0
votes
1answer
26 views

Tiling an $m\times n$ grid.

For natural numbers $m$ and $n$, an $m\times n$ grid of squares can be tiled with tiles of the form completely filling the grid, without overlapping, if and only if $m,n\geq2$ and $6\mid mn$. It ...
1
vote
1answer
36 views

How to prove that a particular polyiamond tiles the Euclidean plane?

I read that among the 24 heptiamonds there is one piece that does not tile the Euclidean plane. My question is the following, given a particular polyiamond how do you prove that the piece does tile ...
1
vote
2answers
72 views

Tiling a $23 \times 23$ square by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles

A $23 \times 23$ square is tiled by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles. Prove that at least one 1 x 1 tile must be used. Find such a tiling with exactly one $1 \times 1$ tile. Hint: ...
2
votes
1answer
36 views

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4x3 and 5x4 rectangles, What is ...
2
votes
1answer
165 views

Minimum square partitions for 4x3 and 5x4 rectangles

Motivation: Tiling an orthogonal polygon with squares Followup question: What is the first $w$ such that a rectangle, $R_{w\times w−1}$ is minimally-square-partitioned by less than $w$ squares.. Yes, ...
3
votes
0answers
160 views

Squaring rectangles

it is a nice high-school exercise to prove that a square can be tiled with n squares if and only if n=1, 4 or is any integer greater or equal to 6. A direct consequence is that any rectangle that can ...
0
votes
0answers
71 views

Elementary proof that the complexity function of the Fibonacci word is $p(n)=n+1$

The Fibonacci word $f$ (or Fibonacci tiling) is a bi-infinite sequence (with prescribed origin) over the alphabet $\{0,1\}$ which is the unique sequence fixed by the substitution $1\mapsto 10, ...
10
votes
1answer
182 views

$2013\times2013$ Board with no trominoes.

Let A be a $2013\times2013$ board with $k$ black squares and containing no $L$ shaped black trominoes(in any rotation) and such that if any white square is dyed black then $A$ contains a black $L$ ...
-2
votes
1answer
169 views

Proof a $2^n$ by $2^n$ board can be filled using L shaped trominoes and 1 monomino

Suppose we have an $2^n$x$2^n$ board. Prove you can use any rotation of L shaped trominoes and a monomino to fill the board completely. You can mix different rotations in the same tililng.
0
votes
1answer
119 views

“Simmetric” connected k-regular bipartite graph

Let $G$ be a k-regular bipartite graph with $k > 0$. Then it is known that the two sets which partition the vertex set of $G$ have the same cardinality. However I am interested in connected ...
14
votes
3answers
1k views

Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.

Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that ...
1
vote
1answer
91 views

Tiling a minimal perimeter region with $n$ unit squares

Suppose I have $n$ identical unit squares and I want to use them all to tile a region with minimal perimeter $p(n)$. For instance I guess $p(n^2)=4n$, by arranging them im a $n\times n$ square. Is ...
6
votes
2answers
292 views

Tiling pythagorean triples with minimal polyominoes

Given a Pythagorean triple $(a,b,c)$ satisfying $a^2+b^2=c^2$, how to calculate the least number of polyominoes of total squares $c^2$, needed, such that both the square $c^2$ can be build by piecing ...
28
votes
1answer
1k views

Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
1
vote
1answer
211 views

Cube nets hexomino tilings.

I am looking for an ~12x12 rectangle (small holes and small obtrusions are okay) made entirely of cube net hexominos. It is my understanding that perfect rectangles, in general, are not possible ...
1
vote
3answers
371 views

Tiling with polyominos

How to prove or disprove that if a polyomino tiles the plane, it must also be able to perfectly tile some larger polyomino, which also tiles the plane? A polyomino is finite set of unit squares ...
2
votes
1answer
140 views

Optimal polyomino induced coloring

Which polyominos (with orientation) of $n$ squares, requires the least number of different colors, $c(n)$, such that if this polyomino is placed anywhere on an optimally colored infinite square grid ...
6
votes
2answers
413 views

Checkerboard-Coloring $\mathbb{Z}^2$

If every square of the unit square lattice in the plane is colored black or white according to a set of rules, is there a way to find the maximum asymptotic ratio $r_n$ of the number of black squares ...
2
votes
3answers
728 views

Summation in a recurrence relation

edited to reflect advice from the comments: While working on a generalization of a tiling problem, I generated a recurrence relation to describe the total number of possible tilings. The relation ...
10
votes
3answers
411 views

a tiling puzzle/question

My teacher gave us a riddle that goes like this: You have a 7x7 square and 16 3x1 tiles. Of the 16 tiles, 15 are straight and 1 is crocked ("L" shaped). When you tile the square with these tiles you ...
10
votes
2answers
1k views

Tiling a 3 by 2n rectangle with dominoes

I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences ...