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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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21 votes
2 answers
631 views

3³+4³+5³=6³ Puzzle

A classic puzzle asks us to break a 6x6x6 cube into the smallest number of pieces which can be reassembled into 3 physically separate cubes of sizes 3, 4, & 5. 3³+4³+5³ =27+64+125 =216 =6³ An 8-...
DMC_Run's user avatar
  • 211
11 votes
4 answers
999 views

Make a square table top with the minimal needed amount of straight cuts

inspired by : Make a square table top with six (or fewer) pieces A carpenter has three pieces of beautiful wood, measuring 12 inches, 15 inches, and 16 inches square, respectively. They want to use ...
Retudin's user avatar
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10 votes
2 answers
505 views

Make a square table top with six (or fewer) pieces

A man had three pieces of beautiful wood, measuring 12 inches, 15 inches, and 16 inches square respectively. He wanted to cut these into the fewest pieces possible that would fit together and form a ...
Will.Octagon.Gibson's user avatar
24 votes
2 answers
4k views

A pizza dilemma

You are a waiter at a restaurant. The restaurant is known for its signature dish: the Donut Pizza. The Donut Pizza is a 5-inch square pizza with a 1-inch square hole in the middle. After several ...
mathlander's user avatar
  • 1,261
38 votes
4 answers
3k views

Pythagorean pentagons

To follow up on the theme of so called "pythagorean" dissections, here is one more for you to chew on. I hope you don't get bored. The pentagons above have sides respectively 3, 4 and 5. ...
Florian F's user avatar
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-1 votes
7 answers
565 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
6 votes
3 answers
471 views

Dissections of the "hat" aperiodic monotile

We all know the "hat" monotile by now, right? It can obviously be dissected into 8 congruent kites, or 4 congruent pentagons. Can it be dissected at all into 2 congruent shapes? What about 3?...
Quuxplusone's user avatar
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3 votes
1 answer
203 views

A dissection puzzle where you're allowed to use dilation

You may be familiar with Dudeney's famous dissection of the equilateral triangle into a square. (A nice physical version is demonstrated here.) His dissection uses four pieces. I believe this to be ...
Akiva Weinberger's user avatar
14 votes
5 answers
3k views

Dividing a square field into 5 equal regions

A farmer has a 10m x 10m field that has fences around the perimeter. What is the least number of 1m fences he needs to add to divide the field into 5 regions of equal area?
Dmitry Kamenetsky's user avatar
13 votes
2 answers
516 views

Put three pieces of cake into a round box

You're about to cut three pieces from a large cake to put in a round box of radius 1. If the pieces must be congruent triangles, and cannot overlap, what shape gives you the maximum amount of cake?
Eric's user avatar
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8 votes
6 answers
627 views

Breaking the Heart geometrically

The King of Geometro nation has 2 very smart wives. On the Geometro Wives day he gets a nice heart shaped cake made. It has a number of icing flowers on it. The King wants to split the cake in half so ...
DrD's user avatar
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5 votes
1 answer
1k views

Three lines to get twenty triangles

Shown below are five squares. Starting at any point, draw three straight lines without lifting the pen, and create exactly twenty (20) triangles. It is understood that this will create some other ...
DrD's user avatar
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8 votes
1 answer
284 views

Ernie and the Christmas Stars

Although Ernie professes to be an atheistic rationalist, he does love the Christmas season. He thinks long and hard to find appropriate gifts, brushes up on his Christmas Carol repertoire, plans a ...
Penguino's user avatar
  • 14k
12 votes
0 answers
890 views

Dissecting a figure into 2, 3, 4, and 5 parts but not 6

This figure is divided in 2, 3 and 4 equal parts of same size and shape, but it is not possible to do it in 5 equal parts of same size and shape. Is it possible to find a figure that can be divided ...
Rodolfo Kurchan's user avatar
5 votes
2 answers
631 views

A piece of paper repeatedly cut into 8 or 12 pieces

You are given a piece of paper. It will be cut into 8 or 12 pieces. Each of those new pieces can be cut again into 8 or 12 pieces or left uncut. This process is (theoretically) repeated as often as ...
ThomasL's user avatar
  • 12.2k
2 votes
3 answers
604 views

Dissecting a figure into three congruent parts in three different ways

Figure 1 is divided in 2 equal parts of same size and shape in 3 different ways Figure 2 is divided in 3 equal parts of same size and shape in 2 different ways Is it possible to find a figure that ...
Rodolfo Kurchan's user avatar
11 votes
2 answers
494 views

Clash of the Robinsons

"Ridiculous!" you think "What can be the odds? Either I'm hallucinating or the amateur writing this story plunged to new depths of incompetence." Both being equally likely you don'...
loopy walt's user avatar
  • 21.3k
9 votes
2 answers
439 views

Ernie and the Cuboidal Crystals

When passing Ernie's letterbox this morning, I found a courier bag about the size of a shoe-box, resting inside (along with a smaller unlabelled bag). I immediately guessed that it was the special ...
Penguino's user avatar
  • 14k
12 votes
1 answer
423 views

Pythagorean triangle dissection

This is a variation of Pythagorean quilts. I will make it short, this time. Pythagoras's theorem also works for triangles. This leads to the following variation: Dissect the triangles of size 5 and ...
Florian F's user avatar
  • 30.6k
7 votes
1 answer
840 views

Cutting off one's nose to spite one's eyes

Disclaimer: to keep graphic depiction of gratuitous violence to a minimum the face to be spited has been deliberately kept abstract. You are required to further reduce any distress this puzzle may ...
loopy walt's user avatar
  • 21.3k
3 votes
1 answer
481 views

Using squares to prove e > 2.7

I loved this puzzle, so thought I'd submit a similar one: The definite integral $\int_{−\infty}^{1} \exp(x)\mathrm dx$ is equal to $e$ . Using two squares of side $1$ and one rectangle size $1\...
Carl Witthoft's user avatar
118 votes
3 answers
12k views

Prove that π > 3

It seems that once upon a time some politicians tried to pass a law fixing the value of π to be exactly 3. The idea being to "make math simpler so that our children can get better at math". ...
Florian F's user avatar
  • 30.6k
1 vote
1 answer
344 views

Rearranging the square

You are given a square piece of paper. You can cut it into pieces and rearrange them to form a new shape. You are allowed to rotate and flip pieces, provided that they are all used. Can you cut the ...
Dmitry Kamenetsky's user avatar
10 votes
5 answers
2k views

5 points on a ball, divide the ball into 2 halves so that one half as exactly 4 points

Suppose that you take a pen and mark five points on a ball. I claim that no matter where you draw those points, I can always slice the ball into two equal halves (two equal and closed hemispheres) ...
Hemant Agarwal's user avatar
3 votes
1 answer
306 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 ...
Florian F's user avatar
  • 30.6k
0 votes
1 answer
364 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 ...
Chris Steinbeck Bell's user avatar
-3 votes
1 answer
109 views

Minimize cuts to an object to obtain weights which can be used to measure 1-40kg

The problem is as follows: Rick has a small store and a two pan scale which allows him to weight the coffee he sells. Certain day he forgotten the weights that he uses in his truck, however in his ...
Chris Steinbeck Bell's user avatar
0 votes
2 answers
203 views

Nested six-point stars: least number of cuts to dissemble

The puzzle is as follows: The figure from below represents a peculiar structure which consists in congruent triangles whose sides intersect and is made of an iron wire. How many cuts passing through ...
Chris Steinbeck Bell's user avatar
1 vote
2 answers
287 views

Least cuts to get 44 rods from a metal grid

The puzzle is as follows: Suppose that you have a metal structure made by brass wire. Assuming that you must get 44 rods of the same size each. What is the least cuts to be made using an electric ...
Chris Steinbeck Bell's user avatar
1 vote
1 answer
231 views

Minimum cuts to make a rectangle into a square, allowing bending

The puzzle is as follows: Mike has a thin sheet of cardboard which is 96 centimeters large by 24 centimeters wide and a guillotine whose maximum cut length is 80 cm. Assuming this guillotine can cut ...
Chris Steinbeck Bell's user avatar
0 votes
0 answers
112 views

Bisecting a 3D object into two equal volume objects - 2

Given the following 3D object and means of an unmarked ruler to draw lines on its surface define a straight cut that will split it into two objects with the same volume. Hint: It seems to have at ...
Moti's user avatar
  • 2,239
9 votes
2 answers
492 views

Cutting a shape into two equal area shapes

Given the following shape - an hexagon ABCDEF of which a parallelogram CDGH is cut out. With a single cut divide the shape into two equal area shapes by means of an unmarked . You may draw lines and ...
Moti's user avatar
  • 2,239
7 votes
4 answers
647 views

Divided by Pie Squared. Aaahhh

I have a machine that can divide a square pie into 9 equal square pieces using 4 blades: The blades can be moved, but there is only one control - which defines the width of the blades in both ...
Lefty's user avatar
  • 1,174
4 votes
1 answer
256 views

Bisecting a 3D object into two equal volume objects

Given the following object - box of which a rectangular pyramid is removed. By means of unmarked ruler, draw lines on the surface of the object to guide cuts of the object into two objects with the ...
Moti's user avatar
  • 2,239
3 votes
1 answer
206 views

Aluminum Foil Folds and Cut

Your task is to convert a diamond shape monomino that is made from aluminum foil into an x-shape pentomino (see figures). You may fold the monomino and make one straight cut with a pair of scissors. ...
TSLF's user avatar
  • 6,656
1 vote
1 answer
158 views

Does a way to find the least amount of pizza cuts for a certain number of slices (11) exist?

I found this problem in a Logic and Reason book from 2000's which seems to be reprinted or adapted from Martin Gardner's book of recreational puzzles. In the beginning it seemed rather easy but I ...
Chris Steinbeck Bell's user avatar
1 vote
1 answer
126 views

Square forming challenge

Cut a square into 3 pieces Rearrange them anyway you want Cuts must be straight (but can begin/end anywhere). Scoring method 1: For each visible square 1 point For each cut -1 point Scoring method 2: ...
Retudin's user avatar
  • 9,278
8 votes
2 answers
419 views

Fit the board into the hole

How can you divide the board into exactly $2$ equally-shaped and sized pieces such that it fits the hole? Bonus: Can you do the same problem in $3$ pieces, such that one of the pieces of the board ...
Anonymous's user avatar
  • 1,754
1 vote
0 answers
146 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 ...
Black Mild's user avatar
3 votes
2 answers
201 views

Dissection puzzle from the Gardner's book. How to define points without tools?

The figure shows a fairly well-known puzzle from the book by Martin Gardner. You need to cut the regular hexagonal star into pieces and fold the square. Question: how to define points (marked in red) ...
Nick's user avatar
  • 1,695
9 votes
1 answer
2k views

How can the white cross be cut into 5 smaller pieces that can be resembled into the two smaller red crosses shown?

How can the white cross be cut into 5 smaller pieces that can be reassembled into the two smaller red crosses shown? Puzzle created by Henry Dudeney on The Strand Newspaper long time ago. Source: Saw ...
rash's user avatar
  • 191
9 votes
1 answer
256 views

Hollow Cube Cuts

The 2x2x2 inches seamless hollow cube with aluminum surfaces can be cut using a box knife. How to cut it into 4 pieces that can be bend to form smaller 1 inch cubes?
TSLF's user avatar
  • 6,656
2 votes
1 answer
116 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 ...
Kikos's user avatar
  • 338
8 votes
1 answer
330 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 ...
greenturtle3141's user avatar
18 votes
1 answer
464 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 ...
Florian F's user avatar
  • 30.6k
83 votes
5 answers
8k 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 ...
Mitsuko's user avatar
  • 2,035
2 votes
4 answers
649 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 ...
math scat's user avatar
  • 1,137
28 votes
5 answers
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 ...
Florian F's user avatar
  • 30.6k
6 votes
1 answer
250 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 ...
Mitsuko's user avatar
  • 2,035
9 votes
2 answers
691 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 ...
subdermatoglyphic's user avatar