84
votes
Is it possible to assemble copies of this shape into a cube?
Assemble four copies of the shape as shown.
Four copies of the assembly create a 6x6x4 cuboid, which can be used to create a 12x12x12 cube.
- 5,493
60
votes
Accepted
Can squares of infinite area always cover a unit square?
We actually only need $\sum a_i^2 >4$ for this to hold*. In particular, note that this condition implies that there is some finite set $I$ of the indexing subset such that $\sum_{i\in I}a_i^2>4$....
- 61.1k
37
votes
Accepted
Hexagons are best for tiling 2D space in terms of perimeter vs area. What's best for 3D space?
This is known as the Kelvin problem; the best known (and conjectured optimal) solution is the Weaire–Phelan structure, but proving this is likely very very hard. I don't know what the best results in $...
- 17.7k
36
votes
Can any number of squares sum to a square?
"Is any $k$ possible?" An easy route to "Yes": You know from the Pythagorean theorem that two squares can add to a perfect square. $$c^2=a^2+b^2$$
$c^2$ must be either odd or even. If odd, it is the ...
- 7,388
35
votes
Accepted
Can a row of five equilateral triangles tile a big equilateral triangle?
I suppose I should post: I solved this on MathOverflow. The answer is YES: a size-45 triangle can be tiled.
I thank two insights from Josh B here: first that a rhombus with side length 15 can be ...
- 4,582
35
votes
Accepted
Convex polygons that do not tile the plane individually, but together they do
There is a tiling of the plane made from regular heptagons and irregular pentagons.
We know that regular heptagons cannot tile the plane.
The irregular pentagon has four equal sides and one shorter ...
- 4,146
34
votes
Can any number of squares sum to a square?
Yes, $k$ can be arbitrary. Define a sequence$$a_1:=3,\,a_{k+1}:=\frac12\left(a_k^2+1\right)$$ of odd positive integers (since $\frac12((2n+1)^2+1)=2(n^2+n)+1$), so$$\begin{align}a_{k+1}^2-a_k^2&=\...
- 116k
31
votes
How few disks are needed to cover a square efficiently?
If different circle sizes are allowed, we may reach an efficiency $>76.3\%$ with just five disks:
With $13$ disks the maximum efficiency is already $>80.4\%$. It is enough to replace each "...
- 355k
28
votes
Accepted
How few disks are needed to cover a square efficiently?
With a bit of Googling I found this paper: Covering a Square with up to 30 Equal Circles, by Kari J. Nurmela and Patric R. J. Östergård.
They used a computer search to find coverings of a square by $...
- 13.8k
25
votes
Accepted
Can any number of squares sum to a square?
This holds far more generally. OP is the special case $S$ = integer squares, which is $\rm\color{#0a0}{closed}$ under multiplication $\,a^2 b^2 = (ab)^2,\,$ and has an element that is a sum of $\,2\,$ ...
- 275k
23
votes
Can any number of squares sum to a square?
Here is a geometric solution (for $k > 5$).
The following are solutions for 6, 7, and 8 squares.
We can replace a square in each of these with four equal size squares to find a tiling with 3 ...
- 3,708
21
votes
Accepted
Tiling a square with rectangles whose sides are powers of two
MAJOR UPDATE:
What I am about to show is not a proof for the minimum number of rectangles. However, in some cases I found the number of rectangles can be less than $f(n)^2$. The smallest $N×N$ grid ...
- 2,624
20
votes
Accepted
How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?
I'm posting a new answer to this question, because the techniques I'm using differ substantially from the previous answer and it was already getting quite long. (Much of this answer was written prior ...
- 17.7k
18
votes
Can a row of five equilateral triangles tile a big equilateral triangle?
Here's the minimal solution, a side-30 triangle. Also posted to MathOverflow here https://mathoverflow.net/questions/267095/can-a-row-of-five-equilateral-triangles-tile-a-big-equilateral-triangle.
- 517
18
votes
How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?
From the OP, I'm using the fact that we can use $79$ triangles to tile a trapezoid with side lengths $11,1,10,1$ and angles of $60$ and $120$ degrees, as well as the parallelogram with side lengths $1$...
- 17.7k
17
votes
Accepted
What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?
Here's my solution with 32 triangles.
How
First, I find all polygons that can be created by attaching the 45-60-75 triangle to a copy of itself, such that an edge coincides. There are twelve unique ...
- 1,148
17
votes
Accepted
Can we form a rectangle with integral lengths using an odd number of copies of this domino?
You can use $11$ copies of that hexomino to make a $6\times 11$ rectangle:
This seems to be the smallest solution. A $5\times 18$ rectangle can be seen in Jean Marie's answer, and here is a $9\times ...
- 13.8k
16
votes
Can squares of infinite area always cover a unit square?
An answer exploiting the idea of Will Jagy outlined in the comments.
If there is some $a_n\geq 1$ there is nothing to prove, so we may assume $0< a_n <1$. There is nothing to prove also if $\...
- 355k
16
votes
Accepted
Are there polyominoes that can't tile the plane, but scaled copies can?
We can achieve this using two polyominoes with one having double the dimensions of the other and the second copy rotated $90^\circ$ and reflected.
The basic element showing the two polyominoes is ...
- 4,146
16
votes
Convex polygons that do not tile the plane individually, but together they do
HINT:
Consider a convex hexagon that can tile the plane. There are three types of tiling hexagons, we take one of type 1, which has two opposite sides parallel and equal
Cut it into two pentagons. ...
- 55k
15
votes
Accepted
Four Dragon Curves are Edge-covering/Plane-tiling
Here’s the substitution rule that generates the dragon curve. It replaces each arrow with two smaller rotated arrows with particular orientations.
If we apply the same substitution rule to this ...
- 8,367
15
votes
Is there a simple perfect squaring of a 1366 by 768 rectangle?
I looked through some of the data on Squaring.Net (actually, copies at the Internet Archive, since the site is currently down), and there is no $1366\times768$ simple perfect squared rectangle of ...
- 4,368
15
votes
How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?
Here’s a trapezoid of ratio $1$ tiled by $195$ triangles, found in a brute-force search. Using three of these to build an equilateral triangle takes $3 \cdot 195 = \mathbf{585}$ triangles.
Old ...
- 8,367
14
votes
Tiling a cylindrical piece of paper
It is impossible :
The number of squares in cylinder is $50^2$
And we color black or white in them, like chess board
Hence at block in b) we have two coloring ways : 3 black and 1 white, 1 black ...
- 20.1k
14
votes
Tiling the plane with consecutive squares
Here is a solution with $n=7$. Edit. I added some comments expressing my belief that $n=7$ may well be the largest solution. (Well, I keep editing my answer, but at the end I came up with a solution ...
- 13.5k
14
votes
Accepted
Is it possible to tile the hat polykite with smaller copies of itself?
No, it is impossible, even allowing different sized hats.
Every internal angle of the hat has a measure of at least $90^\circ$, and there are exactly four right angles, marked A, B, C, and D in the ...
- 78.3k
13
votes
Accepted
Is it possible to cover an $11 \times 12$ rectangle with $19$ rectangles of $1 \times 6$ or $1 \times 7$?
$$\matrix{\color{red}1,0,0,0,0,0,0,\color{red}1,0,0,0,0\\0,\color{red}1,0,0,0,0,0,0,\color{red}1,0,0,0\\0,0,\color{red}1,0,0,0,0,0,0,\color{red}1,0,0\\0,0,0,\color{red}1,0,0,0,0,0,0,\color{red}1,0\\0,...
- 90.7k
12
votes
Accepted
Golden Rectangle into Golden Rectangles
Impossible w/ Rectangles Alone
I believe what you're asking for is impossible using golden rectangles alone. A walk-through of possible horizontal-vertical orientations of the inserted pieces (along ...
- 676
12
votes
Accepted
Rolling icosahedron Hamiltonian path
I solved the icosahedron. There are two nice closed curves that will put a rolling icosahedron through all 120 orientations if the curves are repeated 5 times. The outside edges of the graph are the ...
- 21.4k
Only top scored, non community-wiki answers of a minimum length are eligible