A polyomino is a polyform with the square as its base form. It is constructed by placing a number of identical squares in distinct locations on the plane, keeping the shape connected, and in such a way that at least one edge of each square coincides with an edge of one of the other squares. Polyominoes with from 1 to 6 squares are called respectively monominoes, dominoes, trominoes (or triominoes), tetrominoes, pentominoes and hexominoes. Related to polyominoes are polyiamonds (formed from equilateral triangles), polyhexes (formed from regular hexagons), and other polyforms.
In some contexts, the definition of a polyomino is relaxed or refined. Sometimes it is requested that polyominoes are simply connected, while on other occasions may have holes (in other words, regions which are not tiled with squares but which are unconnected to the exterior of the polyomino). Sometimes polyominoes are generalised to three or more dimensions by aggregating cubes (polycube)or hypercubes.
Polyominoes have been used in popular puzzles since the late 19th century, but were first studied systematically by Solomon W. Golomb and were popularized by Martin Gardner. The game Tetris is based on tetrominoes.
More on [ Polyomino ]

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].
A dissection puzzle - T. Sillke asks for dissections of two heptominoes into squares.
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]
Meta Description: [ This is a schoolproject of KSO Glorieux Ronse (Belgium) about pentominos. There is also a
competition.You can visit the pentominozoo and solve many puzzles ]
A Puzzle by Enrich Friedman - Every square can be dissected into L-ominoes. Can every Pythagorean square? Conjecture needs proof.
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.
Blocking polyominos - Rodolfo Kurchan asks, for each k, what is the smallest polyomino such that k 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.
Eithan's Pentominoes-3D Applet Solver - Solves given Pentominoes 3D puzzles. Solution is displayed in 3-D with disassembly and rotations. General information and data. [requires Java]
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 - Folded paper polyiamonds which can be unfolded to show hidden faces. Make interesting school projects.
Meta Description: [ Magnus Enarssons hemsida. ]
Flexagons - Conrad and Hartline's 1962 article on Flexagons.
Meta Description: [ $FILE ]
Gamepuzzles - Polyomino and polyform games and puzzles manufactured by Kadon Enterprises Inc.
Meta Description: [ Kadon Enterprises, Inc., online store of original, award-winning puzzles and games in lasercut acrylic and wood. Play in online Puzzle Parlor, read a Journal, enjoy children's stories, coloring book, other interesting attractions and features. Annotated links to good resources. ]
George Huttlin's Puzzle Page - George Huttlin shares some ramblings in the world of polyominoes.
Meta Description: [ In introduction to pentominoes and tetrominoes,
a simple font based on pentominoes, some novel pentomino problems,
and some results of computer searches for solutions. ]
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.
Meta Description: [ Lessons, activities, and links by Henri Picciotto on using geometric puzzles to teach mathematics. ]
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.
Meta Description: [ An introduction to hexiamond puzzles and
some results from computer searches for solutuions. ]
Hyperbolic planar tessellations - Don Hatch's page on hyperbolic tesselations with numerous illustrations.
Meta Description: [ tesselations of the hyperbolic plane ]
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].
Meta Description: [ introduction aux pentominos,Mambo - the dynamic portal engine and content management system ]
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.
Mathforum : a pentomino problem - from the 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.)
404
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.
404
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 - Erich Friedman's Introduction to a variety of packing and tiling problems.
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.
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.
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.
Meta Description: [ Pentomino puzzle game provides geometric puzzles for all puzzle enthusiasts ]
Pentomino applet - Fill up a given area using pentomino shapes, rotating and flipping them. Three levels of difficulty.[Java].
Meta Description: [ Welcome to Fwend.com ! Enjoy our free selection of games and puzzles ! Bet you can't handle the SpaceWorm! Better try Crashpad or Blocks. Or maybe you like Word games? We have them, and more. Come and visit us today! ]
Pentomino Applet - Rujith de Silva's applet puzzle offers games of four different sized rectangles. [Java]
Pentomino Covers - Problems on minimal covers.
Pentomino dissection of a square annulus - From Scott Kim's Inversions Gallery.
Pentomino HungarIQa - Kati presents a pentomino puzzle using poly-rhombs instead of poly-squares. [English/French/German/Hungarian]
Meta Description: [ Pentomino hungarIQa. A játék ]
Pentomino Relationships - Symmetries in the families of rectangular solutions.
Pentomino, Homepage - Lorente Philippe's site describes the building blocks, nomenclature, solutions, and numerous games. (French/English)
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.
Pentominopuzzles. - Pentomino solver with download. Windows 95 and later required. [German/English]
Pentominos - Graphics problems, solutions (including animated GIF) and links. (English/German through main page)
Meta Description: [ Playings with Pentominos ]
Pentominos - B. Berchtold's applet helps tile a 6x10 rectangle. [German]
Pentominos Puzzle Solver - David Eck's graphical solver applet uses recursive technique. Source code available. [Java]
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.
404
Polygon Puzzle - Open source polyomino and polyform placement solitaire game.
Meta Description: [ freshmeat maintains the Web's largest index of Unix and cross-platform open source software. Thousands of applications are meticulously cataloged in the freshmeat database, and links to new code are added daily. ]
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.
Meta Description: [ pentomino and hexomino Fuzion puzzles with pentomino shareware game Fuzion that designs puzzles and finds solutions. By Ken Zeltner of ZeezRealm ]
Polyomino problems and variations of a theme - 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.
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 - A brief essay with some references.
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.
404
Random domino tiling of an Aztec diamond - Matthew Blum demonstrates the properties of random domino tiling of an Aztec diamond. Interactive graphics display.
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.
Meta Description: [ Schröder Triangles, Paths, and Parallelogram Polyominoes ]
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.
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.
Meta Description: [ Somatic - a soma puzzle solver ]
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.
The Geometry Junkyard: Polyominoes - Numerous links, sorted alphabetically.
The Mathematics of Polyominoes - Kevin Gong's home page includes articles, programs for Mac, Win and Java.
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 in the works.
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).
The Poly Pages - About various polyforms - polyominoes, polyiamonds, polycubes, and polyhexes.
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).
The tiling puzzle games of OOG - Mr. Confetti presents a Windows and Java game for tangrams, polyominoes, and polyhexes.
Thorleif's SOMA Page - SOMA puzzle site with graphics, newsletter and software.
Meta Description: [ SOMA puzzle ]
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.
Meta Description: [ Tiling and Packing Results ]
Tiling of Pythagorean triplets - Joe Fields suggests that L-decomposition of squares of Pythagorean triplets could always be tiled.
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.
Meta Description: [ Home page of Professor Jonathan L.F. King,
Mathematics Department of The University of Florida.
Articles in Ergodic Theory and Tiling Theory.
Course materials for current and past mathematics classes:
Combinatorics, Measure Theory, Ergodic Theory, Topology, Complex Analysis.
]
404
Tiling UROP Homepage - Undergraduate Research Project in Random Tilings.
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.
| Maths | |
| Next Video | |