Formulae and Growth Rates of Animals on Cubical and Triangular Lattices
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Open Problem Collection
Open problem collection Peter Kagey May 14, 2021 This is a catalog of open problems that I began in late 2017 to keep tabs on different problems and ideas I had been thinking about. Each problem consists of an introduction, a figure which illustrates an example, a question, and a list related questions. Some problems also have references which refer to other problems, to the OEIS, or to other web references. 1 Rating Each problem is rated both in terms of how difficult and how interesting I think the problem is. 1.1 Difficulty The difficulty score follows the convention of ski trail difficulty ratings. Easiest The problem should be solvable with a modest amount of effort. Moderate Significant progress should be possible with moderate effort. Difficult Significant progress will be difficult or take substantial insight. Most difficult The problem may be intractable, but special cases may be solvable. 1.2 Interest The interest rating follows a four-point scale. Each roughly describes what quartile I think it belongs in with respect to my interest in it. Least interesting Problems have an interesting idea, but may feel contrived. More interesting Either a somewhat complicated or superficial question. Very interesting Problems that are particularly natural or simple or cute. Most interesting These are the problems that I care the most about. Open problem collection Peter Kagey Problem 1. Suppose you are given an n × m grid, and I then think of a rectangle with its corners on grid points. I then ask you to \black out" as many of the gridpoints as possible, in such a way that you can still guess my rectangle after I tell you all of the non-blacked out vertices that its corners lie on. -
Treb All De Fide Gra U
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC Grau en Matematiques` T´ıtol:Tilings and the Aztec Diamond Theorem Autor: David Pardo Simon´ Director: Anna de Mier Departament: Mathematics Any academic:` 2015-2016 TREBALL DE FI DE GRAU Facultat de Matemàtiques i Estadística David Pardo 2 Tilings and the Aztec Diamond Theorem A dissertation submitted to the Polytechnic University of Catalonia in accordance with the requirements of the Bachelor's degree in Mathematics in the School of Mathematics and Statistics. David Pardo Sim´on Supervised by Dr. Anna de Mier School of Mathematics and Statistics June 28, 2016 Abstract Tilings over the plane R2 are analysed in this work, making a special focus on the Aztec Diamond Theorem. A review of the most relevant results about monohedral tilings is made to continue later by introducing domino tilings over subsets of R2. Based on previous work made by other mathematicians, a proof of the Aztec Dia- mond Theorem is presented in full detail by completing the description of a bijection that was not made explicit in the original work. MSC2010: 05B45, 52C20, 05A19. iii Contents 1 Tilings and basic notions1 1.1 Monohedral tilings............................3 1.2 The case of the heptiamonds.......................8 1.2.1 Domino Tilings.......................... 13 2 The Aztec Diamond Theorem 15 2.1 Schr¨odernumbers and Hankel matrices................. 16 2.2 Bijection between tilings and paths................... 19 2.3 Hankel matrices and n-tuples of Schr¨oderpaths............ 27 v Chapter 1 Tilings and basic notions The history of tilings and patterns goes back thousands of years in time. -
Some Open Problems in Polyomino Tilings
Some Open Problems in Polyomino Tilings Andrew Winslow1 University of Texas Rio Grande Valley, Edinburg, TX, USA [email protected] Abstract. The author surveys 15 open problems regarding the algorith- mic, structural, and existential properties of polyomino tilings. 1 Introduction In this work, we consider a variety of open problems related to polyomino tilings. For further reference on polyominoes and tilings, the original book on the topic by Golomb [15] (on polyominoes) and more recent book of Gr¨unbaum and Shep- hard [23] (on tilings) are essential. Also see [2,26] and [19,21] for open problems in polyominoes and tiling more broadly, respectively. We begin by introducing the definitions used throughout; other definitions are introduced as necessary in later sections. Definitions. A polyomino is a subset of R2 formed by a strongly connected union of axis-aligned unit squares centered at locations on the square lattice Z2. Let T = fT1;T2;::: g be an infinite set of finite simply connected closed sets of R2. Provided the elements of T have pairwise disjoint interiors and cover the Euclidean plane, then T is a tiling and the elements of T are called tiles. Provided every Ti 2 T is congruent to a common shape T , then T is mono- hedral, T is the prototile of T , and the elements of T are called copies of T . In this case, T is said to have a tiling. 2 Isohedral Tilings We begin with monohedral tilings of the plane where the prototile is a polyomino. If a tiling T has the property that for all Ti;Tj 2 T , there exists a symmetry of T mapping Ti to Tj, then T is isohedral; otherwise the tiling is non-isohedral (see Figure 1). -
Biased Weak Polyform Achievement Games Ian Norris, Nándor Sieben
Biased weak polyform achievement games Ian Norris, Nándor Sieben To cite this version: Ian Norris, Nándor Sieben. Biased weak polyform achievement games. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no. 3 (in progress) (3), pp.147–172. hal- 01188912 HAL Id: hal-01188912 https://hal.inria.fr/hal-01188912 Submitted on 31 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 16:3, 2014, 147–172 Biased Weak Polyform Achievement Games Ian Norris∗ Nándor Siebeny Northern Arizona University, Department of Mathematics and Statistics, Flagstaff, AZ, USA received 21st July 2011, revised 20th May 2014, accepted 6th July 2014. In a biased weak (a; b) polyform achievement game, the maker and the breaker alternately mark a; b previously unmarked cells on an infinite board, respectively. The maker’s goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a; b) game can be built from winning strategies for games involving fewer marks for the maker and the breaker. -
The Fantastic Combinations of John Conway's New Solitaire Game "Life" - M
Mathematical Games - The fantastic combinations of John Conway's new solitaire game "life" - M. Gardner - 1970 Suche Home Einstellungen Anmelden Hilfe MATHEMATICAL GAMES The fantastic combinations of John Conway's new solitaire game "life" by Martin Gardner Scientific American 223 (October 1970): 120-123. Most of the work of John Horton Conway, a mathematician at Gonville and Caius College of the University of Cambridge, has been in pure mathematics. For instance, in 1967 he discovered a new group-- some call it "Conway's constellation"--that includes all but two of the then known sporadic groups. (They are called "sporadic" because they fail to fit any classification scheme.) Is is a breakthrough that has had exciting repercussions in both group theory and number theory. It ties in closely with an earlier discovery by John Conway of an extremely dense packing of unit spheres in a space of 24 dimensions where each sphere touches 196,560 others. As Conway has remarked, "There is a lot of room up there." In addition to such serious work Conway also enjoys recreational mathematics. Although he is highly productive in this field, he seldom publishes his discoveries. One exception was his paper on "Mrs. Perkin's Quilt," a dissection problem discussed in "Mathematical Games" for September, 1966. My topic for July, 1967, was sprouts, a topological pencil-and-paper game invented by Conway and M. S. Paterson. Conway has been mentioned here several other times. This month we consider Conway's latest brainchild, a fantastic solitaire pastime he calls "life". Because of its analogies with the rise, fall and alternations of a society of living organisms, it belongs to a growing class of what are called "simulation games"--games that resemble real-life processes. -
Area and Perimeter
2.2 Area and Perimeter Perimeter is easy to define: it’s the distance all the way round the edge of a shape (land sometimes has a “perimeter fence”). (The perimeter of a circle is called its circumference.) Some pupils will want to mark a dot where they start measuring/counting the perimeter so that they know where to stop. Some may count dots rather than edges and get 1 unit too much. Area is a harder concept. “Space” means 3-d to most people, so it may be worth trying to avoid that word: you could say that area is the amount of surface a shape covers. (Surface area also applies to 3-d solids.) (Loosely, perimeter is how much ink you’d need to draw round the edge of the shape; area is how much ink you’d need to colour it in.) It’s good to get pupils measuring accurately-drawn drawings or objects to get a feel for how small an area of 20 cm2, for example, actually is. For comparisons between volume and surface area of solids, see section 2:10. 2.2.1 Draw two rectangles (e.g., 6 × 4 and 8 × 3) on a They’re both rectangles, both contain the same squared whiteboard (or squared acetate). number of squares, both have same area. “Here are two shapes. What’s the same about them One is long and thin, different side lengths. and what’s different?” Work out how many squares they cover. (Imagine Infinitely many; e.g., 2.4 cm by 10 cm. -
Teacher's Guide to Shape by Shape
Using Puzzles to Teach Problem Solving TEACHER’S GUIDE TO SHAPE BY SHAPE Also includes Brick by Brick, Block by Block, and That-A-Way BENEFITS Shape by Shape vs. Tangrams Shape by Shape is an assembly puzzle that is an exciting way to teach elements of Shape by Shape is closely related to Tangrams, a classic — problem solving puzzle invented in China that became a world wide — visual and thinking craze in the early 18th century. The seven simple — geometry and measurement Tangram shapes can be arranged to create thousands — combinatorial reasoning of different silhouettes, such as people, animals, letters, The game is self-directed, with 60 challenge cards that and everyday objects. Tangrams are used in schools to include hints. The activities in this guide will help your stu- teach problem solving, geometry and fractions. dents get more out of Shape by Shape. ABOUT THE GAME Equipment. Shape by Shape includes 14 puzzle pieces (6 orange and 8 yellow), 60 challenge cards with answers on the backs, and a tray that doubles as storage for the cards. Shape by Shape and Tangrams are both based on the same triangular grid. They are similar enough that most Tangram activities work with Shape by Shape. Shape by Shape, however, is bigger and more complex. We recommend that you introduce math concepts with Tangrams first, then use Shape by Shape as a fol- low-up activity to explore concepts in more depth. Shape by Shape is different enough fromTangrams that both are worth owning. The main differences are: In a frame . -
Heesch Numbers of Unmarked Polyforms
Heesch Numbers of Unmarked Polyforms Craig S. Kaplan School of Computer Science, University of Waterloo, Ontario, Canada; [email protected] Abstract A shape’s Heesch number is the number of layers of copies of the shape that can be placed around it without gaps or overlaps. Experimentation and exhaustive searching have turned up examples of shapes with finite Heesch numbers up to six, but nothing higher. The computational problem of classifying simple families of shapes by Heesch number can provide more experimental data to fuel our understanding of this topic. I present a technique for computing Heesch numbers of non-tiling polyforms using a SAT solver, and the results of exhaustive computation of Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds. 1 Introduction Tiling theory is the branch of mathematics concerned with the properties of shapes that can cover the plane with no gaps or overlaps. It is a topic rich with deep results and open problems. Of course, tiling theory must occasionally venture into the study of shapes that do not tile the plane, so that we might understand those that do more completely. If a shape tiles the plane, then it must be possible to surround the shape by congruent copies of itself, leaving no part of its boundary exposed. A circle clearly cannot tile the plane, because neighbouring circles can cover at most a finite number of points on its boundary. A regular pentagon also cannot be surrounded by copies of itself: its vertices will always remain exposed. However, the converse is not true: there exist shapes that can be fully surrounded by copies of themselves, but for which no such surround can be extended to a tiling. -
By Kate Jones, RFSPE
ACES: Articles, Columns, & Essays A Periodic Table of Polyform Puzzles by Kate Jones, RFSPE What are polyform puzzles? A branch of mathematics known as Combinatorics deals with systems that encompass permutations and combinations. A subcategory covers tilings, or tessellations, that are formed of mosaic-like pieces of all different shapes made of the same distinct building block. A simple example is starting with a single square as the building block. Two squares joined are a domino. Three squares joined are a tromino. Compare that to one proton/electron pair forming hydrogen, and two of them becoming helium, etc. The puzzles consist of combining all the members of a set into coherent shapes, in as many different variations as possible. Here is a table of geometric families of polyform sets that provide combinatorial puzzle challenges (“recreational mathematics”) developed and produced by my company, Kadon Enterprises, Inc., over the last 40 years (1979-2019). See www.gamepuzzles.com. The POLYOMINO Series Figure 1: The polyominoes orders 1 through 5. 36 TELICOM 31, No. 3 — Third Quarter 2019 Figure 2: Polyominoes orders 1-5 (Poly-5); order 6 (hexominoes, Sextillions); order 7 (heptominoes); and order 8 (octominoes)—21, 35, 108 and 369 pieces, respectively. Polyominoes named by Solomon Golomb in 1953. Product names by Kate Jones. TELICOM 31, No. 3 — Third Quarter 2019 37 Figure 3: Left—Polycubes orders 1 through 4. Right—Polycubes order 5 (the planar pentacubes, Kadon’s Quintillions set). Figure 4: Above—the 18 non-planar pentacubes (Super Quintillions). Below—the 166 hexacubes. 38 TELICOM 31, No. 3 — Third Quarter 2019 The POLYIAMOND Series Figure 5: The polyiamonds orders 1 through 7. -
Tiling with Polyominoes, Polycubes, and Rectangles
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2015 Tiling with Polyominoes, Polycubes, and Rectangles Michael Saxton University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Saxton, Michael, "Tiling with Polyominoes, Polycubes, and Rectangles" (2015). Electronic Theses and Dissertations, 2004-2019. 1438. https://stars.library.ucf.edu/etd/1438 TILING WITH POLYOMINOES, POLYCUBES, AND RECTANGLES by Michael A. Saxton Jr. B.S. University of Central Florida, 2013 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Fall Term 2015 Major Professor: Michael Reid ABSTRACT In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z × Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes. ii ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor, Professor Michael Reid, who spent countless hours mentoring me over my time at the University of Central Florida. -
Pentomino -- from Wolfram Mathworld 08/14/2007 01:42 AM
Pentomino -- from Wolfram MathWorld 08/14/2007 01:42 AM Search Site Discrete Mathematics > Combinatorics > Lattice Paths and Polygons > Polyominoes Other Wolfram Sites: Algebra Wolfram Research Applied Mathematics Demonstrations Site Calculus and Analysis Pentomino Discrete Mathematics Integrator Foundations of Mathematics Tones Geometry Functions Site History and Terminology Wolfram Science Number Theory more… Probability and Statistics Recreational Mathematics Download Mathematica Player >> Topology Complete Mathematica Documentation >> Alphabetical Index Interactive Entries Show off your math savvy with a Random Entry MathWorld T-shirt. New in MathWorld MathWorld Classroom About MathWorld Send a Message to the Team Order book from Amazon 12,698 entries Fri Aug 10 2007 A pentomino is a 5-polyomino. There are 12 free pentominoes, 18 one-sided pentominoes, and 63 fixed pentominoes. The twelve free pentominoes are known by the letters of the alphabet they most closely resemble: , , , , , , , , , , , and (Gardner 1960, Golomb 1995). Another common naming convention replaces , , , and with , , , and so that all letters from to are used (Berlekamp et al. 1982). In particular, in the life cellular automaton, the -pentomino is always known as the -pentomino. The , , and pentominoes can also be called the straight pentomino, L-pentomino, and T-pentomino, respectively. SEE ALSO: Domino, Hexomino, Heptomino, Octomino, Polyomino, Tetromino, Triomino. [Pages Linking Here] REFERENCES: Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 110-111, 1987. Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Adding Games, 2nd ed. Wellesley, MA: A K Peters, 2001. Berlekamp, E. -
Tiling with Polyominoes*
JOURNAL OF COMBINATORIAL THEORY 1, 280-296 (1966) Tiling with Polyominoes* SOLOMON W. GOLOMB Electrical Engineering Department, University of Southern California, Los Angeles, California Communicated by Mark Kac INTRODUCTION An n-omino is a set of n rookwise connected unit squares [1]. The simpler figures (monomino, domino, trominoes, tetrorninoes, and pen- tominoes) are shown in Figure I. Fig. 1. The simpler polyominoes. It is interesting to consider the tiling properties of the various poly- ominoes. Certain polyominoes will tile the infinite plane, as illustrated with the X-pentomino in Figure 2. Some polyominoes can tile an infinite strip, as indicated with the F-pentomino in Figure 3. There are polyominoes * This research was supported in part by the United States Air Force under Grant AF-AFOSR-874-66. 280 TILING WITH POLYOMINOES 281 which are capable of tiling a rectangle, as exemplified by the Y-pen- tomino in Figure 4. Also, certain polyominoes are "rep-tiles" [2], i.e., they can be used to tile enlarged scale models of themselves, as shown with the P-pentomino in Figure 5. F_I- Fig. 2. The X-pentomino can be used to tile the plane. Fig. 3. The F-pentomino can be used to tile a strip. Fig. 4. The Y-pentomino can be used Fig. 5. The P-pentomino can be used to tile a rectangle, to tile itself. 282 GOLOMB The objective of this paper is to establish a hierarchy of tiling capa- bilities for polyominoes, and to classify all the simpler polyominoes (through hexominoes) as accurately as possible with respect to this hierarchy.