Contents  > Science  > Math  > Recreations  > Polyominoes

• Gerard's Universal Polyomino Solver - Computes from 1 to 3.38 billion solutions with graphic display to each of the 60+ problems of different sizes and shapes. Pieces vary from pentominoes to heptominoes, sometimes in combination. Table summarizes properties and example solution of each problem. [Java required].

  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

  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

  George Huttlin's Puzzle Page George Huttlin shares some ramblings in the world of polyominoes. http://members.aol.com/huttlin/pentominoes.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/

  Hexiamonds George Huttlin explains and illustrates these shapes composed of 6 equilateral triangles, which in turn tiles different forms. http://members.aol.com/huttlin/hexiamonds.html

  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-Far?se 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: Theme and Variations 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

  Solomon W. Golomb Home Page of the inventor of polyominoes. Includes biography, black and white picture, research interests and publications list. http://commsci.usc.edu/faculty/golomb.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 Mu?iz 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