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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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...)
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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, ...
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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?
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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, ...
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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 ...
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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 ...
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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:
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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?
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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.
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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 ...
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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, ...
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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 ...
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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 ...
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A new way to cut a pizza
Can you cut a pizza (circle) into 12 congruent pieces, such that half of them have crust (circle boundary), while the other half do not? The pieces must have the same shape and area, but can be ...
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Match the colors on the edges of the rectangles - is it possible?
In April 1971 (it says so on the back of the cards) I made the following puzzle which requires one to form a square with adjacent colours being the same.
Can this puzzle be solved and, if yes, what ...
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Seven-Segment Telephone
The ideas and concocted tools I used to solve these two puzzles allowed me to make this puzzle. Credit to TSLF for the inspiration!
Light up some of the edges in the grid on the left such that the ...
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Tiling three pears with three-pair hexominos
The 21 hexominos below are all those that can be made by joining three dominos together (i.e. they have a perfect matching with respect to the graph of their squares) and are not rectangles. The ...
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Number Box Puzzle
All digits from 0-9 as shown are 1 in. thick and within 3x5 in.dimesion except number 1. The task is to put all the numbers inside the square box with smallest dimension nxn without overlapping of ...
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Tiling with Js and Ls
In this puzzle you must tile the plane with identically sized colored L and J tetraminos. To start I will place two of them like so:
Your task will be to tile the entire rest of the plane meeting ...
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Rectangles and squares of trominoes filling a grid
Let's have a board 24 squares by 24 squares. This board is to be filled with trominoes of three different colors. There are equal numbers of trominoes of each color. The board is to be filled with ...
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Tessellation with nonagons and equilateral triangles
What type of convex nonagon is required to tesselate a plane with equilateral triangles and nonagons? All sides of the nonagons are equal.
NOTE: Partial tessellation of a plane should accompany your ...
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Manual tiling with 8 dodecadudes
Here are 8 dodecadudes. (Drawn from the very numerous dodecadrafters, made from 12 half equilateral triangles, dodecadudes are a subset of 770 pieces with sharp points and narrow necks excluded). ...
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Minimize 1×3 tiles on a 5×5 table to block any more 1×3 tiles
What is the minimum number of 1×3 tiles that can be put on a 5×5 table so that no more 1×3 tiles can be put on it?
Borders of a tiles are parallel to sides of the table.
It is 5 but I can not prove ...
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Polly O'Mino's Hexcellent Adventure
An entry in Fortnightly Topic Challenge #52: Polyominoes.
I had heard that my good friend Ingrid Deduction was back in town, so I popped round to her apartment today, only to find she'd got herself a ...
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Surrounding an L-shaped tromino
You are given an L-shaped tromino, shown below. Can you surround it by 10 more L-shaped trominoes? The new trominoes can be rotated and must touch the original tromino at an edge or a corner.
This ...
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Ammann chair tiling puzzle
The Amman chair is an interesting shape that can be dissected in two pieces that are smaller copies of the original. The sizes of the two pieces are different. The ratio between the areas of the ...
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What is the perimeter of a pentomino which can tile this heart-shaped board?
The puzzle is as follows:
The figure from below shows a board that is made up of 30 squares whose sides are of 2 centimeters each. Such board must be split in six congruent pieces. These must be made ...
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Filling a 26x36 grid with trominoes
Let's have 312 L-trominoes of three different colors, 104 trominoes of each color. Can you fill a 26x36 grid with these trominoes without two of the same colors touching side-to-side anywhere? In ...
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Covering a 15x15 grid with rectangles
You are given a 15x15 grid and asked to cover it with rectangles whose dimensions are a power of 2. For example you can use rectangles 8x1 and 4x4, but not 2x3. The rectangles must cover every cell of ...
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