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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. -
Donald Knuth Fletcher Jones Professor of Computer Science, Emeritus Curriculum Vitae Available Online
Donald Knuth Fletcher Jones Professor of Computer Science, Emeritus Curriculum Vitae available Online Bio BIO Donald Ervin Knuth is an American computer scientist, mathematician, and Professor Emeritus at Stanford University. He is the author of the multi-volume work The Art of Computer Programming and has been called the "father" of the analysis of algorithms. He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar,[4] Knuth created the WEB and CWEB computer programming systems designed to encourage and facilitate literate programming, and designed the MIX/MMIX instruction set architectures. As a member of the academic and scientific community, Knuth is strongly opposed to the policy of granting software patents. He has expressed his disagreement directly to the patent offices of the United States and Europe. (via Wikipedia) ACADEMIC APPOINTMENTS • Professor Emeritus, Computer Science HONORS AND AWARDS • Grace Murray Hopper Award, ACM (1971) • Member, American Academy of Arts and Sciences (1973) • Turing Award, ACM (1974) • Lester R Ford Award, Mathematical Association of America (1975) • Member, National Academy of Sciences (1975) 5 OF 44 PROFESSIONAL EDUCATION • PhD, California Institute of Technology , Mathematics (1963) PATENTS • Donald Knuth, Stephen N Schiller. "United States Patent 5,305,118 Methods of controlling dot size in digital half toning with multi-cell threshold arrays", Adobe Systems, Apr 19, 1994 • Donald Knuth, LeRoy R Guck, Lawrence G Hanson. -
Solving Sudoku with Dancing Links
Solving Sudoku with Dancing Links Rob Beezer [email protected] Department of Mathematics and Computer Science University of Puget Sound Tacoma, Washington USA African Institute for Mathematical Sciences October 25, 2010 Available at http://buzzard.pugetsound.edu/talks.html Example: Combinatorial Enumeration Create all permutations of the set f0; 1; 2; 3g Simple example to demonstrate key ideas Creation, cardinality, existence? There are more efficient methods for this example Rob Beezer (U Puget Sound) Solving Sudoku with Dancing Links AIMS October 2010 2 / 37 Brute Force Backtracking BLACK = Forward BLUE = Solution RED = Backtrack root 0 1 2 0 1 3 3 0 2 1 0 2 3 3 0 3 1 0 3 1 0 2 0 0 1 2 2 0 1 3 0 2 1 3 0 2 3 0 3 1 3 0 3 3 1 0 2 1 0 0 0 1 2 0 1 0 2 1 0 2 0 3 1 0 3 1 0 2 0 0 1 2 3 0 0 2 0 0 3 0 1 0 2 2 0 1 0 1 2 0 2 0 2 2 0 3 0 3 2 root 1 0 2 0 1 0 0 1 0 2 0 0 2 0 3 0 0 3 2 0 1 1 0 2 3 0 1 0 1 3 0 2 0 2 3 0 3 0 3 2 1 0 1 0 2 . 0 1 1 0 1 3 0 0 2 1 0 2 3 0 0 3 1 0 3 2 1 1 0 0 . -
Representing Sudoku As an Exact Cover Problem and Using Algorithm X and Dancing Links to Solve the Represented Problem
Bhavana Yerraguntla Bhagya Dhome SUDOKU SOLVER Approach: Representing Sudoku as an exact cover problem and using Algorithm X and Dancing Links to solve the represented problem. Before we start, let us try to understand what Exact Cover is; Exact Cover: Given a set S and another set X where each element is a subset to S, select a set of subsets S* such that every element in X exist in exactly one of the selected sets. This selection of sets is said to be a cover of the set S. The exact cover problem is a decision problem to find an exact cover. E.g. Let S = {A, B, C, D, E, F} be a collection of subsets of a set X = {1, 2, 3, 4, 5, 6, 7} s t: A = {1, 4, 7} B = {1, 4} C = {4, 5, 7} D = {3, 5, 6} E = {2, 3, 6, 7} F = {2, 7} Then the set of subsets S* = {B, D, F} is an exact cover. Before we represent the Sudoku problem into an exact cover, let’s explore an example smaller. The basic approach towards solving an exact cover problem is simple 1) List the constraints that needs to be satisfied. (columns) 2) List the different ways to satisfy the constraints. (rows) Simple example: Imagine a game of Cricket (a batting innings), the batting order is almost filled with three batsmen A, B, C are yet to be placed and with three exact spots unfilled. One opener, one top order and one middle order. Each batsman must play in one order, and no two batsman must be in the same order. -
Polyominoes with Maximum Convex Hull Sascha Kurz
University of Bayreuth Faculty for Mathematics and Physics Polyominoes with maximum convex hull Sascha Kurz Bayreuth, March 2004 Diploma thesis in mathematics advised by Prof. Dr. A. Kerber University of Bayreuth and by Prof. Dr. H. Harborth Technical University of Braunschweig Contents Contents . i List of Figures . ii List of Tables . iv 0 Preface vii Acknowledgments . viii Declaration . ix 1 Introduction 1 2 Proof of Theorem 1 5 3 Proof of Theorem 2 15 4 Proof of Theorem 3 21 5 Prospect 29 i ii CONTENTS References 30 Appendix 42 A Exact numbers of polyominoes 43 A.1 Number of square polyominoes . 44 A.2 Number of polyiamonds . 46 A.3 Number of polyhexes . 47 A.4 Number of Benzenoids . 48 A.5 Number of 3-dimensional polyominoes . 49 A.6 Number of polyominoes on Archimedean tessellations . 50 B Deutsche Zusammenfassung 57 Index 60 List of Figures 1.1 Polyominoes with at most 5 squares . 2 2.1 Increasing l1 ............................. 6 2.2 Increasing v1 ............................ 7 2.3 2-dimensional polyomino with maximum convex hull . 7 2.4 Increasing l1 in the 3-dimensional case . 8 3.1 The 2 shapes of polyominoes with maximum convex hull . 15 3.2 Forbidden sub-polyomino . 16 1 4.1 Polyominoes with n squares and area n + 2 of the convex hull . 22 4.2 Construction 1 . 22 4.3 Construction 2 . 23 4.4 m = 2n − 7 for 5 ≤ n ≤ 8 ..................... 23 4.5 Construction 3 . 23 iii iv LIST OF FIGURES 4.6 Construction 4 . 24 4.7 Construction 5 . 25 4.8 Construction 6 . -
Solving Sudoku with Dancing Links
Solving Sudoku with Dancing Links Rob Beezer [email protected] Department of Mathematics and Computer Science University of Puget Sound Tacoma, Washington USA Stellenbosch University October 8, 2010 Available at http://buzzard.pugetsound.edu/talks.html Example: Combinatorial Enumeration Create all permutations of the set f0; 1; 2; 3g Simple example to demonstrate key ideas Creation, cardinality, existence? There are more efficient methods for this example Rob Beezer (U Puget Sound) Solving Sudoku with Dancing Links Stellenbosch U October 2010 2 / 37 Brute Force Backtracking BLACK = Forward BLUE = Solution RED = Backtrack root 0 1 2 0 1 3 3 0 2 1 0 2 3 3 0 3 1 0 3 1 0 2 0 0 1 2 2 0 1 3 0 2 1 3 0 2 3 0 3 1 3 0 3 3 1 0 2 1 0 0 0 1 2 0 1 0 2 1 0 2 0 3 1 0 3 1 0 2 0 0 1 2 3 0 0 2 0 0 3 0 1 0 2 2 0 1 0 1 2 0 2 0 2 2 0 3 0 3 2 root 1 0 2 0 1 0 0 1 0 2 0 0 2 0 3 0 0 3 2 0 1 1 0 2 3 0 1 0 1 3 0 2 0 2 3 0 3 0 3 2 1 0 1 0 2 . 0 1 1 0 1 3 0 0 2 1 0 2 3 0 0 3 1 0 3 2 1 1 0 0 . -
A Flowering of Mathematical Art
A Flowering of Mathematical Art Jim Henle & Craig Kasper The Mathematical Intelligencer ISSN 0343-6993 Volume 42 Number 1 Math Intelligencer (2020) 42:36-40 DOI 10.1007/s00283-019-09945-0 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self- archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy For Our Mathematical Pleasure (Jim Henle, Editor) 1 have argued that the creation of mathematical A Flowering of structures is an art. The previous column discussed a II tiny genre of that art: numeration systems. You can’t describe that genre as ‘‘flowering.’’ But activity is most Mathematical Art definitely blossoming in another genre. Around the world, hundreds of artists are right now creating puzzles of JIM HENLE , AND CRAIG KASPER subtlety, depth, and charm. We are in the midst of a renaissance of logic puzzles. A Renaissance The flowering began with the discovery in 2004 in England, of the discovery in 1980 in Japan, of the invention in 1979 in the United States, of the puzzle type known today as sudoku. -
Bastian Michel, "Mathematics of NRC-Sudoku,"
Mathematics of NRC-Sudoku Bastian Michel December 5, 2007 Abstract In this article we give an overview of mathematical techniques used to count the number of validly completed 9 × 9 sudokus and the number of essentially different such, with respect to some symmetries. We answer the same questions for NRC-sudokus. Our main result is that there are 68239994 essentially different NRC-sudokus, a result that was unknown up to this day. In dit artikel geven we een overzicht van wiskundige technieken om het aantal geldig inge- vulde 9×9 sudokus en het aantal van essentieel verschillende zulke sudokus, onder een klasse van symmetrie¨en,te tellen. Wij geven antwoorden voor dezelfde vragen met betrekking tot NRC-sudoku's. Ons hoofdresultaat is dat er 68239994 essentieel verschillende NRC-sudoku's zijn, een resultaat dat tot op heden onbekend was. Dit artikel is ontstaan als Kleine Scriptie in het kader van de studie Wiskunde en Statistiek aan de Universiteit Utrecht. De begeleidende docent was dr. W. van der Kallen. Contents 1 Introduction 3 1.1 Mathematics of sudoku . .3 1.2 Aim of this paper . .4 1.3 Terminology . .4 1.4 Sudoku as a graph colouring problem . .5 1.5 Computerised solving by backtracking . .5 2 Ordinary sudoku 6 2.1 Symmetries . .6 2.2 How many different sudokus are there? . .7 2.3 Ad hoc counting by Felgenhauer and Jarvis . .7 2.4 Counting by band generators . .8 2.5 Essentially different sudokus . .9 3 NRC-sudokus 10 3.1 An initial observation concerning NRC-sudokus . 10 3.2 Valid transformations of NRC-sudokus . -
A New Mathematical Model for Tiling Finite Regions of the Plane with Polyominoes
Volume 15, Number 2, Pages 95{131 ISSN 1715-0868 A NEW MATHEMATICAL MODEL FOR TILING FINITE REGIONS OF THE PLANE WITH POLYOMINOES MARCUS R. GARVIE AND JOHN BURKARDT Abstract. We present a new mathematical model for tiling finite sub- 2 sets of Z using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear pro- gramming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/ m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB. 1. Introduction and motivation 2 Consider a planar square lattice Z . We refer to each unit square in the lattice, namely [~j − 1; ~j] × [~i − 1;~i], as a cell.A polyomino is a union of 2 a finite number of edge-connected cells in the lattice Z . We assume that the polyominoes are simply-connected. The order (or area) of a polyomino is the number of cells forming it. The polyominoes of order n are called n-ominoes and the cases for n = 1; 2; 3; 4; 5; 6; 7; 8 are named monominoes, dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, and octominoes, respectively. -
Coverage Path Planning Using Reinforcement Learning-Based TSP for Htetran—A Polyabolo-Inspired Self-Reconfigurable Tiling Robot
sensors Article Coverage Path Planning Using Reinforcement Learning-Based TSP for hTetran—A Polyabolo-Inspired Self-Reconfigurable Tiling Robot Anh Vu Le 1,2 , Prabakaran Veerajagadheswar 1, Phone Thiha Kyaw 3 , Mohan Rajesh Elara 1 and Nguyen Huu Khanh Nhan 2,∗ 1 ROAR Lab, Engineering Product Development, Singapore University of Technology and Design, Singapore 487372, Singapore; [email protected] (A.V.L); [email protected] (P.V); [email protected] (M.R.E.) 2 Optoelectronics Research Group, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 3 Department of Mechatronic Engineering, Yangon Technological University, Insein 11101, Myanmar; [email protected] * Correspondence: [email protected] Abstract: One of the critical challenges in deploying the cleaning robots is the completion of covering the entire area. Current tiling robots for area coverage have fixed forms and are limited to cleaning only certain areas. The reconfigurable system is the creative answer to such an optimal coverage problem. The tiling robot’s goal enables the complete coverage of the entire area by reconfigur- ing to different shapes according to the area’s needs. In the particular sequencing of navigation, it is essential to have a structure that allows the robot to extend the coverage range while saving Citation: Le, A.V.; energy usage during navigation. This implies that the robot is able to cover larger areas entirely Veerajagadheswar, P.; Thiha Kyaw, P.; with the least required actions. This paper presents a complete path planning (CPP) for hTetran, Elara, M.R.; Nhan, N.H.K. -
Posterboard Presentation
Dancing Links and Sudoku A Java Sudoku Solver By: Jonathan Chu Adviser: Mr. Feinberg Algorithm by: Dr. Donald Knuth Sudoku Sudoku is a logic puzzle. On a 9x9 grid with 3x3 regions, the digits 1-9 must be placed in each cell such that every row, column, and region contains only one instance of the digit. Placing the numbers is simply an exercise of logic and patience. Here is an example of a puzzle and its solution: Images from web Nikoli Sudoku is exactly a subset of a more general set of problems called Exact Cover, which is described on the left. Dr. Donald Knuth’s Dancing Links Algorithm solves an Exact Cover situation. The Exact Cover problem can be extended to a variety of applications that need to fill constraints. Sudoku is one such special case of the Exact Cover problem. I created a Java program that implements Dancing Links to solve Sudoku puzzles. Exact Cover Exact Cover describes problems in h A B C D E F G which a mtrix of 0’s and 1’s are given. Is there a set of rows that contain exactly one 1 in each column? The matrix below is an example given by Dr. Knuth in his paper. Rows 1, 4, and 5 are a solution set. 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 We can represent the matrix with toriodal doubly-linked lists as shown above. -
Combinatorialalgorithms for Packings, Coverings and Tilings Of
Departm en t of Com m u n ication s an d Networkin g Aa lto- A s hi k M a thew Ki zha k k ep a lla thu DD 112 Combinatorial Algorithms / 2015 for Packings, Coverings and Tilings of Hypercubes Combinatorial Algorithms for Packings, Coverings and Tilings of Hypercubes Hypercubes of Tilings and Coverings Packings, for Algorithms Combinatorial Ashik Mathew Kizhakkepallathu 9HSTFMG*agdcgd+ 9HSTFMG*agdcgd+ ISBN 978-952-60-6326-3 (printed) BUSINESS + ISBN 978-952-60-6327-0 (pdf) ECONOMY ISSN-L 1799-4934 ISSN 1799-4934 (printed) ART + ISSN 1799-4942 (pdf) DESIGN + ARCHITECTURE Aalto Un iversity Aalto University School of Electrical Engineering SCIENCE + Department of Communications and Networking TECHNOLOGY www.aalto.fi CROSSOVER DOCTORAL DOCTORAL DISSERTATIONS DISSERTATIONS Aalto University publication series DOCTORAL DISSERTATIONS 112/2015 Combinatorial Algorithms for Packings, Coverings and Tilings of Hypercubes Ashik Mathew Kizhakkepallathu A doctoral dissertation completed for the degree of Doctor of Science (Technology) to be defended, with the permission of the Aalto University School of Electrical Engineering, at a public examination held at the lecture hall S1 of the school on 18 September 2015 at 12. Aalto University School of Electrical Engineering Department of Communications and Networking Information Theory Supervising professor Prof. Patric R. J. Östergård Preliminary examiners Dr. Mathieu Dutour Sikirić, Ruđer Bošković Institute, Croatia Prof. Aleksander Vesel, University of Maribor, Slovenia Opponent Prof. Sándor Szabó, University of Pécs, Hungary Aalto University publication series DOCTORAL DISSERTATIONS 112/2015 © Ashik Mathew Kizhakkepallathu ISBN 978-952-60-6326-3 (printed) ISBN 978-952-60-6327-0 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 (printed) ISSN 1799-4942 (pdf) http://urn.fi/URN:ISBN:978-952-60-6327-0 Unigrafia Oy Helsinki 2015 Finland Abstract Aalto University, P.O.