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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Cutting a 27×27 square into incomparable rectangles
How can I cut a 27×27 square into 8 incomparable rectangles?
A rectangle with width w and height x is incomparable with a rectangle of width y and height z iff $w<y\land x>z$ or $w>y\land x&...
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Combine 8x8 square grid and 15x15 square grid into 17x17 square
You have an $8\times8$ square grid and a $15\times15$ square grid. You want to dissect them and combine them into a $17\times17$ square. What is the minimum number of pieces required?
Only cuts along ...
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Dissection: $7^2 + 1^2 = (5\sqrt{2})^2$ and related problems
What is the minimum number of pieces needed to dissect a $7 \times 7$ square, plus a $1 \times 1$ square tile, to form a $5\sqrt{2} \times 5\sqrt{2}$ square? The $7 \times 7$ may only be cut by line ...
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7x10 floor and a 8x8 and a 6x1 carpet, only one cut allowed
We have a 8x8 carpet, we can do only one cut of any complexity and also we have an uncuttable 6x1 carpet. We need to fill a 7x10 space using those 2. The 8x8 can only be cut into two pieces.
We have ...
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Dividing shape into 2 congruent pieces
How do I divide this shape into 2 congruent polygons?
Source: Penrose Tiles to Trapdoor Ciphers...and the Return of Dr. Matrix by Martin Gardner, Chapter 12 (no solutions in the book though).
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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-...
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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 ...
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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 ...
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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 ...
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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.
...
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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....
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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?...
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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 ...
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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?
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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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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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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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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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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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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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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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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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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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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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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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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\...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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