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 Science > Math > Recreations > Polyominoes

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Science > Math > Recreations > Polyominoes

Polyominoes:

Animal Enumerations

Enumeration on regular tilings of the Euclidean and Hyperbolic planes.

http://www.ieeta.pt/~tos/animals.html

Anna's Pentomino Page

Anna Gardberg makes pentominoes out of sculpey and agate.

http://www.geom.uiuc.edu/~summer95/gardberg/pent.html

Arnab's Pentominos Puzzle

Fast Pentominos puzzle solver, works on DOS/Windows platform. Free downloads.

http://www.geocities.com/hirak_99/goodies/pento.html

Graand

Free Classified Ads Portal

www.graand.com

Blocking Polyominos

Rodolfo Kurchan searchss the smallest polyomino such that a particular number of copies can form a blocked pattern. With solutions.

http://www.eldar.org/~problemi/pfun/blocked.html

Canonical Polygons

Ronald Kyrmse investigates grid polygons in which all side lengths are one or sqrt(2).

http://sti.br.inter.net/rkyrmse/canonic-e.htm

Christopher Monckton's Eternity Puzzle

Rules, the solution by Alex Selby and Oliver Riordan, other resources and links. The puzzle is made up of 209 pieces of polydrafters, each one is a combination of 12-30/60/90 triangles.

http://mathpuzzle.com/eternity.html

Counting Horizontally Convex Polyominoes

Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.8. Defines and counts horizontal convexity.

http://www.cs.uwaterloo.ca/journals/JIS/HICK2/chcp.html

Cynthia Lanius' Lesson: Polyominoes Introduction

From tetris to hexominoes, Cynthia explains them in color.

http://math.rice.edu/~lanius/Lessons/Polys/poly1.html

Dancing Links

Don Knuth discusses implementation details of polyomino search algorithms (compressed PostScript format).

http://www-cs-faculty.stanford.edu/~knuth/papers/dancing-color.ps.gz

Equilateral Pentagons

Jorge Luis Mireles Jasso investigates these polygons and dissects various polyominos into them. Animations show cases of infinite solutions.

http://www.geocities.com/jorgeluismireles/equilaterals/

Eternity Page

Alex Selby's page with a description of his solution method, with illustrations in .png and .pdf files.

http://www.archduke.demon.co.uk/eternity/index.html

Flexagons

Conrad and Hartline's 1962 article on Flexagons.

http://delta.cs.cinvestav.mx/~mcintosh/comun/flexagon/flexagon.html

Gamepuzzles

Polyomino and polyform games and puzzles manufactured by Kadon Enterprises Inc.

http://www.gamepuzzles.com/

The Geometry Junkyard: Polyominoes

Numerous links, sorted alphabetically.

http://www.ics.uci.edu/~eppstein/junkyard/polyomino.html

Gerard's Pentomino Page

Illustrates the 12 shapes. symmetrical combinations.

http://www.xs4all.nl/~gp/pentomino.html

Golygons

Harry J. Smith's explains polyominoes with consecutive integer side lengths.

http://www.geocities.com/hjsmithh/Golygons/

Golygons by Mathworld

What they are, and how to find them.

http://mathworld.wolfram.com/Golygon.html

Harold McIntosh's Flexagon Papers

Including copies of the original 1962 Conrad-Hartline papers. Abstract, html-pages, or .pdf documents.

http://delta.cs.cinvestav.mx/~mcintosh/oldweb/pflexagon.html

Henri Picciotto's Geometric Puzzles in the Classroom

Polyform puzzle lessons for math educators to use with their students, including polyominoes, supertangrams, and polyarcs.

http://www.picciotto.org/math-ed/puzzles/

Information on Pentomino Puzzles

At the Combinatorial Object Server.

http://www.theory.csc.uvic.ca/~cos/inf/misc/PentInfo.html

Isoperimetric Polygons

Livio Zucca tiles polygons of equal perimeter, or isoperiploes.

http://www.geocities.com/liviozuc/polyedges.html

Java pentominoes

Thery families web site with pentomino solver. (English/French)[Java].

http://www.thery.free.fr/index.php?option=com_content&task=view&id=18&Itemid=44

Knight's Move Tessellations

Dan Thomasson looks at tesselations with numerous unexpected shapes traced out by knight moves.

http://www.borderschess.org/KTtess.htm

Lego Pentominos

Eric Harshbarger. This puzzle maker says that the hard part was finding legos in enough different colors.

http://www.ericharshbarger.org/lego/pentominoes.html

Livio Zucca's polyomino-covered cube

Colorful illustrations demonstrate how closed surfaces could be covered by polyominoes.

http://www.geocities.com/liviozuc/pag3_eng.html

Logical Art and the Art of Logic

Pentomino pictures, software and other resources by Guenter Albrecht-Buehler.

http://www.basic.northwestern.edu/g-buehler/pentominoes/

The Mathematics of Polyominoes

Kevin Gong offers download of his polyominoes games shareware for Windows and Mac. 100 boards are included. A Java version is under development.

http://www.kevingong.com/Polyominoes/

Mathforum : a Pentomino Problem

Geometry Forum: Lists the pentominoes; fold them to form a cube; play a pentomino game. (project of the month, 1995)

http://mathforum.org/pom/project2.95.html

Mathforum : Minimal Domino Tiling

Tiling a square without cutting it into two.(Problem of the week 826, Spring 1997)

http://mathforum.org/wagon/spring97/p826.html

Mathforum : Tiling Rectangles from Ell

Stan Wagon asks which rectangles can be tiled with an ell-tromino.

http://mathforum.org/wagon/spring98/p856.html

Maximum Convex Hulls of Connected Systems of Segments and of Polyominoes

Bezdek, Brass, and Harborth. Abstract to an article which places bounds on the convex area needed to contain a polyomino. (Contributions to Algebra and Geometry Volume 35 (1994), No. 1, 37-43.)

http://www.maths.soton.ac.uk/EMIS/journals/BAG/vol.35/no.1/b35h1har.abs

Miroslav Vicher's Puzzles Pages

Polyforms (polyominoes, and polyiamonds) graphics, tables and resources (English/Czech).

http://www.vicher.cz/puzzle/

My Polyomino Page

Michael Reid's numerous articles on polyominoes and tilnig, with references and links.

http://www.math.ucf.edu/~reid/Polyomino/index.html

Packing Polyominoes

Mark Michell investigates packing pentominoes into rectangles of various non-integer aspect ratios in order to obtain the largest possible pieces using straight cuts.

http://www.users.bigpond.com/themichells/packing_pentominoes.htm

Packing Shapes

Erich Friedman's Introduction to a variety of packing and tiling problems.

http://www.stetson.edu/~efriedma/packing.html

Pairwise Touching Hypercubes

Erich Friedman's problem of the month asks how to partition the unit cubes of an a*b*c-unit rectangular box into as many connected polycubes as possible with a shared face between every pair of polycubes. Answers provided.

http://www.stetson.edu/~efriedma/mathmagic/0903.html

Pentamini Pentaminos Pentominoes

A container of mathematical games, gadgets and software. (English/Italian)

http://www.geocities.com/liviozuc/

Pento

Amamas Software offers a pentomino solving software.

http://ourworld.compuserve.com/homepages/DavidandPenny/Pento.htm

Pento-Mania

Pentomino based puzzle game lets children solve and create geometric puzzles. Win32 software, try or buy.

http://www.virtu-software.com/PentoMania/

Pentomino Applet

Rujith de Silva's applet puzzle offers games of four different sized rectangles. Source code available. [Java]

http://www.cs.cmu.edu/~desilva/pento/pento.html

Pentomino Applet

Fill up a given area using pentomino shapes, rotating and flipping them. Three levels of difficulty.[Java].

http://www.fwend.com/pentomino.htm

Pentomino Covers

Problems on minimal covers.

http://www.xprt.net/~munizao/polycover/

The Pentomino Dictionary by Gilles Esposito-Farèse

English words that can be written using the pentomino name letters FILNPTUVWXYZ and other related curiosities, including a homage to Georges Perec. (English/French).

http://www2.iap.fr/users/esposito/pento.html

Pentomino Dissection of a Square Annulus

From Scott Kim's Inversions Gallery.

http://www.scottkim.com/inversions/gallery/golomb.html

Pentomino Homepage

Lorente Philippe's site describes the building blocks, nomenclature, solutions, and numerous games. (French/English)

http://membres.lycos.fr/pentomino/index.html

Pentomino HungarIQa

Kati presents a pentomino puzzle using poly-rhombs instead of poly-squares. [English/French/German/Hungarian]

http://www.pentomino.tvnet.hu/

Pentomino Puzzles.

Pentomino solver with download. Windows 95 and later required. [German/English]

http://www.exi-online.de/html/eintritt_e.html

Pentominoes

Expository paper by R. Bhat and A. Fletcher. Covers pre-Golomb discoveries. the triplication problem and other aspects.

http://www.andrews.edu/~calkins/math/pentos.htm

Pentominoes : an Introduction

Centre for Innovation in Mathematics Teaching presents colourful examples of many tiling problems, duplication, triplication, etc.

http://www.cimt.plymouth.ac.uk/resources/puzzles/pentoes/pentoint.htm

Pentominos

B. Berchtold's applet helps tile a 6x10 rectangle. [German]

http://www.mathematik.ch/anwendungenmath/pento/

Pentominos

Graphics problems, solutions (including animated GIF) and links. (English/German through main page)

http://www.mathematische-basteleien.de/pentominos.htm

Pentominos Puzzle Solver

David Eck's graphical solver applet uses recursive technique. Source code available. [Java]

http://math.hws.edu/xJava/PentominosSolver/

The Poly Pages

About various polyforms - polyominoes, polyiamonds, polycubes, and polyhexes.

http://www.recmath.com/PolyPages/

Polyform and Dissection Puzzle Links

Christian Eggermont's link page.

http://web.inter.nl.net/users/C.Eggermont/Links.new/Puzzles/Polyforms.and.dissection/index.noframe.shtml

Polyform Spirals

Jorge Luis Mireles explains finite and infinite spirals made up of polyforms.

http://www.geocities.com/jorgeluismireles/spirals/

Polyforms

Ed Pegg Jr.'s site has pages on tiling, packing, and related problems involving polyominos, polyiamonds, polyspheres, and related shapes.

http://www.mathpuzzle.com/polyom.htm

Polygon Puzzle

Open source polyomino and polyform placement solitaire game.

http://freshmeat.net/projects/hextk/

Polyiamond Exclusion

Colonel Sicherman asks what fraction of the triangles need to be removed from a regular triangular tiling of the plane, in order to make sure that the remaining triangles contain no copy of a given polyiamond.

http://www.monmouth.com/~colonel/xpoly/xpoly.html

Polyiamonds

Mathforum. This Geometry problem of the week asks whether a six-point star can be dissected to form eight distinct hexiamonds.

http://mathforum.org/pow/solutio4.html

Polyomino and Polyhex Tiling

Joseph Myer's tables of polyominoes and of polyomino tilings, in Postscript format.

http://www.srcf.ucam.org/~jsm28/tiling/

Polyomino Applet

Wil Laan's applet searches for solution of packing hexominoes into more than 45 different shapes.[Java]

http://home.quicknet.nl/mw/prive/wil.laan/puzzle/cornucopia.html

Polyomino Enumeration

K. S. Brown examines the number of polyominoes up to order 12 for various cases involving rotation or reflections. Equations linking the cases are proposed.

http://www.mathpages.com/home/kmath039.htm

Polyomino Fuzion Game

Puzzles using pentominoes and hexominoes. Fuzion, game that designs and (semi-)automatically finds solutions. Links.

http://home.earthlink.net/~kenzelt/

Polyominoes

Describes a numerical invariant that can be used to classify polyominoes.

http://members.tripod.com/~modularity/pol.htm

Polyominoes

Introduction to Tetrominoes, Pentominoes, Hexominoes, Heptominoes, Octominoes, Fixed (translation only) Polyominoes. Numerous Links.

http://www.geocities.com/alclarke0/PolyPages/Polyominoes.html

Polyominoes

Jankok presents information about filling rectangles, other polygons, boxes, etc., with dominoes, trominoes, tetrominoes, pentominoes, solid pentominoes, hexiamonds, and whatever else people have invented as variations of a theme. References included.

http://homepages.cwi.nl/~jankok/etc/Polyomino.html

Polyominoids

Jorge Luis Mireles Jasso presents connected sets of squares in a 3d cubical lattice. Includes a Java applet as well as non-animated description.

http://www.geocities.com/jorgeluismireles/polyominoids/

Polypolygon Tilings

S. Dutch discusses polyominoes, poliamonds, and polypolygons with special attention to tiling characteristics.

http://www.uwgb.edu/dutchs/symmetry/polypoly.htm

Primes of a 14-omino

Michael Reid shows that a 3x6 rectangle with a 2x2 bite removed can tile a (much larger) rectangle. It is open whether it can do this using an odd number of copies.

http://www.math.ucf.edu/~reid/Polyomino/14omino02_rect.html

Puzzle Fun

Newsletter edited by Rodolfo Kurchan about pentominoes and other math problems.

http://www.eldar.org/~problemi/pfun/pfun.html

Rectifiable Polyomino

Karl Dahlke explains and demonstrates tiling. Includes C-program source.

http://www.eklhad.net/polyomino/

Schröder Triangles, Paths, and Parallelogram Polyominoes

A paper on their enumeration by Elisa Pergola and Robert A. Sulanke.

http://diamond.boisestate.edu/~sulanke/PAPER1/PergolaSulanke/PergolaSulanke.html

Six Squares Problem

This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube.

http://mathforum.org/pow/solution22.html

The Soma Cube

Soma-solving program in QBASIC by Courtney McFarren.

http://www.geocities.com/abcmcfarren/soma/soma.htm

Somatic

A solver for arbitrary polyomino and polycube puzzles. Binary code and source downloads available.

http://www.moerig.com/somatic/

Sqfig and Sqtile

Eric Laroche presents computer programs for generating polyominoes and polyomino tilings. Includes source codes in C, and binaries.

http://www.lrdev.com/lr/c/sqfig.html

Taniguchi's Programs

Windows software to solve polyiamond and sliding block puzzles.

http://homepage2.nifty.com/yuki-tani/index_e.html

Tesselating Locking Polyominos

Bob Newman examines the history of the subject and presents his minimal solutions.

http://www.noggs.dsl.pipex.com/ts/

Thorleif's SOMA Page

SOMA puzzle site with graphics, newsletter and software.

http://www.fam-bundgaard.dk/SOMA/SOMA.HTM

The Three Dimensional Polyominoes of Minimal Area

L. Alonso and R. Cert's abstract of a paper published in vol. 3 of the Elect. J. Combinatorics. Full paper available in different formats (.pdf, postscript, tex etc).

http://www.combinatorics.org/Volume_3/Abstracts/v3i1r27.html

Three Nice Pentomino Coloring Problems

Alexandre Owen Muñiz presents the Icehouse set which lends itself to different polyomino coloring games.

http://xprt.net/~munizao/mathrec/pentcol.html

Tiling a Square With Eight Congruent Polyominoes

Michael Reid's abstract of a paper in the "Journal of Combinatorial Theory, Series A".

http://www.math.ucf.edu/~reid/Research/Eight/

Tiling of Pythagorean Triplets

Joe Fields suggests that L-decomposition of squares of Pythagorean triplets could always be tiled.

http://www2.math.uic.edu/~fields/puzzle/puzzle.html

The Tiling Puzzle Games of OOG

Mr. Confetti presents a Windows and Java game for tangrams, polyominoes, and polyhexes.

http://www.mcmprod.com/

Tiling Rectangles and Half Strips with Congruent Polyominoes

Michael Reid's abstract of paper in the "Journal of Combinatorial Theory, Series A".

http://www.math.ucf.edu/~reid/Research/Halfstrip/

Tiling Stuff

Jonathan King examines problems of determining whether a given rectangular brick can be tiled by certain smaller bricks. Includes numerous articles in .pdf format.

http://www.math.ufl.edu/~squash/tilingstuff.html

Tiling with Notched Cubes

Robert Hochberg and Michael Reid exhibit an unboxable reptile: a polycube that can tile a larger copy of itself, but can't tile any rectangular block. Abstract of article to "Discrete Mathematics".

http://www.math.ucf.edu/~reid/Research/Notched/

Unbalanced Anisohedral Tiling

Joseph Myers and John Berglund found a polyhex that must be placed in two different ways in a tiling of a plane, such that one placement occurs twice as often as the other.

http://www.angelfire.com/mn3/anisohedral/unbalanced.html

Unbeatable Tetris

Java applet demonstres that this tetromino-packing game is a forced win for the side dealing the tetrominoes. Complete with mathematical proof. [Java]

http://www.geom.uiuc.edu/java/tetris/

Unfolding the Tesseract

Peter Turney lists the 261 polycubes that can be folded in four dimensions to form the surface of a hypercube, and provides animations of the unfolding process.

http://www.apperceptual.com/tesseract.html

What is a Golygon?

Harry Smith describes Dr. Dewdney's article in the July 1990 Scientific American's Mathematical Recreations column.

http://www.geocities.com/hjsmithh/Golygons/GolyWhat.html

Xominoes

Livio Zucca finds a set of markings for the edges of a square that lead to exactly 100 possible tiles, and asks how to fit them into a 10x10 grid.

http://www.geocities.com/liviozuc/xominoes.html