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[1803.08530] A Coloring Book Approach to Finding Coordination Sequences
An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. We illustrate the method by first applying it to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-3^ tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8 ,12, 16, ..., the same as for a vertex in the familiar square (or 4^4) tiling. We thought that such a simple fact should have a simple proof, and this article is the result. We also apply the method to obtain coordination sequences for the 3^,, 4.8^2, 3.12^2, and 3^4.6 uniform tilings, as well as the snub-632 and bew tilings. In several cases the results provide proofs for previously conjectured formulas.
combinatorics  feature-construction  representation  rather-interesting  enumeration  to-write-about  mathematical-recreations  consider:random-graphs  consider:non-tilings 
15 days ago by Vaguery
Exploring the golden ratio Φ
This post is really just things I've learned from other people, but things that surprised me
mathematical-recreations  phi  geometry  to-write-about 
7 weeks ago by Vaguery
The Metonymy of Matrices - Scientific American Blog Network
As a tool, the matrix is so powerful that it is easy to forget that it is a representation of a function, not a function itself. A matrix truly is just the array of numbers, but I think in this context, most mathematicians are metonymists. (Metonymers? Metonymnistes?) We think of the matrix as the function itself, and it’s easy to lose sight of the fact that it's only notation. Matrices don’t even have to encode linear transformations. They are used in other contexts in mathematics, too, and restricting our definition to linear transformations can shortchange the other applications (though for my money, the value of the matrix as a way of representing linear transformations dwarfs any other use they have).
matrices  representation  functions  mathematical-recreations  to-write-about 
10 weeks ago by Vaguery
Orthogonal polygons | The Math Less Traveled
Quite a few commenters figured out what was going on, and mentioned several nice (equivalent) ways to think about it. Primarily, the idea is to draw all possible orthogonal polygons, that is, polygons with only right angles, organized by the total number of vertices. (So, for example, the picture above shows all orthogonal polygons with exactly ten vertices.) However, we have to be careful what we mean by the phrase “all possible”: there would be an infinite number of such polygons if we think about things like the precise lengths of edges. So we have to say when two polygons are considered the same, and when they are distinct. My rules are as follows:
mathematical-recreations  looking-to-see  geometry  polyominoes  to-write-about 
11 weeks ago by Vaguery
Cooking the books – Almost looks like work
Since Christmas, at my house we’ve been cooking with 5 ingredients or fewer thanks to the acquisition of Jamie Oliver’s new book, the contents of which are mostly available online here. The recipes are unanimously very tasty, but that’s besides the point. The real mark of culinary excellence (in my humble opinion) is how efficiently one can buy ingredients to make as many of the recipes as possible in one shopping trip. Let’s investigate while the lamb is on.

Each of the 135 recipes in the book consists of 5 ingredients, some of which overlap. It is therefore not necessary to purchase 675 ingredients, there are actually only 239 unique ones. (Yes, I did spend a Sunday morning typing 675 individual ingredients into a spreadsheet.)

The question is then this:

In which order should I buy my ingredients to maximise the number of possible recipes as a function of number of ingredients?

Let’s start simply, and look at the frequency of occurrence of the ingredients.
mathematical-recreations  looking-to-see  cooking  data-analysis  leading-questions  rather-interesting 
11 weeks ago by Vaguery
Exactly how bad is the 13 times table? | The Aperiodical
Along the way, OEIS editor Charles R Greathouse IV added this intriguing conjecture:

Conjecture: a(n)≤N
for all n
. Perhaps N
can be taken as 81
number-theory  mathematical-recreations  open-questions  to-write-about  consider:feature-discovery 
april 2018 by Vaguery
mathrecreation: Tigers and Treasure
The graph below shows all 196 possible puzzles. The puzzles that lead to contradictions are coloured as white squares, the puzzles that do not lead to contradictions are coloured in black. We can see a white strip across the bottom: these are all the puzzles where door 2 is labelled with statement 1. Can you find the statement that always leads to contradictions for door 1? There are 131 puzzles that are "good" in the sense that they do not lead to contradictions.
mathematical-recreations  logic  looking-to-see  to-write-about  consider:robustness  nudge-targets 
april 2018 by Vaguery
The moving sofa problem — Dan Romik's home page
The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what is the shape of largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked.
mathematical-recreations  computational-geometry  engineering-design  constraint-satisfaction  rather-interesting  nudge-targets  to-write-about 
april 2018 by Vaguery
Home | Open Problem Garden
Welcome to the Open Problem Garden, a collection of unsolved problems in mathematics. Here you may:

Read descriptions of open problems.
Post comments on them.
Create and edit open problems pages (please contact us and we will set you up an account. Unfortunately, the automatic process is too prone to spammers at this moment.)
open-problems  mathematical-recreations  mathematics  computer-science  to-write-about  nudge-targets 
march 2018 by Vaguery
[1112.2896] On the structure of Ammann A2 tilings
We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of [B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete and Computational Geometry 20 (1998) 265-279]. By the same techniques we show that Ammann A2 tilings are not robust in the sense of [B. Durand, A. Romashchenko, A. Shen. Fixed-point tile sets and their applications, Journal of Computer and System Sciences, 78:3 (2012) 731--764].
tiling  aperiodic-tiles  combinatorics  rather-interesting  mathematical-recreations 
march 2018 by Vaguery
[1710.04495] Of puzzles and partitions: Introducing Partiti
We introduce Partiti, the puzzle that will run in Mathematics Magazine in 2018, and use the opportunity to recall some basic properties of integer partitions.
mathematical-recreations  puzzles  number-theory  rather-interesting  to-write-about 
march 2018 by Vaguery
[1705.03973] Transreality puzzle as new genres of entertainment technology
The author considers a class of mechatronic puzzles falling in the mixed-reality category, present examples of such devices, and propose a way to categorize them. Close relationships of such devices with the Tangible User Interface are described. The device designed by the author as an illustration of a mixed reality puzzle is presented.
puzzles  augmented-reality  rather-interesting  to-write-about  mathematical-recreations 
march 2018 by Vaguery
This page is dedicated to that most elegant of puzzles (combining mathematical and word play) which as been one of my interests, on and off, for my entire adult life. If you've never seen an alphametic, I'll show you what the fuss is all about. If you have, I will try to regale you some of my own creations that have pushed the envelope of alphametic possibilities to new and bizarre heights.
mathematical-recreations  puzzles  nudge-targets 
march 2018 by Vaguery
Alphametic Puzzle Generator
This page will allow you to search for alphametic puzzles (for a description of what they are and for some examples, see my alphametics page) among a group of words. These puzzles are rare, but given a number of words to use, it's not unusual to find one or two. I've also written a web-based solver for these puzzles.
puzzles  mathematical-recreations  nudge-targets  to-write-about 
march 2018 by Vaguery
[1510.02875] Exploring mod2 n-queens games
We introduce a two player game on an n x n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the number of queens already on the board which can attack that square modulo two. The game is explored along with some variations and its complexity.
mathematical-recreations  chess  enumeration  game-theory  rather-interesting  nudge-targets  consider:looking-to-see 
february 2018 by Vaguery
[1608.01562] Convex domino towers
We study convex domino towers using a classic dissection technique on polyominoes to find the generating function and an asymptotic approximation.
domino-tiling  mathematical-recreations  rather-interesting  combinatorics  enumeration  nudge-targets  consider:open-questions 
february 2018 by Vaguery
[1608.01563] On the enumeration of k-omino towers
We describe a class of fixed polyominoes called k-omino towers that are created by stacking rectangular blocks of size k×1 on a convex base composed of these same k-omino blocks. By applying a partition to the set of k-omino towers of fixed area kn, we give a recurrence on the k-omino towers therefore showing the set of k-omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity with parameters similar to those given in a classical result of Kummer.
mathematical-recreations  combinatorics  enumeration  polyominoes  to-write-about 
february 2018 by Vaguery
[1801.02262] Magic Polygons and Their Properties
Magic squares are arrangements of natural numbers into square arrays, where the sum of each row, each column, and both diagonals is the same. In this paper, the concept of a magic square with 3 rows and 3 columns is generalized to define magic polygons. Furthermore, this paper will examine the existence of magic polygons, along with several other properties inherent to magic polygons.
mathematical-recreations  magic-squares  combinatorics  number-theory  rather-interesting  to-write-about  nudge-targets  consider:looking-to-see 
february 2018 by Vaguery

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