All Questions
22
questions
5
votes
1
answer
156
views
Tile homotopy and T-tetromino packing of rectangles
From my old question (Which rectangles can be tiled with L-trominos, when only two orientations are allowed?), I learned a very interesting way to deal with tiling problems. I was wondering about T-...
10
votes
2
answers
481
views
Which rectangles can be tiled with L-trominos, when only two orientations are allowed?
This is a question that I got after reading this: https://www.cut-the-knot.org/Curriculum/Games/LminoRect.shtml. (This link already gave me the same result as theorem 1.1 of the article https://www....
2
votes
1
answer
288
views
Definition of aperiodic tiling
I think I got confused with the definition of aperiodic tiling. Look at the following example:
First, try to find a "1-dimensional aperiodic tiling". Start with the string 0, then make the ...
- 3,519
5
votes
2
answers
203
views
Edgematching tiles
Consider a 3×3 grid. Now, look at the patterns which generate 1 to 7 dots around the edges, taking into account rotations and reflections. Turns out there are 49 patterns, as seen in the set below ...
- 21.4k
8
votes
2
answers
203
views
Is it possible to cover an $11 \times 12$ rectangle with $19$ rectangles of $1 \times 6$ or $1 \times 7$?
Is it possible to cover an $11 \times 12$ rectangle with $19$ rectangles of $1 \times 6$ or $1 \times 7$?
Attempt:
There should be $132$ unit squares to be covered. Since there are $19$ rectangles ...
- 4,855
1
vote
1
answer
332
views
Generating all possible Domino tilings on a $4 \times 4$ grid
I have a task to write a program which generates all possible combinations of tiling domino on a $4 \times 4$ grid. I have found many articles about tilings, but it is for me quite difficult and I ...
- 11
4
votes
2
answers
625
views
Number of different fault-free $2 \times 1$ domino tilings on a $5 \times 6$ rectangle
Fifteen $2 \times 1$ dominoes can be used to tile a $5 \times 6$ rectangle. In tiling the rectangle we might generate what are known as fault-lines. A fault-line is any horizontal or vertical line ...
- 11.8k
4
votes
0
answers
206
views
Derive formula for number of tilings of an $m \times n$ board.
I have tried to find the derivation of the formula for the number of tilings of an $m \times n$ board with $2 \times 1$ tiles which is the following.
$$\prod_{k=1}^{m}\prod_{l=1}^{n} \left(4\cos^2{\...
- 275
1
vote
1
answer
646
views
Rectangling the rectangle
There is a classic problem of 'squaring the square', or constructing a perfect squared square, which is a unit square cut into a finite number of smaller squares whose sidelengths are all different. ...
- 3,343
3
votes
2
answers
950
views
Puzzle: Cut regular tetrahedron into distinct-sized regular tetrahedra?
The classic problem of constructing a perfect squared square, i.e. tiling a square with a finite number of squares of pairwise distinct sidelengths (which must be rational multiples of the tiled ...
- 3,343
12
votes
2
answers
350
views
Divide a square into rectangles where each occurs once in each orientation
A $26 \times 26$ square divides into different rectangles so that each occurs exactly twice in different orientations.
I've also found a solution for the $10 \times 10$ square, but no others. Are ...
- 21.4k
4
votes
0
answers
102
views
Sequential square plane-tiling patches
Using squares of size $1$ to $n$, make a patch that can tile the plane. Solutions for $n=2, 3, 4, 5$ are easy to find.
Example for 6:
Example for 7:
Example for 8:
Example for 9:
I ...
- 21.4k
7
votes
0
answers
266
views
Sequential square packings
There are various studies for packing sequential squares of size $1$ to $n$. We can try to find the smallest square they will pack into, as in tightly packed squares. We can find the smallest square ...
- 21.4k
6
votes
0
answers
217
views
Perfect Mondrian Triangle Dissections
In the Mondrian Art Problem, a square is divided into non-congruent integer-sided rectangles so that the largest area and smallest area are as close as possible.
A lattice square can be divided ...
- 21.4k
7
votes
2
answers
794
views
Same Diagonal Dissection
Divide a rectangle into smaller rectangles with two criteria:
All sub-rectangles must have different sizes.
All sub-rectangles must have diagonals with length 1.
What is the smallest possible ...
- 21.4k