Questions tagged [tiling]
36 questions
26
votes
2 answers
Is Dominosa NP-Hard?
Dominosa is a relatively new puzzle game. It is played on an $(n+1)\times(n+2)$
grid. Before the game begins, the domino bones $\left(0,0\right),\left(0,1\right),\ldots,\left(n,n\right)$
are placed on the grid (constituting a perfect tiling).…
Yoav bar sinai
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19
votes
3 answers
How many cookies in the cookie box? -- Tiling stars
With holiday season coming up I decided to make some cinnamon stars. That was fun (and the result tasty), but my inner nerd cringed when I put the first tray of stars in the box and they would not fit in one layer:
Almost! Is there a way they could…
Raphael
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19
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1 answer
Are 'zero-one' jigsaw puzzles NP-complete?
I'm interested in a slight variant of tiling, the 'jigsaw' puzzle: each edge of a (square) tile is labeled with a symbol from $\{1\ldots n, \bar{1}\ldots\bar{n}\}$, and two tiles can be placed adjacent to each other iff the symbol on one tile's…
Steven Stadnicki
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13
votes
2 answers
Tiling an orthogonal polygon with squares
Given an orthogonal polygon (a polygon whose sides are parallel to the axes), I want to find the smallest set of interior-disjoint squares, whose union equals the polygon.
I found several references to slightly different problems, such as:
Covering…
Erel Segal-Halevi
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7
votes
0 answers
Algorithms to generate random nowhere-neat rectangulation?
I want to generate random rectangular partition of a given $m*n$ rectangle under the constraint that it must be nowhere-neat partition. Nowhere-neat partition means that a dissection of a rectangle into smaller rectangles such that the original…
Mohammad Al-Turkistany
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5
votes
1 answer
How to convert a Turing Machine program to a tiling using Wang Tiles?
This is a cross-post from a post on MathSE due to lack of answers.
To illustrate my question I provide the following example.
The website Online Turing Machine provides a Turing Machine simulator. The following program adds 1 to any binary…
nilo de roock
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5
votes
1 answer
What is the minimum square partition of an almost-square rectangle?
This question is motivated by an older question about tiling an orthogonal polygon with squares.
It is a generalisation of my former question about how to prove that the minimum square partition of a 3X2 rectangle has 3 squares).
Let:
An…
Realz Slaw
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5
votes
1 answer
Efficient algorithm to compute the Heesch number of a shape
The Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it. For example the following shape (in the center) has a Heesch number of 4, because we can get at most 4 layers to surround it:
Given a…
Dmitry Kamenetsky
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5
votes
4 answers
3xN tiling problem with blocks of size 3x1 or 2x2
I know there are a number of different tiling problems and some of them have been discussed here:
Number of ways of tiling a 3*N board with 2*1 dominoes problem
Domino and Tromino Combined Tiling
DP tiling a 2xN tile with L shaped tiles and 2x1…
Amir
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4
votes
1 answer
Domino and Tromino Combined Tiling
If I have a nx2 grid which I need to fill using 2x1 dominoes and L shaped trominoes in any combination, how many different combinations are possible?
I am aware that when only 2x1 dominoes are used then the cell definition using dynamic programming…
Roshan SA
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3
votes
1 answer
DP tiling a 2xN tile with L shaped tiles and 2x1 tiles?
https://www.iarcs.org.in/inoi/online-study-material/topics/dp-tiling.php
The second question in the above link requires us to fill an 2xN grid with tiles of dimension 2x1 and an L shaped tile.
Question 1) In the recursive formula for g(n):
g(n) =…
Dhruv Chadha
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3
votes
1 answer
Domino tiling of a 2xN rectangle in O(ln n)
I solved this problem using Dynamic Programming in $\mathcal{O}(n)$ time. I found that is equivalent to the Fibonacci Numbers.
$F(0) = F(1) = 1$
$F(n) = F(n-1)+F(n-2)$
Where the $F(n-1)$ term is from fixing the left most domino vertically, and the…
Black Arrow
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3
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1 answer
For tiling simply connected regions with shapes beyond just rectangles, is there a lower # of tile shapes needed for NP-completeness?
In "TILING SIMPLY CONNECTED REGIONS WITH RECTANGLES" by Igor Pak and Jed Yang, they show there is a set of "no more than $10^6$ rectangles" such that the problem of tiling an arbitrary simply connected region with these rectangle shapes is…
user2566092
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3
votes
1 answer
n-polygon lattice datastructure?
I'm trying to simulate a boardgame what can be played on a board with an arbitrary lattice, anything from triangles to heptagons to 37-sided regular polygons is allowed. Moreover the shape of the board is also arbitrary.
I need to represent this…
yellowbird
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3
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0 answers
Packing rectangles to generate a sprite sheet
I am writing a sprite sheet generator tool in adobe AIR, and I have to force with the question: How to pack a collection of 2D rectangles to smallest possible 2D rectangle with power of two. (like 1024x444).
I think that this question is NP-Hard. I…
Ilya Gazman
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