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

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.

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

A Dissection Puzzle
T. Sillke asks for dissections of two heptominoes into squares.

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

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 Ferrers Shapes
Alon, Bóna, and Spencer show that one can't cover very much of an n by p(n) rectangle with staircase polyominoes (where p(n) is the number of these shapes).

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 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 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.

Polyomino Tiling
Joseph Myers classifies the n-ominoes up to n=15 according to how symmetrically they can tile the plane.

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

Random Domino Tiling of an Aztec Diamond
Matthew Blum's undergraduate project demonstrates the properties of random domino tiling of an Aztec diamond. Interactive graphics display included.

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 & 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.

Square into Similar Triangles
T.Sillke discusses the dissection problem.

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 and Packing Results of Torsten Sillke
Polyominoes, polycubes and polyspheres.

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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