastically unpacked and disassembled the puzzle when she got it. (It came put-together ...
justfortherec's user avatar
2 votes
2 answers
205 views

Find the ratio of green to red circles for given patterns

Just a small visual task. Find the ratio of green to red circles, for infinite planes, filled with the following patterns:
Alexey Birukov's user avatar
6 votes
1 answer
675 views

Solving a 5x5 pentomino with only certain shapes

I have a physical Pentomino puzzle lying around, which contains 2 F pieces, one Y piece, one T piece and one W piece. The area into which the squares are supposed to fit in is just a little under 6 ...
Cheese Macken's user avatar
8 votes
1 answer
508 views

Pentomino - is there any solution with the straight-bar piece in the middle of a rectangle?

Is there a solution for a straight-bar piece not touching the edge of the rectangle? By rectangle solution, I mean either one of the patterns 15x4, 12x5, or 10x6. By straight-bar piece, I mean the ...
Przemyslaw Remin's user avatar
0 votes
0 answers
45 views

Impossible tiling of board using dominoes [duplicate]

prove that no matter how you tile a 6 x 6 board using 2 x 1 tiles, there would always be a vertical or horizontal line separating the board. Separation here means that no tile would be cutting across ...
Rishabh Jain's user avatar
3 votes
1 answer
246 views

Tetromino tilings of a 4x5 rectangle with minimal diversity

This is a relatively easy manual tiling puzzle. In fact the tiling is all done for you, you just have to specify how many of each of the 26 given tilings to use. The puzzle is: Using N complete sets ...
theonetruepath's user avatar
2 votes
1 answer
123 views

Tiling with dominoes then with tetrominoes

For reference, here are the five tetrominoes: Suppose you join ten dominoes to make a polyomino made of twenty unit squares. Is it possible that you can tile the polyomino using all the five given ...
Will Octagon Gibson's user avatar
4 votes
1 answer
234 views

Are there Stars on the Knight Sky?

If you watched the Queen's Gambit you'll know that there are lots of upside-down chess boards all over the world's ceilings. If not just take my word for it. The same is true of the actual sky only ...
loopy walt's user avatar
  • 20.9k
8 votes
1 answer
499 views

Tiling a 5x5 square with five pentominoes AND a Latin square

Suppose that a 5x5 square has been tiled with five (not necessarily distinct) pentominoes. Is it true that there will necessarily exist at least one Latin square of size 5x5 (using the numbers 1,2,3,...
Will Octagon Gibson's user avatar
4 votes
1 answer
188 views

Tiling twelve 5 x 10 rectangles with ten sets of the twelve pentominoes

I have ten copies of each of the twelve pentominoes. Can I use all of them to completely tile twelve 5 x 10 rectangles?
Bernardo Recamán Santos's user avatar
2 votes
1 answer
121 views

What size square grid can you tile?

A tiling of an n × n square grid is formed using 4 × 1 tiles. What are the possible values of n? A tiling has no gaps or overlaps, and no tile goes outside the region being tiled.
Simd's user avatar
  • 7,816
47 votes
1 answer
4k views

Which country flags can you make in Tetris?

Your friend is playing Tetris. In her version of the game, the pieces use the standard colors and can drop in any possible order without restrictions. In the order shown below, the colors are light ...
noedne's user avatar
  • 15.4k
2 votes
1 answer
227 views

Progressive Daedalian Opus

The 1990 Game Boy game Daedalian Opus is essentially a series of 36 pentomino puzzles. In the first level, however, you only have three pentominos; the rest are introduced in the levels after that, ...
Parcly Taxel's user avatar
  • 7,359
2 votes
1 answer
200 views

Five 3:1 rectangles tiling a square

Can you fully tile a square with 5 rectangles such that: Every rectangle has 3:1 ratio, ie., their length is triple their width. No part of any rectangle is outside the square. No two rectangles ...
Dmitry Kamenetsky's user avatar
4 votes
2 answers
427 views

Seven 2:1 rectangles covering a square

Can you fully cover a square with 7 rectangles such that: Every rectangle has 2:1 ratio, ie., length double its width. No part of any rectangle is outside the square. No two rectangles overlap. Note ...
Dmitry Kamenetsky's user avatar