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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2
votes
1answer
95 views

Making *9* congruent triangles from the pieces of a triangle dissection

Working on the making 7 congruent triangles from the pieces of a triangle dissection question I realized it's possible to do even better! So here it is for extra points: Use six lines to cut a ...
7
votes
1answer
172 views

Making 7 congruent triangles from the pieces of a triangle dissection

I got this challenging geometrical conundrum from a Russian geometrical magazine. It states: (A. Soifer) Use six lines to cut a triangle into parts such that it is possible to compose seven ...
17
votes
1answer
339 views

Sharing a field among 4 sons

A wealthy famer has a large estate in the shape of an irregular squarish octogon. In the middle he has a rectangular retention basin for storing water. He is getting old and discusses with his wife ...
73
votes
5answers
6k views

Can you fold a square into a square of one-fifth the area?

I love origami, and it recently gave me an idea for a very hard but beautiful puzzle. I'm really curious whether anyone here can solve it. So here's the puzzle. You are given a large perfectly square ...
2
votes
4answers
604 views

Picture puzzle: Leaf to Heart

Background Once upon a time there lived a poor man. He was so poor that he had nothing but a low-quality scissor that could only cut three times. On Valentines day, he wanted to make a card for his ...
25
votes
5answers
2k views

Pythagorean quilts

The King requests Pythagoras to his palace to discuss an important matter. After the usual formal greetings the King asks: - I have been told that you have a marvelous formula about adding squares ...
5
votes
1answer
141 views

How can I cut a cube so that all its vertices except for two mutually opposite vertices are equally distanced from the plane of the cut?

A friend of mine has been struggling with a solid geometry problem and, knowing my imagination skills developed by playing gomokunarabe and renju, has asked me to help her, but the problem has proved ...
9
votes
2answers
607 views

A rectangle in a rectangular hole

I have a carpet of 240 inches by 120 inches, but my floor, which it needs to cover, is 180 inches by 160 inches. How can I do this by cutting the carpet into exactly two pieces? Source: Rational ...
12
votes
2answers
2k views

Cut this shape into 3 pieces and fit them together to form a square

A shape is drawn on a sheet of squared paper as shown in the picture below. The shape is then cut from the sheet and given to you. You are asked to first make a straight cut across the shape and then ...
15
votes
3answers
863 views

Four fanatics and one checkerboard

Four checkers-playing fanatics eagerly pair up for two simultaneous games of checkers but somehow find themselves with just one board. They do have enough checkers for two games, so it is time to act. ...
18
votes
1answer
864 views

Restore the square

Triangle ABE is cut from square ABCD (centre E) and placed at one side. Restore the square by a single action different from the exact reversal of this transformation. A solution is to cut triangle ...
26
votes
1answer
555 views

You find a piece of paper in your bag

In your bag you find a piece of paper with a size of 5 x 5. You want to make it a 6 x 4 but you may only make 1 continuous cut and reposition the pieces. You are not allowed to bend or twist the ...
13
votes
4answers
803 views

Sharing a Cake with 7, 8, or 9

There may be 7, 8, or 9 guests at a party. The guests will share a round cake (shaped like a cylinder). Define a cut to be any plane or half plane that intersects the cake, i.e. straight cuts only. ...
32
votes
1answer
2k views

Dissecting the exotic bulbfish

Can you cut the following black shape into exactly three pieces, and then rearrange those pieces into a square?
0
votes
3answers
99 views

4x7 rectangle divided into 7 different rectangles

Can you divide a 4x7 rectangle into 7 rectangles all of different area? Can you find multiple solutions? Good luck! P.S. @Deusovi wanted me to make puzzles that have an "aha moment", so here is my ...
0
votes
2answers
157 views

Bake and share Fair and square

The chef ask each of the 4 judges to make a single slice on the whole round cake, so they'll all have a 1/5th piece to take. How the judges do it for fairness sake?
12
votes
1answer
775 views

Ernie and the Superconducting Boxes

I was in anticipation all last week. Ernie, who had been travelling for several months, was finally coming home. So over the weekend I dropped in on him. He was bursting with news. "Some wonderful ...
4
votes
1answer
148 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: ...
10
votes
3answers
353 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
1k 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 ...
19
votes
2answers
662 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
557 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
144 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
253 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
150 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
203 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
286 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
251 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
413 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
395 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
170 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
121 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
299 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
333 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
128 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
849 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 ...
7
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
764 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
528 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 ...
8
votes
1answer
506 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
869 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
2k 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.
28
votes
5answers
792 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 ...
7
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 ...
4
votes
1answer
481 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
380 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
147 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, ...
17
votes
5answers
1k 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
321 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
194 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, ...