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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13 votes
2 answers
435 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?
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7 votes
6 answers
595 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 ...
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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 ...
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7 votes
1 answer
189 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 ...
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10 votes
0 answers
437 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 ...
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5 votes
2 answers
541 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 ...
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2 votes
3 answers
456 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 ...
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10 votes
2 answers
467 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'...
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9 votes
2 answers
325 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 ...
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  • 12.4k
10 votes
1 answer
340 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 ...
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6 votes
1 answer
814 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 ...
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3 votes
1 answer
373 views

Using squares to prove e > 2.7

Edited to replace $\exp(-x)$ with $\exp(x)$. My apologies. I loved this puzzle, so thought I'd submit a similar one: The definite integral $\int_{−\infty}^{1} \exp(x)dx$ is equal to $e$ . Using ...
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106 votes
2 answers
10k 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". ...
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1 vote
1 answer
241 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 ...
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9 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) ...
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1 vote
0 answers
185 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 ...
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0 votes
1 answer
196 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 ...
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-3 votes
1 answer
93 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 ...
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0 votes
2 answers
190 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 ...
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1 vote
2 answers
275 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 ...
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1 vote
1 answer
152 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 ...
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0 votes
0 answers
100 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 ...
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9 votes
2 answers
315 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 ...
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7 votes
4 answers
618 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 ...
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4 votes
1 answer
234 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 ...
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  • 2,166
3 votes
1 answer
171 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. ...
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  • 6,596
1 vote
1 answer
113 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 ...
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1 vote
1 answer
113 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: ...
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  • 4,767
8 votes
2 answers
375 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 ...
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  • 1,744
1 vote
0 answers
137 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 ...
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3 votes
2 answers
168 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) ...
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8 votes
1 answer
1k 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 ...
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  • 181
9 votes
1 answer
235 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?
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  • 6,596
2 votes
1 answer
111 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 ...
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8 votes
1 answer
234 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 ...
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17 votes
1 answer
384 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 ...
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77 votes
5 answers
7k 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 ...
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  • 1,597
2 votes
4 answers
631 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 ...
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  • 1,149
26 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 ...
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5 votes
1 answer
186 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 ...
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  • 1,597
9 votes
2 answers
638 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 ...
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12 votes
2 answers
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 ...
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  • 1,597
14 votes
3 answers
913 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. ...
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  • 21.3k
19 votes
1 answer
905 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 ...
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  • 66.2k
26 votes
1 answer
590 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 ...
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  • 263
13 votes
4 answers
1k 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. ...
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  • 411
34 votes
1 answer
3k views

Dissecting the exotic bulbfish

Can you cut the following black shape into exactly three pieces, and then rearrange those pieces into a square?
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0 votes
2 answers
120 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 ...
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0 votes
2 answers
164 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?
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  • 6,596
13 votes
1 answer
857 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 ...
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