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Snakes in the Plane
Snakes in the Plane by Paul Church A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2008 c 2008 Paul Church I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of poly- gons. I present a brief background of tiling theory and past work, with particular empha- sis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new “boundary-based” technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite lan- guage of directions, to replace the “area-based” approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge as- semblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier’s recursive generation algo- rithm, and enumerates polysnakes using a novel approach. -
002-Contents.Pdf
CubeRoot Contents Contents Contents Purple denotes upcoming contents. 1 Preface 2 Signatures of Top Cubers in the World 3 Quotes 4 Photo Albums 5 Getting Started 5.1 Cube History 5.2 WCA Events 5.3 WCA Notation 5.4 WCA Competition Tutorial 5.5 Tips to Cubers 6 Rubik's Cube 6.1 Beginner 6.1.1 LBL Method (Layer-By-Layer) 6.1.2 Finger and Toe Tricks 6.1.3 Optimizing LBL Method 6.1.4 4LLL Algorithms 6.2 Intermediate 进阶 6.2.1 Triggers 6.2.2 How to Get Faster 6.2.3 Practice Tips 6.2.4 CN (Color Neutrality) 6.2.5 Lookahead 6.2.6 CFOP Algorithms 6.2.7 Solve Critiques 3x3 - 12.20 Ao5 6.2.8 Solve Critiques 3x3 - 13.99 Ao5 6.2.9 Cross Algorithms 6.2.10 Xcross Examples 6.2.11 F2L Algorithms 6.2.12 F2L Techniques 6.2.13 Multi-Angle F2L Algorithms 6.2.14 Non-Standard F2L Algorithms 6.2.15 OLL Algorithms, Finger Tricks and Recognition 6.2.16 PLL Algorithms and Finger Tricks 6.2.17 CP Look Ahead 6.2.18 Two-Sided PLL Recognition 6.2.19 Pre-AUF CubeRoot Contents Contents 7 Speedcubing Advice 7.1 How To Get Faster 7.2 Competition Performance 7.3 Cube Maintenance 8 Speedcubing Thoughts 8.1 Speedcubing Limit 8.2 2018 Plans, Goals and Predictions 8.3 2019 Plans, Goals and Predictions 8.4 Interviewing Feliks Zemdegs on 3.47 3x3 WR Single 9 Advanced - Last Slot and Last Layer 9.1 COLL Algorithms 9.2 CxLL Recognition 9.3 Useful OLLCP Algorithms 9.4 WV Algorithms 9.5 Easy VLS Algorithms 9.6 BLE Algorithms 9.7 Easy CLS Algorithms 9.8 Easy EOLS Algorithms 9.9 VHLS Algorithms 9.10 Easy OLS Algorithms 9.11 ZBLL Algorithms 9.12 ELL Algorithms 9.13 Useful 1LLL Algorithms -
Kurze Geschichte Des Würfels (Unknown Author)
Kurze Geschichte des Würfels (unknown author) ........................................................................................ 1 Erno Rubik .......................................................................................................................................... 1 Die Herstellung des Original-Rubik-Würfels in Ungarn ................................................................ 3 Die Rubik-Würfel-Weltmeisterschaft ............................................................................................... 6 A Rubik's Cube Chronology (Mark Longridge) .............................................................................................. 8 From five thousand to fifteen millions ....................................................................................................... 11 Toy-BUSINESS KONSUMEX .......................................................................................................................... 14 HISTORY (Nagy Olivér) ................................................................................................................................ 15 Kurze Geschichte des Würfels (unknown author) Jede Erfindung hat ein offizielles Geburtsdatum. Das Geburtsdatum des Würfels ist 1974, das Jahr, in dem der erste funktionsfähige Prototyp entstand und die erste Patentanmeldung entworfen wurde. Der Geburtsort war Budapest, die Hauptstadt Ungarns. Der Name des Erfinders ist inzwischen überall bekannt. Damals war Erno Rubik ein Dozent an der Fakultät für Innenarchitektur an der Akademie -
Mathematics of the Rubik's Cube
Mathematics of the Rubik's cube Associate Professor W. D. Joyner Spring Semester, 1996{7 2 \By and large it is uniformly true that in mathematics that there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." John von Neumann COLLECTED WORKS, VI, p. 489 For more mathematical quotes, see the first page of each chapter below, [M], [S] or the www page at http://math.furman.edu/~mwoodard/mquot. html 3 \There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man's life to know them the little new that each man gets from life is very costly and the only heritage he has to leave." Ernest Hemingway (From A. E. Hotchner, PAPA HEMMINGWAY, Random House, NY, 1966) 4 Contents 0 Introduction 13 1 Logic and sets 15 1.1 Logic................................ 15 1.1.1 Expressing an everyday sentence symbolically..... 18 1.2 Sets................................ 19 2 Functions, matrices, relations and counting 23 2.1 Functions............................. 23 2.2 Functions on vectors....................... 28 2.2.1 History........................... 28 2.2.2 3 × 3 matrices....................... 29 2.2.3 Matrix multiplication, inverses.............. 30 2.2.4 Muliplication and inverses............... -
Cube Lovers: Index by Date 3/18/17, 2 09 PM
Cube Lovers: Index by Date 3/18/17, 209 PM Cube Lovers: Index by Date Index by Author Index by Subject Index for Keyword Articles sorted by Date: Jul 80 Alan Bawden: [no subject] Jef Poskanzer: Complaints about :CUBE program. Alan Bawden: [no subject] [unknown name]: [no subject] Alan Bawden: [no subject] Bernard S. Greenberg: Cube minima Ed Schwalenberg: Re: Singmeister who? Bernard S. Greenberg: Singmaster Allan C. Wechsler: Re: Cubespeak Richard Pavelle: [no subject] Lauren Weinstein: confusion Alan Bawden: confusion Jon David Callas: [no subject] Bernard S. Greenberg: Re: confusion Richard Pavelle: confusion but simplicity Allan C. Wechsler: Short Introductory Speech Richard Pavelle: the cross design Bernard S. Greenberg: Re: the cross design Alan Bawden: the cross design Yekta Gursel: Re: Checker board pattern... Bernard S. Greenberg: Re: Checker board pattern... Michael Urban: Confusion Bernard S. Greenberg: Re: the cross design Bernard S. Greenberg: Re: Checker board pattern... Bernard S. Greenberg: Re: Confusion Bernard S. Greenberg: The Higher Crosses Alan Bawden: The Higher Crosses Bernard S. Greenberg: Postscript to above Bernard S. Greenberg: Bug in above Ed Schwalenberg: Re: Patterns, designs &c. Alan Bawden: Patterns, designs &c. Alan Bawden: 1260 Richard Pavelle: [no subject] Allan C. Wechsler: Re: Where to get them in the Boston Area, Cube Language. Alan Bawden: 1260 vs. 2520 Alan Bawden: OOPS Bill McKeeman: Re: Where to get them in the Boston Area, Cube Language. Bernard S. Greenberg: General remarks Bernard S. Greenberg: :cube feature http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/ Page 1 of 45 Cube Lovers: Index by Date 3/18/17, 209 PM Alan Bawden: [no subject] Bernard S. -
Rubiku Kuubiku Lahendamine Erinevaid Algoritme Kasutades
TALLINNA TEHNIKAÜLIKOOL Infotehnoloogia teaduskond Arvutitehnika instituut Eero Rahamägi 123837 RUBIKU KUUBIKU LAHENDAMINE ERINEVAID ALGORITME KASUTADES Bakalaureusetöö Juhendaja: Sergei Kostin PhD Teadur Tallinn 2017 Autorideklaratsioon Kinnitan, et olen koostanud antud lõputöö iseseisvalt ning seda ei ole kellegi teise poolt varem kaitsmisele esitatud. Kõik töö koostamisel kasutatud teiste autorite tööd, olulised seisukohad, kirjandusallikatest ja mujalt pärinevad andmed on töös viidatud. Autor: Eero Rahamägi 21.05.2017 2 Annotatsioon Käesolevas bakalaureuse lõputöös kirjeldatakse erinevaid Rubiku kuubiku lahendamise algoritme. Analüüs ning võrdlus erinevate algoritmide vahel; kiirust ning optimaalsust arvesstades käikude arvu lühendamisel kuupi lahendades. Võrdlust aitab realiseerida ka prototüüpne programm mis lahendab kuubiku valitud algoritme kasutades. Töö eesmärk on anda lugejale selge, arusaadav ning detailne ülevaade Rubiku kuubiku algoritmidest analüüsides ning kirjeldades erinevaid algoritme ning nende seotust keerukuse ning optimaalsuse lähtepunktist. Samuti annan ülevaate kasutatavast prototüüpsest programmist mis on võimeline lahendama Rubiku kuubikut suvalisest algolekust. Lugejale esitatakse programne prototüüp visuaalsel kujul. Lõputöö on kirjutatud eesti keeles ning sisaldab teksti 31 leheküljel, 3 peatükki, 6 joonist, 1 tabel. 3 Abstract Solving Rubiks Cube using different algorithms This Bachelors thesis describes different Rubik’s cube solving algorithms. Analysis and comparison between various algorithms; considering speed and optimality in decreasing the amount of steps it takes to complete a cube. A prototype program will be presented as a tool to help put the thesis into practice using selected algorithms. However creating this kind of application can be quite complex taking into consideration the sheer size of algorithms and different patterns, searching methods and the complexity of the programming language itself. Developing this kind of program requires handling many different parts of code and algorithms simultaneously. -
Gemblo and Polyhex Blokus
with John Gough < [email protected]> Gemblo and Polyhex Blokus Blokus joined together whole-edge to whole-edge. Gemblo is one of the possible polyhex Blokus is an abstract strategy board game games that might be invented. Why for 2 to 4 players. It uses polyominoes obvious? Because both Blokus and Blokus WHicH are plane geometric lgures formed Trigon used families of shapes made from by joining one or more equal squares edge regular polygons that tessellate. How many to edge refer &igure . 0olyominoes HaVe are there? been described as being “a polyform whose cells are squaresv and they are classiled Gemblo according to how many cells they have. ie. number of cells. Wikipedia, 2017 March In 2005, Gemblo was created by Justin Oh 19). Polyominoes and other mathematically in Korea. It is an abstract strategy board patterned families of shapes were invented game with translucent, coloured pieces, by Solomon Golomb, and popularised by each of which is made up of one to lve hex- Martin Gardner. agons. The six colours are clear, red, yellow, green, blue, and purple. Despite the obvious similarities to Blokus, Oh has stated that when he was inventing Gemblo he did not know about Blokus. The name is based on the word ‘gem’, alluding to the gem-like colours used for the play- ing pieces and ‘blocking’—a central feature of playing. The size of the playing board depends on the number of people playing. Figure 1: Examples of polyominoes With players using a hexagonal shaped board that has sides of 8 unit-hexagons. Blokus—using squares—has a sister- game called Blokus Trigon that uses poly- Playing Gemblo iamonds, those families of shapes made s Gemblo can be played as a one- from unit-sided equilateral triangles joined person puzzle or solitaire, or by whole-edge to whole-edge. -
Dino Cube / Rainbow Cube / Brain Twist 06/02/2007 02:36 PM
Dino Cube / Rainbow Cube / Brain Twist 06/02/2007 02:36 PM Dino Cube / Rainbow Cube / Brain Twist http://www.geocities.com/jaapsch/puzzles/dinocube.htm Page 1 of 5 Dino Cube / Rainbow Cube / Brain Twist 06/02/2007 02:36 PM Picture kindly supplied by Hendrik Haak The Dino Cube is a cube shaped puzzle, and like the Skewb, it has eight axes of rotation centred around the corners. Its cutting planes go diagonally through the square faces, cutting off triangular pyramidal corners. There are twelve moving pieces, one on each edge of the cube. It is called a dino cube because it originally had pictures of dinosaurs on the sides. Versions with 6 colours (pictured above) were also made, as well as with only 4 or 2 colours. The second puzzle pictured above is the Rainbow Cube. It has the shape of a cuboctahedron. It is very much like a Dino Cube in which the corners have been cut off, giving it 8 triangular faces as well as 6 square ones. The puzzle still has only 12 moving pieces, but now there are also stationary centres in the triangular faces. There are two colour schemes. One has 14 colours, the other has only 7 colours with opposite faces the same colour. As the puzzle with the 7 colour scheme does not have any identical pieces, the two colour schemes give puzzles of the same difficulty. The Brain Twist is a new puzzle by Hoberman. The third picture above shows what it looks like when unfolded in its star shape. -
History and Current State of Recreational Mathematics and Its Relation to Serious Mathematics
Charles University in Prague Faculty of Mathematics and Physics DOCTORAL THESIS Tereza B´artlov´a History and current state of recreational mathematics and its relation to serious mathematics Department of Mathematical Analysis Supervisor of the doctoral thesis: prof. RNDr. LuboˇsPick, CSc., DSc. Study programme: Mathematics Study branch: General Questions of Mathematics and Information Science Prague 2016 I declare that I carried out this doctoral thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University in Prague has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act. In Prague on June 28, 2016 Tereza B´artlov´a i Title: History and current state of recreational mathematics and its relation to serious mathematics Author: Tereza B´artlov´a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. LuboˇsPick, CSc., DSc., Department of Mathematical Analysis Abstract: The present thesis is devoted to the study of recreational mathematics, with a particular emphasis on its history, its relation to serious mathematics and its educational benefits. The thesis consists of five papers. In the first one we investigate the history of recreational mathematics. We focus on the development of mathematical problems throughout history, and we try to point out the people who had an important influence on the progress of recreational mathematics. The second article is dedicated to Edwin Abbott Abbott and his book called Flatland. -
Edgy Puzzles Karl Schaffer Karl [email protected]
Edgy Puzzles Karl Schaffer [email protected] Countless puzzles involve decomposing areas or volumes of two or three-dimensional figures into smaller figures. “Polyform” puzzles include such well-known examples as pentominoes, tangrams, and soma cubes. This paper will examine puzzles in which the edge sets, or "skeletons," of various symmetric figures like polyhedra are decomposed into multiple copies of smaller graphs, and note their relationship to representations by props or body parts in dance performance. The edges of the tetrahedron in Figure 1 are composed of a folded 9-gon, while the cube and octahedron are each composed of six folded paths of length 2. These constructions have been used in dances created by the author and his collaborators. The photo from the author’s 1997 dance “Pipe Dreams” shows an octahedron in which each dancer wields three lengths of PVC pipe held together by cord at the two internal vertices labeled a in the diagram on the right. The shapes created by the dancers, which might include whimsical designs reminiscent of animals or other objects as well as mathematical forms, seem to appear and dissolve in fluid patterns, usually in time to a musical score. L R a R L a L R Figure 1. PVC pipe polyhedral skeletons used in dances The desire in the dance company co-directed by the author to incorporate polyhedra into dance works led to these constructions, and to similar designs with loops of rope, fingers and hands, and the bodies of dancers. Just as mathematical concepts often suggest artistic explorations for those involved in the interplay between these fields, performance problems may suggest mathematical questions, in this case involving finding efficient and symmetrical ways to construct the skeletons, or edge sets, of the Platonic solids. -
How to Solve the Rubik's Cube 03/11/2007 05:07 PM
How to Solve the Rubik's Cube 03/11/2007 05:07 PM Rubik's Revolution Rubik's Cubes & Puzzles Rubik Cube Boston's Wig Store Everything you wanted to know Rubiks Cube 4x4, Keychain & Huge selection of Rubik Cube Great selection & service Serving about the all new electronic Twist In Stock Now-Free Shipping items. the Boston area Rubik’s cube Over $75 eBay.com www.mayswigs.com www.rubiksrevolution.com AwesomeAvenue.biz Ads by Goooooogle Advertise on this site How to Solve the Rubik's Cube This page is featured under Recreation:Games:Puzzles:Rubik's Cube:Solutions in Yahoo! My Home Page | My Blog | My NHL Shootout Stats 2006-2007 There are three translations of this page: Danish (Dansk) (Word Document), Japanese (日本語) (HTML) and Portuguese (Português) (HTML). If you want to translate this page, go ahead. Send me an email when you are done and I will add your translation to this list. So you have a Rubik's Cube, and you've played with it and stared at it and taken it apart...need I go on any further? The following are two complete, fool-proof solutions to solving the cube from absolutely any legal position. Credit goes not to me, but to David Singmaster, who wrote a book in 1980, Notes on Rubik's Magic Cube, which explains pretty much all of what you need to know, plus more. Singmaster wrote about all of these moves except the move for Step 2, which I discovered independently (along with many other people, no doubt). I've updated this page to include a second solution to the cube. -
A Rubik's Cube Chronology
http://cubeman.org/cchrono.txt 02/10/2007 11:35 AM A Rubik's Cube Chronology ------------------------- Researched and maintained by Mark Longridge (c) 1996-2004 Pre-Rubik --------- Feb 2, 1960 William Gustafson files patent for Manipulatable Toy Mar 12, 1963 Gustafson receives US patent 3,081,089 1970 Uwe Meffert invents a model for research of energy flow in different shape solids (pyraminx) Apr 9, 1970 Frank Fox applies for UK patent for spherical 3x3x3 Mar 4, 1970 Larry Nichols files patent for Twizzle (2x2x2 cube) Apr 11, 1972 Nichols receives US patent 3,655,201 Jan 16, 1974 Frank Fox receives UK patent 1,344,259 Post-Rubik ---------- Spring 1974 Erno Rubik gets idea to make the cube Summer 1974 Erno Rubik solves the cube (arguably the first solver) Jan 30, 1975 Rubik applies for patent on cube Oct 12, 1976 Terutoshi Ishige Japanese Patent 55-8192 for 3x3x3 1977 Rubik's Cube starts distribution in Hungary Mar 28, 1977 Erno Rubik receives Hungarian Patent HU00170062 Aug 1978 Bela Szalai first sees cube in Hungary, later manufactures cube in U.S. (Logical Games Inc) Sept 1979 Ideal Toy buys exclusive rights to the cube for one million dollars Jan 4, 1980 Victor Toth wins pioneering cube contest in 55 sec. July 1980 MIT cube lovers group starts up Sept 1980 Omni prints article on cube in Games column Jan 12, 1981 Steven Hanson and Jeffrey Breslow file US Patent for Missing Link March 1981 Scientific American's 1st article on cube March 1981 Uwe Meffert patents the pyraminx Mar 23, 1981 First mention of Rubik's Cube in Time Magazine May 1981 Reader's Digest prints cube story July 30, 1981 Walter Moll receives German patent for Dodecahedron July 31, 1981 Cube contest featuring James G.