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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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19 votes
2 answers
531 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-...
5 votes
2 answers
228 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 ...
11 votes
4 answers
998 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 ...
10 votes
2 answers
503 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 ...
18 votes
1 answer
1k views

The Challenge Square

Q: Can you divide this shape into 4 equal parts, and then form a square? This puzzle was published by Henry Dudeney who received the puzzle from Edward B. Escott.
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 ...
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 ...
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. ...
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) ...
32 votes
6 answers
4k views

Dissection Puzzle - The Umbrella Stand

You own a square-shaped table. You want to drill a small hole in the center to place an umbrella stand. Unfortunately, you're a little drunk: Alas. Fortunately, not all is lost. You are sober now, ...
7 votes
8 answers
7k views

Divide 3 cakes into 4 equal parts?

You have three cakes of diameter 20cm, 16cm and 12cm respectively as shown in the figure. They all have the same height. Now your task is to divide these 3 cakes into 4 equal parts and you are ...
12 votes
0 answers
887 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 ...
-1 votes
7 answers
562 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....
8 votes
2 answers
2k views

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

The figure below consists of 5 equal squares in the form of a Greek 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 ...
12 votes
3 answers
1k views

Dissecting a square

You are asked to dissect an $N \times N$ square into polyomino pieces such that each piece shares a portion of its boundary with exactly $D$ other pieces, and no piece has area exceeding $N$. This can ...
14 votes
2 answers
1k views

Find smallest rectangle divided into figures so each figure has 5 neighbours

The following 3x4 rectangle can be cut into pieces along grid lines, so that each piece has exactly three neighbors: Problem: Find the smallest rectangle on the integer grid that can be cut into ...
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?...
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\...
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 ...
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?
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". ...
16 votes
2 answers
10k views

Placing 2x1 dominoes on a chessboard with two corners removed

Suppose you have a checkerboard with two opposite corner squares removed, like this: Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
8 votes
2 answers
1k 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 ...
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?
7 votes
6 answers
626 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 ...
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 ...
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 ...
8 votes
1 answer
283 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 ...
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 ...
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 ...
9 votes
3 answers
446 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.)                   &...
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'...
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 ...
5 votes
1 answer
584 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-sized copies of itself.   Each copy is √2 = 1.414... times as large ...
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 ...
1 vote
1 answer
343 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 ...
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 ...
0 votes
1 answer
363 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 ...
-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 ...
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 ...
1 vote
1 answer
230 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 ...
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 ...
9 votes
2 answers
491 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 ...
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 ...
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 ...
8 votes
7 answers
12k views

Maximum number of pieces of equal area you can obtain by cutting a pizza a certain number of times

If I cut a perfectly circular pizza through its center 6 times at 30° angles, I get 12 pieces of equal area. If I don't have to cut through the center, I can cut in a grid shape to divide the pizza ...
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. ...
1 vote
1 answer
157 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 ...
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: ...
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 ...