Questions tagged [dissection]

A geometric puzzle where a given figure has to be cut into a number of pieces subject to a number of constraints.

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4
votes
1answer
138 views

Dystopian Tax Collection

The year is 2081, and... oh, what can I say? Dystopian stories have been done to death. I have a much more practical problem, though. I need to... gasp... pay my taxes. I owe five different taxes: ...
8
votes
2answers
203 views

What is the largest number of cubes that can be cut?

Consider a cube made up of 27 unit cubes. If you consider a plane going through the middle of the larger cube it cuts through a number of the unit cubes. The number of cubes that are cut depends on ...
6
votes
2answers
763 views

Cutting a cross made of 5 equal squares by 2 straight cut into 4 figure to together form a square

A figure consists of 5 equal squares in the form of a cross. Please show how to divide it with two straight cuts into 4 equal pieces which will fit together to form a square. A MSE told me I need to ...
18
votes
2answers
610 views

Piece of Cake for King Solomon

Today (7th of July, 2018) marks 40 years of independence for the Solomon Islands. To celebrate, I have decided to bake a cake! And what a pretty cake it's going to be: When viewed from the top, ...
12
votes
1answer
453 views

Ernie and the Case of the Singing Sisters

On my way to work each morning I pass a news-agency that usually has a sandwich-board displaying the latest headlines of a somewhat disreputable tabloid newspaper (not that I would ever buy such a ...
2
votes
1answer
135 views

Tile a square with five rectangles with 10 distinct edges

The baby brother of: Cutting a square into seven rectangles Tile a square with five rectangles. Select the lengths of the edges of the rectangles from the set $1$ through $10$, with no length ...
5
votes
2answers
218 views

20 right isosceles triangles into a square

Similar: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Five graded difficulty isosceles right triangle into square tilings Two difficult "Seventeen right isosceles ...
2
votes
2answers
140 views

Two difficult “Seventeen right isosceles triangles into a square” tilings

Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Five graded difficulty isosceles right triangle into square tilings V.hard problem, 20 right isosceles triangles ...
5
votes
2answers
195 views

Five graded difficulty isosceles right triangle into square tilings

Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Two difficult "Seventeen right isosceles triangles into a square" tilings V.hard problem, 20 right ...
12
votes
2answers
260 views

Unlucky tiling: Arrange thirteen right isosceles triangles into a square

Link to next puzzle in this series:Five graded difficulty isosceles right triangle into square tilings Two difficult "Seventeen right isosceles triangles into a square" tilings V.hard ...
6
votes
2answers
228 views

Challenging 15 rectangle tiling problem

This will test you, a computer will definitely help. Just one set of $1:2$ aspect ratio rectangles this time, but $15$ of them. Short side only is listed. The challenge is to arrange them into a ...
5
votes
1answer
380 views

Unfair tiling puzzle

Your goal is to make two squares of the same size from a set of rectangles. Each of the rectangles has an aspect ratio of $1:2$. Select two sets of rectangles from the list: ...
4
votes
1answer
332 views

Hard tiling puzzle

Your goal is to make two squares of the same size from a set of rectangles. Each of the rectangles has an aspect ratio of 1:2. Select two sets of rectangles from the list: ...
4
votes
1answer
166 views

Slightly bigger 1:2 hand tiling, getting a little harder

Pretty sure an expert hand tiler would be able to find this. Please post a spoiler only if you hand tile it... Tile a $60\times60$ square with $1:2$ rectangles of sizes ...
3
votes
1answer
114 views

Dissect a square into 1:2 non-congruent integer-sided rectangles

(Similar to the recent 3:1 and my 3:2 rectangle question) Tile a square completely with rectangles which have aspect ratio 1:2, integral side lengths and all different sizes. In other words selected ...
0
votes
2answers
296 views

Common among shapes

What is common to the following real, physical objects: A cardboard/piece of paper carved as letter E (with all the segments of equal size) A cardboard/ piece of paper shaped as two full cycles of a ...
9
votes
2answers
326 views

How to find the layout of the plots?

A peasant had a square garden of $100 × 100$ meters divided into $100$ equal square plots. In the testament, he left to each of his $7$ male grandchildren a connected region of $10$ plots, forming ...
2
votes
2answers
121 views

Right angled triangle to all acute angled triangles

What is the minimum number of cuts needed to dissect a right angled triangle into acute-angled triangles ?
13
votes
3answers
832 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 ...
8
votes
6answers
5k views

Obtain four equal parts with a single cut

It's easy to obtain four equal parts from a rectangle with three cuts: just make four strips. It's also easy to obtain four equal parts from a rectangle with two cuts: make a horizontal and a vertical ...
9
votes
3answers
716 views

Election day in Rectangularea (Crack 'em, pack 'em, stack 'em, elect 'em)

Remember when Rectangularea gained parliamentary representation? The regrettably segregated population broke down into parties as follows. Gerrymandericans (g)...
0
votes
1answer
376 views

Dissecting a triangle into 4 pieces [closed]

How can I dissect a triangle into 4 piece and form a square? I took A right triangle with sides 3,4,5.i was able to get a rectangle of side 3 and 2,if I dissect that I was able to get 3 squares of ...
7
votes
1answer
381 views

Skeleton sudoku, the second

I've already set one skeleton Sudoku. Here's a rather harder one. (Nonetheless, it should be within the realms of what can be done by hand; I solved it manually myself to verify that there was only ...
8
votes
2answers
703 views

A spartan skeleton Sudoku

A skeleton crossword is a crossword where the black and white squares aren't given; you have to deduce them. Here's a generalisation of that idea to the Sudoku puzzle. Here's a 4 by 4 grid, with four ...
4
votes
2answers
1k views

Can you divide this figure into two equal parts? [duplicate]

This is very simple when I see the answer, but I was not able to answer it(by thinking of single line), So posting for you.
27
votes
5answers
708 views

Dissecting the holey octomino into a square

This is a pure dissection problem, with no added twists. Cut the holey octomino (i.e., a square with the middle third removed) into several pieces, and reassemble those pieces into a square with no ...
8
votes
9answers
1k views

One Morning at the Coffeehouse

The three bears are regular customers at the Goldilock's (see illustration). When Goldilock brought the cake as ordered by the Bear family, Little bear asked her if she can divide the big round cake ...
3
votes
1answer
454 views

Infinitely simple polygon solipsism

           Solipsism — The self is all that can be known to exist. Above is a simple polygonal region divided into infinitely many different-...
9
votes
3answers
363 views

Unreflected infinitely simple polygon reflexivity

(This was retrofitted to more tightly match a surprise solution and to allow for another puzzle with the original intent.)                   &...
2
votes
1answer
134 views

How to improve “Universal dissection”?

Yesterday I've posted quite easy puzzle: Universal dissection. Now the actual problem. When we deal with 8x8 board with 1 missing cell it doesn't matter whether we allow to flip parts or not, ...
16
votes
4answers
935 views

Universal dissection

Alice has a squared paper 8 by 8. She cuts out one 1x1 square from it, at row N, column M. Bob cuts the rest of the paper into pieces. Once he is done, Alice asks Bob to put the pieces together in a ...
4
votes
2answers
286 views

1 Cube 1 Square 1 Line

Lets take a 1'x 1'x 1' Cube rock and a 1' x 1' Square paper. With 1 straight Line cut of a scissors we can create a hole on the foldable Square paper where the Cube rock can pass through without ...
9
votes
4answers
183 views

Shifting halved Square along Diagonal through Translations only, without Rotations

A square is halved along a diagonal, creating equal halves A and B. Prove that it is not possible to dissect one half into a finite number of pieces, and through translations only (i.e. no rotations, ...
13
votes
4answers
758 views

Checker-board Problems

Question 1: Here is a $4\times4$ black and white checker-board. You may fold the checker-board in any direction and times. You may cut the checker-board along a straight line once. The cut needs to ...
6
votes
8answers
11k views

Maximum Pieces of Cake in Four Cuts

I had read somewhere that the if we cut a cake four times (Two horizontal cuts and two vertical cuts, in whichever order), the maximum number of pieces we can get is 27. No solution was given there. ...
9
votes
3answers
1k views

Four Triangles Five Shapes puzzle

I discovered here the Four Triangles Five Shapes puzzle (from Emrehan Halici). From the $4$ triangles, we can form: — $3$ different parallelograms — a square — an isoceles triangle The sizes and ...
4
votes
1answer
330 views

Can I Haz My Eye Center'd 2?

I have noticed Can I Haz My Eye Center'd? puzzle, which I know in slightly different formulation: You have a disk on a table and a point A on it. You need to cut the disk into smallest possible ...
16
votes
3answers
3k views

Can I Haz My Eye Center'd?

Alice has painted a cat on a white disk such that the left eye of the cat is exactly on the center of the disk. Inspired by it, Bob grabbed a white disk of the same size and sat down to paint on it a ...
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
198 views

Tiling a $13 \times 11$ rectangle with squares

What is the smallest number of integer-sided squares required to tile a $13 \times 11$ rectangle without overlaps?
5
votes
1answer
204 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: ...
3
votes
1answer
143 views

Professor Knowfair's square dissection assignment

While at the public library with professor Knowfair, he also told me his plans for this week's assignment. He said: Dissecting a triangle is all fine and dandy, but squares are the real deal. For ...
1
vote
1answer
148 views

Professor Knowfair's triangle dissection assignment

Last Sunday I met with Professor Knowfair at the public library. We caught up and he told me about the most recent assignment he gave his students. He said: We had been studying medians and how ...
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?
6
votes
1answer
220 views

Squaring a cross one more time

This is the lateral-thinking fifth part to this question. In this cross shape, all twelve sides are the same length and all angles are right angles. What is the smallest number of cuts that can ...
18
votes
3answers
2k views

Tiling a rectangle with nine squares

A rectangle is tiled by nine squares with side lengths $2,5,7,9,16,25,28,33$ and $36$ (without overlapping and without gaps). What are the side lengths of the rectangle? What does the tiling look ...
17
votes
1answer
1k views

Squaring a cross

In this cross shape, all twelve sides are the same length and all angles are right angles. How many cuts does it take to divide the cross into pieces that can be rearranged to form each of the ...
11
votes
1answer
753 views

Professor Halfbrain and the dissection of a rectangle

Professor Halfbrain has spent his entire weekend with cutting rectangles into smaller rectangles. In particular, he proved the following deep theorem on such dissections. Professor Halfbrain's ...
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 "...
8
votes
1answer
378 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?