Problems that involve partitioning a geometric figure into smaller pieces with certain conditions on them (equal area, equal shape, possible to be rearranged into another given figure, etc.)
Questions tagged [dissection]
68 questions
44
votes
1 answer
Cutting $n$ circular cakes of different radii but equal heights into $p>n$ equal shares
You probably know the following problem:
We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only cut one piece of one cake. What is the minimal…
Hippalectryon
- 7,830
40
votes
4 answers
Dividing an equilateral triangle into N equal (possibly non-connected) parts
It’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles.
But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an awesome way to do it — see the picture below (note…
Grigory M
- 18,082
32
votes
0 answers
How many "prime" rectangle tilings are there?
Given two tilings of a rectangle by other rectangles, say that they are equivalent if there is a bijection from the edges, vertices, and faces of the tilings which preserves inclusion. For instance, the following two tilings are equivalent (some…
RavenclawPrefect
- 18,328
28
votes
2 answers
Can a figure be divided into 2 and 3 but not 6 equal parts?
Is there a two dimensional shape (living in a plane) that can be divided into $2$ and $3$ but not $6$ equal parts of same size and shape? This question is a simpler take on this puzzling.SE question.
If such a shape exists, the $3$ parts can't be…
Eod J.
- 525
25
votes
1 answer
Do side-rational triangles of the same area admit side-rational dissections?
Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my question is as in the title:
Do any two rational…
Steven Stadnicki
- 53,147
19
votes
2 answers
Can an equilateral triangle be dissected into 5 congruent convex pieces?
There is a rather surprising dissection of an equilateral triangle into 5 congruent pieces:
However, these pieces aren't very "nice", consisting of 2 or 6 connected components depending on how one…
RavenclawPrefect
- 18,328
17
votes
2 answers
Splitting equilateral triangle into 5 equal parts
Is it possible to divide an equilateral triangle into 5 equal (i.e., obtainable
from each other by a rigid motion) parts?
Jaska
- 1,299
16
votes
2 answers
Union of two disjoint congruent polygons is centrally symmetric. Must the polygons differ by a 180 degree rotation?
Let $P$ be a polygon with $180^\circ$ rotational symmetry. Let $O$ be the center of $P$ and suppose $P$ is dissected into congruent polygons $A$ and $B$. Must the $180^\circ$ rotation around $O$ send $A$ to $B$?
This conundrum appears to have…
greenturtle3141
- 1,066
16
votes
2 answers
A regular tetrahedron can be dissected into $1,2,3,4,6,8,12,$ or $24$ congruent pieces. Is this it?
By placing a tetrahedron on a face and making vertical cuts centered at the "top" vertex, it is easy to dissect the tetrahedron into $1, 2, 3,$ or $6$ congruent pieces.
By cutting the tetrahedron into four identical pyramids meeting at the center,…
RavenclawPrefect
- 18,328
15
votes
0 answers
Dissection of equilateral triangle into square with flipping pieces - variation of Henry Dudeney's problem
Is it possible to divide the equilateral triangle into 4 pieces and assemble a square with these four pieces, provided that one or two pieces are flipped over to the other side? In other words, I want a mirror reflection of one or two pieces. If…
15
votes
2 answers
Generalizing the Borsuk problem: How much can we shrink a planar set of diameter 1 by cutting it into $k$ pieces?
Borsuk's problem asks whether a bounded set in $\mathbb{R}^n$ can be split into $n+1$ sets of strictly smaller diameter. While true when $n=1,2,3$, it fails in dimension $64$ and higher; I believe all other $n$ are open as of this writing.
However,…
RavenclawPrefect
- 18,328
14
votes
3 answers
Is it always possible to cut out a piece of the square with $\frac{1}{5}$ of its area?
Let there be a square that has $n+1$ notches on each edge (corners included) to divide each edge into $n$ equal parts. We can make cuts on the square from notch to notch. Is it always possible to cut out a connected piece with area $\frac{1}{5}$…
mathlander
- 4,097
11
votes
0 answers
Are there "close" solutions to Hilbert's third problem?
Hilbert's third problem (or a modern formulation thereof) asks whether two polyhedra $P,Q$ of equal volume are equidecomposable by cutting $P$ into finitely many polyhedral pieces and rearranging them to obtain $Q$. (In the two-dimensional case,…
RavenclawPrefect
- 18,328
10
votes
1 answer
Is there only one way to divide an equilateral triangle into congruent fourths?
Suppose we wish to divide an equilateral triangle into fourths, such that each piece is congruent. (Let's also require connectedness.) One way to do this is to connect the medians, forming one inverted triangle in the center and three at the…
Akiva Weinberger
- 25,412
10
votes
1 answer
Transforming a 8x8, 4x4 and 1x1 square into a 9x9 square
Good day to all of you!
I have a puzzle which I just cannot solve. I attached a photo of it. The task is to transform the shape on the left into a 9x9 square (on the right) using ONLY 2 "cuts" - dividing it into 3 separate objects. Mirroring,…
