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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.
      Anna's Pentomino Page: - Anna Gardberg makes pentominoes out of sculpey and agate.
      Arnab's Pentominos Puzzle: - Fast Pentominos puzzle solver, works on DOS/Windows platform. Free downloads.
      Blocking Polyominos: - Rodolfo Kurchan searchss the smallest polyomino such that a particular number of copies can form a blocked pattern. With solutions.
      Canonical Polygons: - Ronald Kyrmse investigates grid polygons in which all side lengths are one or sqrt(2).
      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.
      Counting Horizontally Convex Polyominoes: - Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.8. Defines and counts horizontal convexity.
      Cynthia Lanius' Lesson: Polyominoes Introduction: - From tetris to hexominoes, Cynthia explains them in color.



      Dancing Links: - Don Knuth discusses implementation details of polyomino search algorithms (compressed PostScript format).
      Eithan's Pentominoes-3D Applet Solver: - Solves given Pentominoes 3D puzzles. Solution is displayed in 3-D with disassembly and rotations.[Java applet.
      Equilateral Pentagons: - Jorge Luis Mireles Jasso investigates these polygons and dissects various polyominos into them. Animations show cases of infinite solutions.
      Eternity Page: - Alex Selby's page with a description of his solution method, with illustrations in .png and .pdf files.
      Flexagons: - Conrad and Hartline's 1962 article on Flexagons.
      Flexagons: - Folded paper polyiamonds which can be unfolded to show hidden faces. Make interesting school projects.
      Gamepuzzles: - Polyomino and polyform games and puzzles manufactured by Kadon Enterprises Inc.
      The Geometry Junkyard: Polyominoes: - Numerous links, sorted alphabetically.
      George Huttlin's Puzzle Page: - George Huttlin shares some ramblings in the world of polyominoes.


      Gerard's Pentomino Page: - Illustrates the 12 shapes. symmetrical combinations.
      Golygons: - Harry J. Smith's explains polyominoes with consecutive integer side lengths.
      Golygons by Mathworld: - What they are, and how to find them.
      Harold McIntosh's Flexagon Papers: - Including copies of the original 1962 Conrad-Hartline papers. Abstract, html-pages, or .pdf documents.
      Henri Picciotto's Geometric Puzzles in the Classroom: - Polyform puzzle lessons for math educators to use with their students, including polyominoes, supertangrams, and polyarcs.
      Hepto: - Some packings of the 108 heptominoes (with unit thickness) into various blocks.
      Hexiamonds: - George Huttlin explains and illustrates these shapes composed of 6 equilateral triangles, which in turn tiles different forms.
      Hyperbolic Planar Tessellations: - Don Hatch's page on hyperbolic tesselations with numerous illustrations.
      Information on Pentomino Puzzles: - At the Combinatorial Object Server.


      Isoperimetric Polygons: - Livio Zucca tiles polygons of equal perimeter, or isoperiploes.
      Java pentominoes: - Thery families web site with pentomino solver. (English/French)[Java].
      Knight's Move Tessellations: - Dan Thomasson looks at tesselations with numerous unexpected shapes traced out by knight moves.
      Lego Pentominos: - Eric Harshbarger. This puzzle maker says that the hard part was finding legos in enough different colors.
      Livio Zucca's polyomino-covered cube: - Colorful illustrations demonstrate how closed surfaces could be covered by polyominoes.
      Logical Art and the Art of Logic: - Pentomino pictures, software and other resources by Guenter Albrecht-Buehler.
      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.
      Mathforum : a Pentomino Problem: - Geometry Forum: Lists the pentominoes; fold them to form a cube; play a pentomino game. (project of the month, 1995.
      Mathforum : Minimal Domino Tiling: - Tiling a square without cutting it into two.(Problem of the week 826, Spring 1997.


      Mathforum : Tiling Rectangles from Ell: - Stan Wagon asks which rectangles can be tiled with an ell-tromino.
      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.
      Miroslav Vicher's Puzzles Pages: - Polyforms (polyominoes, and polyiamonds) graphics, tables and resources (English/Czech).
      My Polyomino Page: - Michael Reid's numerous articles on polyominoes and tilnig, with references and links.
      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.
      Packing Shapes: - Erich Friedman's Introduction to a variety of packing and tiling problems.
      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.
      Pentamini Pentaminos Pentominoes: - A container of mathematical games, gadgets and software. (English/Italian.
      Pento: - Amamas Software offers a pentomino solving software.


      Pento-Mania: - Pentomino based puzzle game lets children solve and create geometric puzzles. Win32 software, try or buy.
      Pentomino Applet: - Rujith de Silva's applet puzzle offers games of four different sized rectangles. Source code available. [Java.
      Pentomino Applet: - Fill up a given area using pentomino shapes, rotating and flipping them. Three levels of difficulty.[Java].
      Pentomino Covers: - Problems on minimal covers.
      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).
      Pentomino Dissection of a Square Annulus: - From Scott Kim's Inversions Gallery.
      Pentomino Homepage: - Lorente Philippe's site describes the building blocks, nomenclature, solutions, and numerous games. (French/English.
      Pentomino HungarIQa: - Kati presents a pentomino puzzle using poly-rhombs instead of poly-squares. [English/French/German/Hungarian.
      Pentomino Puzzles.: - Pentomino solver with download. Windows 95 and later required. [German/English.


      Pentomino Relationships: - Symmetries in the families of rectangular solutions.
      Pentominoes: - Expository paper by R. Bhat and A. Fletcher. Covers pre-Golomb discoveries. the triplication problem and other aspects.
      Pentominoes : an Introduction: - Centre for Innovation in Mathematics Teaching presents colourful examples of many tiling problems, duplication, triplication, etc.
      A Pentominoes Project from Belgium: - Secondary School project about pentominoes and fun with math. History, descriptions, and problems. Bi-monthly pentomino competition. A solver is available. [English, French, Dutch.
      Pentominos: - B. Berchtold's applet helps tile a 6x10 rectangle. [German.
      Pentominos: - Graphics problems, solutions (including animated GIF) and links. (English/German through main page.
      Pentominos Puzzle Solver: - David Eck's graphical solver applet uses recursive technique. Source code available. [Java.
      The Poly Pages: - About various polyforms - polyominoes, polyiamonds, polycubes, and polyhexes.
      Polyform and Dissection Puzzle Links: - Christian Eggermont's link page.


      Polyform Spirals: - Jorge Luis Mireles explains finite and infinite spirals made up of polyforms.
      Polyforms: - Ed Pegg Jr.'s site has pages on tiling, packing, and related problems involving polyominos, polyiamonds, polyspheres, and related shapes.
      Polygon Puzzle: - Open source polyomino and polyform placement solitaire game.
      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.
      Polyiamonds: - Mathforum. This Geometry problem of the week asks whether a six-point star can be dissected to form eight distinct hexiamonds.
      Polyomino and Polyhex Tiling: - Joseph Myer's tables of polyominoes and of polyomino tilings, in Postscript format.
      Polyomino Applet: - Wil Laan's applet searches for solution of packing hexominoes into more than 45 different shapes.[Java.
      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.
      Polyomino Fuzion Game: - Puzzles using pentominoes and hexominoes. Fuzion, game that designs and (semi-)automatically finds solutions. Links.


      Polyominoes: - Describes a numerical invariant that can be used to classify polyominoes.
      Polyominoes: - Introduction to Tetrominoes, Pentominoes, Hexominoes, Heptominoes, Octominoes, Fixed (translation only) Polyominoes. Numerous Links.
      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.
      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.
      Polypolygon Tilings: - S. Dutch discusses polyominoes, poliamonds, and polypolygons with special attention to tiling characteristics.
      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.
      Puzzle Fun: - Newsletter edited by Rodolfo Kurchan about pentominoes and other math problems.
      Rectifiable Polyomino: - Karl Dahlke explains and demonstrates tiling. Includes C-program source.
      Schröder Triangles, Paths, and Parallelogram Polyominoes: - A paper on their enumeration by Elisa Pergola and Robert A. Sulanke.


      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.
      Solomon W. Golomb: - Home Page of the inventor of polyominoes. Includes biography, black and white picture, research interests and publications list.
      The Soma Cube: - Soma-solving program in QBASIC by Courtney McFarren.
      Soma Cube Applet: - Mehta and Ward Alberg explains the soma cube and provides an applet for practice. Source codes included. [Java.
      Somatic: - A solver for arbitrary polyomino and polycube puzzles. Binary code and source downloads available.
      Sqfig and Sqtile: - Eric Laroche presents computer programs for generating polyominoes and polyomino tilings. Includes source codes in C, and binaries.
      Taniguchi's Programs: - Windows software to solve polyiamond and sliding block puzzles.
      Tesselating Locking Polyominos: - Bob Newman examines the history of the subject and presents his minimal solutions.
      Thorleif's SOMA Page: - SOMA puzzle site with graphics, newsletter and software.


      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).
      Three Nice Pentomino Coloring Problems: - Alexandre Owen Muñiz presents the Icehouse set which lends itself to different polyomino coloring games.
      Tiling a Square With Eight Congruent Polyominoes: - Michael Reid's abstract of a paper in the "Journal of Combinatorial Theory, Series A".
      Tiling of Pythagorean Triplets: - Joe Fields suggests that L-decomposition of squares of Pythagorean triplets could always be tiled.
      The Tiling Puzzle Games of OOG: - Mr. Confetti presents a Windows and Java game for tangrams, polyominoes, and polyhexes.
      Tiling Rectangles and Half Strips with Congruent Polyominoes: - Michael Reid's abstract of paper in the "Journal of Combinatorial Theory, Series A".
      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.
      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".
      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.


      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.
      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.
      What is a Golygon?: - Harry Smith describes Dr. Dewdney's article in the July 1990 Scientific American's Mathematical Recreations column.
      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.


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