All Questions
Tagged with co.combinatorics tiling
94 questions
2
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Counting problem, tiling rectangle with two types right isosceles triangle
How many ways are there to tile a rectangle of size $m\times n$ with two types of isosceles triangle, type 1 having area $\frac{1}{2}$ and type 2 having area 1?
I know with only type 1 there are $2^{...
CommunityBot
- 1
6
votes
5
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543
views
Tiling with ten-fold symmetry and (unoriented) Penrose tiles?
Consider tilings of the plane made out of rhombi of side 1 and either angles $\pi/10$ and $2\pi/5$ or angles $\pi/5$ and $3\pi/10$. If we give a certain orientation to the edges and respect that ...
- 2,370
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0
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292
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Tilings in finite (not necessarily Abelian) groups
Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that
$$ G = \bigsqcup_{b\in B} bA.$$
...
- 1,354
1
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1
answer
138
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Recognizability/unique composition property for substitution tiling
This may be a very basic question, but I have not found an answer to it so far in my search. The question is whether there is an "algorithmic" way to check unique-composition/recognizability ...
CommunityBot
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1
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Looking for an efficient way of maximising 'pair scores' for subset of 30 selected from 50 to 10 000 objects
Context: I have a tiling program that uses a directed breadth first search algorithm. It is 'directed' by what I call 'pair scoring'. There are $N$ polyforms (pieces) used in the tiling. I have an $N\...
CommunityBot
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2
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2
answers
227
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On cutting tetrahedrons into mutually congruent pieces
Simple observations: A regular tetrahedron can be cut into 2 mutually congruent pieces (in 3 obvious ways which are all basically the same way, giving one and same pair of congruent pieces). The ...
- 6,101
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1
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98
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Chromatic tiling complexity and the chromatic number conjecture
Let $T$ be a finite set of tiles in $\mathbb{R}^d$. A tiling of $\mathbb{R}^d$ by $T$ is a collection of disjoint translates of tiles in $T$ whose union is $\mathbb{R}^d$. A tiling is $k$-chromatic if ...
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2
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0
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62
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On convex polygons that can be cut into convex and mutually congruent pieces in exactly one way
Observations: any thin isosceles triangle has exactly 1 partition into 2 congruent pieces - only 1 line, bisector of its apex, does it.
By attaching a right triangle with base 1 and altitude 2 to an ...
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79
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Are there triangles that can be cut into 7 mutually congruent connected polygons?
First question below had appeared in a note at Triangles that can be cut into mutually congruent and non-convex polygons
Following the results of Beeson quoted in the answer at Subdivision of ...
- 5,979
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Tiling rectangle with trominoes — an invariant
There are two types of trominoes, straight shapes and L-shaped. Suppose a rectangle $R$ admits at least one tiling using trominoes, with an even number of L-trominoes.
EDIT: we do not admit ALL ...
CommunityBot
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0
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138
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Hyponontiling Wang tiles
Call a finite collection of tiles that can tile the plane if we have to use each tile at least once tiling.
Is there a collection of at least 3 tiles that is not tiling, but such that after removing ...
- 18.7k
15
votes
4
answers
887
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Tiling a rectangle with all simply connected polyominoes of fixed size
For which values of $n$ can we tile some rectangle with one copy of each free simply-connected $n$-omino (that is, each polyomino with $n$ squares that has no holes)?
It appears that it is possible ...
- 4,713
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0
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175
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Can a square be partitioned into mutually non-congruent triangles all of same area and perimeter?
It is known that the plane cannot be tiled by pair-wise non-congruent triangles all having same area and same perimeter (https://arxiv.org/abs/1711.04504).
Question: Can a square be partitioned into ...
- 5,979
95
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5
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4k
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Can a row of five equilateral triangles tile a big equilateral triangle?
Can rotations and translations of this shape
perfectly tile some equilateral triangle?
I originally asked this on math.stackexchange where it was well received and we made some good progress. Here's ...
- 82.7k
12
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0
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168
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Can the optimal packing density in $\mathbb{Z}^d$ be irrational?
For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
- 1,209
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2
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148
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Decidability of (restricted) periodicity of Wang tilings
Consider a Wang tiling (given a subset of $C^4$ for a finite set $C$ of colours, e.g.). It's well-known to be undecidable whether there exists a tiling, and also whether there exists a periodic tiling....
- 63.9k
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1
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212
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Does $\mathbb{Z}\times\mathbb{Z}$ have an aperiodic monotile?
For any set $S\subseteq \mathbb{Z}\times\mathbb{Z}= \mathbb{Z}^2$ and $a\in \mathbb{Z}^2$, we set $a+S = \{a+s: s\in S\}$, where $+$ is the componentwise addition in $\mathbb{Z}^2$. Moreover, for any ...
- 6,652
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2
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Does any set of dominoes tile some common figure?
Let $D_1,\dots,D_n \subset \mathbb{Z}^2$ be two-point sets, i.e. 'dominoes' (unlike common dominoes, these are not necessarily connected, but I couldn't come up with a better name).
Does there always ...
- 1,375
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1
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397
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How much of an aperiodic tiling is needed to force aperiodicity?
Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
- 13.4k
3
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2
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397
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An "incomplete" tiling?
Given an $m\times n$ chess board, we place $p$ $2\times 1$ dominoes on the board so that they don't overlap. How many ways can we place them?
When each square of the board is covered by a domino this ...
- 82.7k
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4
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2k
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Rhombus tilings with more than three directions
The point of this question is to construct a list of references on the following subject: Fix vectors $v_1$, $v_2$, ..., $v_g$ in $\mathbb{R}^2$, all lying in a half plane in that cyclic order. I am ...
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1
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For which $n$ does a y-formed $n$-polyomino tile a $n \times n \times n$-cube?
I got from my children as a gift a puzzle consisting of 25 y-shaped 5-polyominoes that form a $5 \times 5 \times 5$-cube (see picture).
I'm wondering for which $n$ does a y-formed $n$-polyomino tile a ...
- 22.5k
1
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2
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329
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Sufficient conditions for periodic tiling by Wang tiles
I'm recently interested in whether a sub-shift of finite type contains a doubly-periodic problem, when the set of configurations is of the sort $\mathcal{A}^{\mathbb{Z}^2}$. When $Q_2=\{0,1\}^2$, and ...
- 633
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657
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Unique domino tiling
Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset $S$ of the $xy$-plane is star-convex if there ...
- 63.9k
3
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1
answer
230
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Tiling planar integer lattice by finite point sets
I am interested in the following question.
Are there nice characterizations of the finite sets $S\subseteq \mathbb{Z}\times\mathbb{Z}$ that tile $ \mathbb{Z}\times\mathbb{Z}$ by translations (i.e. $\...
- 3,068
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209
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Aperiodic tilings of the plane by squares and rhombi
Consider tilings of the plane by unit squares and by rhombi of unit side length and angles $\pi/3$, $2\pi/3$. It is easy to come up with periodic tilings of the plane - consider the following:
(from ...
- 20.2k
2
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0
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171
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Square-and-equilateral-triangle aperiodic tiling with $\leq 4$ prototiles?
There exist aperiodic tilings composed of square and equilateral-triangle tiles of unit side length: see https://tilings.math.uni-bielefeld.de/substitution/square-triangle/ and https://hal.archives-...
- 20.2k
5
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2
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382
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What is known about tiling a rectangle in an irreducible way by smaller rectangles?
Given two naturals $s<t$. Is there always a square (or at least a bigger rectangle) that can be tiled with $s\times t$ rectangles in an irreducible way (i.e. any grid line splitting it cuts at ...
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1
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Polyomino that can tile itself
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ ...
- 1,462
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0
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108
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number of ways to cover an $m × n$ rectangle
Given a positive integer $k\ge2$, let be $f_k(m,n)$ the number of ways to cover an $m × n$ rectangle with $mn/k$ tiles ( $1×k$ or $k×1$)
$f_2(m,n)$ is kasteleyn formula
$f_k(m,n)$?
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1
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Maximum number of colors for an optimal tiling which guarantees infinite paths
This question is a more applicable version of the question I've asked in mathexchange recently:
What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square
block ...
- 4,219
21
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1
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Monomer-Dimer tatami tilings need better relationships with other math. Summary of results
A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the tatami condition if no four tiles meet at any point. (Or you can think of it as the removal of a matching from ...
- 857
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0
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104
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Three dimensional Cairo Pentagonal Tiling
The Cairo pentagonal tiling is an interesting tessellation of the two-dimensional plane by irregular pentagons, which is given by taking two irregular hexagonal tilings, congruent but perpendicular to ...
- 121
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1
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Is there a formula for a number of one-sided N-ominoes?
As we all know, Polyominoes are shapes which consist of certain number of squares connected together. A famous videogame - Tetris - has a gameplay based around tetraminoes - polyominoes with 4 squares ...
- 151k
2
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1
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What is the average component size of a coloring?
Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
- 4,589
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2
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308
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For what n and t can a square be partitioned into n similar rectangles in t congruence classes?
It is known that a square can be partitioned into three similar rectangles, all mutually non-congruent. I don't think it's possible with four. With what numbers of rectangles can this be achieved? And ...
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Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
It is a nice exercise with rational generating functions (or equivalently, ...
- 32.1k
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6
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4k
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Does every polyomino tile R^n for some n?
This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer.
A polyomino is usually defined to ...
- 1,944
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0
answers
92
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Possible cardinalities of spherical tiling
Suppose that we have a tiling of $n$-dimensional (I want to get answer for $n = 4$, but general result would be nice!) sphere by isometric tiles strictly contained inside the right-angled simplex. ...
- 13.6k
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83
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Distance spectra of uniform tilings
Let a uniform tiling be defined by a vertex configuration $(n_1.n_2.\cdots.n_k)^m$, which is either spherical, Euclidean or hyperbolic. Assume that the tiling is vertex-transitive, especially that ...
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1
answer
202
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Overlaying two domino-like constructions such that all individual pairs of domino-like cells in the overlay have matching symbols
Imagine I have two $n$ x $m$ assemblies of $P = (n*m)$ unit square cells on the plane, $(c_{(a,1)}, ..., c_{(a,P)}) \in A$ and $(c_{(b,1)}, ..., c_{(b,P)}) \in B$, where every cell, $c_k$, in a ...
- 63.9k
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Triangling the triangle
Is it possible to tile an equilateral integer-sided triangle with smaller equilateral integer-sided triangles, with no congruent triangles touching? This has been answered in the negative for the case ...
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0
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142
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Tradeoffs in translation-invariant tilings of $\mathbb{R}^3$
Suppose I tile $\mathbb{R}^3$ in a ($\mathbb{Z}^3$-)translation-invariant manner. If we insist on the tiling being regular, then we are left with only the cubic tiling. However, suppose that we ...
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7
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1
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354
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Changing tiles by swapping squares
In an $n\times n$ table, initially there is a $1\times n$ tile in each row. A swap is an operation that involves choosing two tiles, move one square from the first to the second tile, and ...
- 2,671
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0
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137
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Aperiodic tile with rational area
Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
- 745
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2
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454
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Random Walk on Pentagonal Tiling
I’ve recently been looking at closed walks on tilings of the plane in which the “player” can move from one tile to any of its edge-adjacent neighbors. In particular, I’m trying to find asymptotic ...
- 14.2k
7
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3
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981
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Tiling a square with rectangles whose areas or perimeters are 1, 2, 3, ..., N
For which positive integers N does there exist a square that can be completely tiled with N rectangles of integer sides whose areas or perimeters are precisely 1, 2, 3, ..., N?
- 613
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0
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Periodic tilings of the plane with fundamental domain given by $k$ squares of prescribed size
Given $k$ strictly positive real numbers $l_1,\dots,l_k$, can one decide the existence of a periodic tiling of the plane whose fundamental domain is the union of $k$ squares
of length $l_1,\dots,l_k$?...
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0
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157
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Tiling rectangles using all squares of sides 1, 2, 3, ..., n
Do integers n greater than 2 exist such that all the squares of sides 1, 2, 3, ..., n can be partitioned into two or more sets (none a singleton) each of whose squares can be used to tile a rectangle?
- 3,707
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Hooks, monomers, dimers and Young diagrams: Part II
As promised, I've upgraded my last question.
Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, ...
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