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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2
votes
1answer
137 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 ...
4
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2answers
317 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 ...
49
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3answers
15k views

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 ...
11
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2answers
490 views

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 ...
5
votes
1answer
231 views

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 ...
11
votes
1answer
215 views

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 ...
4
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1answer
269 views

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 ...
14
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4answers
1k views

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 ...
-1
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2answers
182 views

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 ...
-1
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2answers
135 views

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 ...
8
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1answer
353 views

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). ...
9
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5answers
1k views

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 ...
23
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1answer
793 views

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 ...
2
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2answers
112 views

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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0answers
181 views

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 ...
0
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1answer
173 views

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 ...
-2
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1answer
116 views

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 ...
6
votes
2answers
490 views

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 ...
15
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5answers
831 views

Domino tiling on 8x8 checkerboard with four squares removed

I once posted this problem on the (now deleted) Area 51 Math Puzzles proposal. It was well-received there, but obviously I didn't get an answer. I still don't know the answer, and I'm not even sure if ...
3
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2answers
167 views

Generalized rectangular tilings with no "fault lines"

I recently came across this question: One rectangle, indivisible The goal is, by tiling 2x1 rectangles, to create a larger rectangle that cannot be split into 2 smaller rectangles. But my question is ...
-3
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1answer
150 views

a 17X17 grid filled with trominoes of three different colors

Let's have an 17x17 grid. We can fill this grid with 96 trominoes of three different colors, 32 trominoes of each color. On this particular grid the empty single square is the position A1. By visual ...
29
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1answer
938 views

A flag-packing problem

An entry in Fortnightly Topic Challenge #45: Flags You are provided with a 9x9 grid of squares and 21 minimalistic flags (pictured below, all shown to scale). TASK: Assign a colour (Black, Blue, ...
7
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2answers
361 views

Pentomino tiling on wrap-around 5x5 grids

It is known that P pentominoes cannot tile a 5x5 square board. Q1: If the east and west edges of the 5x5 square board are "wrapping around" (if you move a piece through one of the edges, the ...
4
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1answer
279 views

Grids with trominoes

Let's have two 8x8 grids. By visual inspection we see they are filled with trominoes of three different colors. There are 7 trominoes of each color. On the grids the trominoes are not allowed to touch ...
2
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1answer
311 views

Forming pairs of trominoes on an 8X8 grid

On an 8x8 grid I put 21 trominoes of thee different colors. Each group of 7 trominoes has one color. By visual inspection we see the trominoes cover the whole surface except the single empty square ...
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1answer
188 views

Elections in the United States of Alfagonia

Elections were held in the 45 electoral districts of the United States of Alfagonia. The Green Party won the election in 23 of the 45 districts. Alfagonia is made up of nine states of five districts, ...
12
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2answers
455 views

The art of computer programming

EDIT: I know we are not supposed to edit in new requirements after first posting but as far as I understand it this requirement is implicit in all questions here: Explain your answer! At least a ...
1
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0answers
132 views

Any comparison between some variations of T puzzles? [closed]

I spent time to experience some variations of the classical T puzzles in here - a kind of dissection/tiling puzzle (Gardner's T, Nob's T, and Asymmetric T). They are 4-piece tangrams. They all give ...
10
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1answer
329 views

A chessboard tiling with corners removed in 3D

A famous problem asks whether an 8x8 chessboard with two opposite corners deleted can be tiled with dominoes, where a domino is a rectangle congruent to two adjacent squares of the board. Now, let C ...
4
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2answers
253 views

Double tiling congruent triangles with little else in common

When you really want to tile more than one layer but triple tiling is just too much of a good thing, surely the happy medium is double tiling. How may a mosaic of more than 900 sections be double ...
8
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1answer
166 views

L-tromino pair!

Amy is playing with different polyominoes. She suddenly thinks of a problem as follows. Choose two positive integers $m,n$. If we can use only L-trominos to tessellate a $m\times n$ rectangle with no ...
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0answers
132 views

Building a cube from small bricks such that no lines can be pushed through between the seams

There is a puzzle on fault-free rectangles tiled by dominoes. It is rather known (it is described in Martin Gardner’s “Mathematical puzzles and diversions”, see here) and rather old (it is known at ...
7
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0answers
122 views

Hand tiling - 4x inflated pentominoes with pentominoes and tetrominos

This is a manual tiling puzzle. Difficulty level always hard to estimate, but I would say a few minutes per solution for an accomplished solver, all the way to just about impossible for someone like ...
8
votes
3answers
944 views

Fillomino Tiling...how many 1's?

Suppose a 'Fillomino tiling', much like a completed Fillomino puzzle, consists of a set of polyominoes covering a region without gaps nor overlaps, with no two n-ominoes of the same size touching ...
8
votes
1answer
167 views

Sliced sudoku: rearrange the tiles and then solve it

Rearrange the tiles without rotating nor flipping them to form a $9 \times 9$ sudoku. Then solve the sudoku according to the standard rules. Credits: the sudoku is one of the twelve schemas freely ...
20
votes
3answers
697 views

Filling the plane with two colors

In this puzzle you must tile the plane with colored T-tetraminos. I will start by laying down 3 of them for you like so: Your task will be to tile the entire rest of the plane meeting the following ...
14
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2answers
1k views

Make a hexiamond star by hand

Using some or all of the hexiamonds (pictured), make a star. You may flip pieces. The usual tiling rules apply, no overlaps, no gaps. Use only one or none of each piece. Answer is unique. Target shape ...
16
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1answer
674 views

A cube build with cuboids

You are given 27 pieces of 1x2x4 cuboids. Is it possible to build a 6x6x6 cube using those 27 cuboids?
9
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1answer
382 views

Tile-laying for beginners

Coldport is celebrating having a new town hall by laying a tiled design in the public square in front of the building. The design is also square: $5\times 5$ tiles square, in fact, and features three ...
8
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2answers
346 views

Hand tiling puzzle demonstrating Eisenstein triple $c^2 = a^2 -ab + b^2$

An Eisenstein triple is related to 60 degree triangles and a special case of the cosine law. But we need not worry about that except to note that a specific example of an Eisenstein triple is $7^2 = 5^...
22
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3answers
2k views

Can you put L trominos to fill the figure?

In the above picture, there are 24 squares. Can you only use L trominos to fill the figure? If yes, give an example. Otherwise, please explain why. An L tromino is like this:
10
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3answers
325 views

Four hand tiled squares demonstrating a Pythagorean Quadruple

Demonstrating the Pythagorean Quadruple $6\times6 + 6\times6 + 7\times7 = 11\times11$ Using the pieces shown in the $11\times11$ square: The objective: Arrange the pink pieces (four enneominoes) ...
17
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3answers
539 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. ...
7
votes
1answer
311 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 ...
5
votes
3answers
169 views

Fit as many overlapping generators as possible

Rimworld is a tile-based videogame. One of its constructibles in the wind generator: The wind generator itself occupies a space of 7x2 and can be placed facing the 4 cardinal directions. In order ...
8
votes
1answer
614 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 ...
11
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2answers
415 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 ...
3
votes
1answer
197 views

Max 4x1 pattern fit within 11x11 area

Rimworld is a tile-based videogame. There is a constructible called a sun lamp, which provides light for indoor farming: . As you can see, the area covered by the lamp is 11 * 11, minus 1 for the ...
7
votes
2answers
211 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 ...
5
votes
2answers
837 views

The Pentomino Snake

The premise of the puzzle is quite simple. Here's how to set it up. Draw a 5x5 grid of squares. Write the number 1 in the middle. Make a "snake" of numbers up to 25 so that each number is ...