digplanet beta 1: Athena
Share digplanet:


Applied sciences






















The 35 free hexominoes

A hexomino (or 6-omino) is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge.[1] The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different free hexominoes. When reflections are considered distinct, there are 60 one-sided hexominoes. When rotations are also considered distinct, there are 216 fixed hexominoes.[2][3]


The figure shows all possible free hexominoes, coloured according to their symmetry groups:

  • 20 hexominoes (coloured grey) have no symmetry. Their symmetry group consists only of the identity mapping.
  • 6 hexominoes (coloured red) have an axis of mirror symmetry aligned with the gridlines. Their symmetry group has two elements, the identity and a reflection in a line parallel to the sides of the squares.
  • 2 hexominoes (coloured green) have an axis of mirror symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection.
  • 5 hexominoes (coloured blue) have point symmetry, also known as rotational symmetry of order 2. Their symmetry group has two elements, the identity and the 180° rotation.
  • 2 hexominoes (coloured purple) have two axes of mirror symmetry, both parallel to the gridlines (thus one horizontal axis and one vertical axis). Their symmetry group has four elements. It is the dihedral group of order 2, also known as the Klein four-group.

If reflections of a hexomino are considered distinct, as they are with one-sided hexominoes, then the first and fourth categories above would each double in size, resulting in an extra 25 hexominoes for a total of 60. If rotations are also considered distinct, then the hexominoes from the first category count eightfold, the ones from the next three categories count fourfold, and the ones from the last category count twice. This results in 20 × 8 + (6+2+5) × 4 + 2 × 2 = 216 fixed hexominoes.

Packing and tiling[edit]

Each of the 35 hexominoes satisfies the Conway criterion; hence every hexomino is capable of tiling the plane.[4]

Although a complete set of 35 hexominoes has a total of 210 squares, it is not possible to pack them into a rectangle. (Such an arrangement is possible with the 12 pentominoes which can be packed into any of the rectangles 3 × 20, 4 × 15, 5 × 12 and 6 × 10.) A simple way to demonstrate that such a packing of hexominoes is not possible is via a parity argument. If the hexominoes are placed on a checkerboard pattern, then 11 of the hexominoes will cover an even number of black squares (either 2 white and 4 black or vice versa) and 24 of the hexominoes will cover an odd number of black squares (3 white and 3 black). Overall, an even number of black squares will be covered in any arrangement. However, any rectangle of 210 squares will have 105 black squares and 105 white squares.

However, there are other simple figures of 210 squares that can be packed with the hexominoes. For example, a 15 × 15 square with a 3 × 5 rectangle removed from the centre has 210 squares. With checkerboard colouring, it has 106 white and 104 black squares (or vice versa), so parity does not prevent a packing, and a packing is indeed possible.[5] It is also possible for two sets of pieces to fit a rectangle of size 420, or for the set of 60 one-sided hexominoes (18 of which cover an even number of black squares) to fit a rectangle of size 360.[6]

Polyhedral nets for the cube[edit]

All 11 unfoldings of the cube

A polyhedral net for the cube is necessarily a hexomino, with 11 hexominoes actually being nets. They appear on the right, again coloured according to their symmetry groups.


  1. ^ Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8. 
  2. ^ Weisstein, Eric W. "Hexomino". From MathWorld – A Wolfram Web Resource. Retrieved 2008-07-22. 
  3. ^ Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics 36: 191–203. doi:10.1016/0012-365X(81)90237-5. 
  4. ^ Rhoads, Glenn C. (2003). Planar Tilings and the Search for an Aperiodic Prototile. PhD dissertation, Rutgers University. 
  5. ^ Mathematische Basteleien: Hexominos (English)
  6. ^ Hexomino Constructions

External links[edit]

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Hexomino — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
184 videos foundNext > 

Hexomino game AI at different levels. (Like tetris, but with 6-block pieces.)

In this video, I showcase how Pierre Dellacherie's Tetris AI fares with a mixed tetromino+pentomino+hexomino piece set. It is insanely difficult even for a human.

Hexomino Meaning

Video shows what hexomino means. A polyomino made up of six squares.. hexomino synonyms: 6-omino. Hexomino Meaning. How to pronounce, definition ...

Conway's game of life files pattern 9 Pi Heptomino And A Hexomino Pi Parent

The pi heptomino is a fairly commoly occuring pattern in conway's game of life, either as a stand-alone pattern or as debris created by part of a larger pattern.

Hexomino Rush

This is a board game I made for my Fundamentals of Game Design class.

Conway's game of life files pattern 11 Century Hexomino

The century is another commonly occuring pattern in john conway's ogame of life. As the name may suggest, this pattern survives for about 100 generations (103 ...

Mathematics Geometry Topology n-Polytopes Polygons Polyominos Hexominos

Hexominos!! Very important figures in understanding how to play the game Tac Tix, by Piet Hein. Here are all 35 Hexominos, and at the end, the word ...

March 2013 Time Trial: "Hexominos" - 55.8 solution

A winning solution to the first Time Trial level from March 2013 played by jblewis (in MS mode). Level "Hexominos" designed by Miika Toukola. For more details ...

Block Puzzle For Iphone

http://itunes.apple.com/us/app/block-puzzles/id511736378 Block puzzle is a addictive polyomino puzzle game. There are domino ,tromino, tetromino, pentomino ...


CUBOS DESPLEGADOS La figura adjunto representa en cubo que muestra sólo tres caras y se va a desplegar para mostrar todos las caras en un mismo plano ...

Conway's Game Of Life Files Pattern 49: Century Eater

This pattern is a classic example of an eater, which is any pattern which can interact with another pattern in such a way that the other pattern gets removed and ...

184 videos foundNext > 

4 news items

DenverBroncos.com (blog)
Mon, 28 Apr 2014 13:37:52 -0700

ENGLEWOOD, Colo. – It will be a long time before the disappointment around the way the 2013 season ended subsides for Broncos players and fans. But this offseason, the players are using that defeat as a motivator for their offseason workouts.
Tue, 23 Jan 2007 16:00:00 -0800

The milk chocolate version has 12 pieces, each a pentomino (5 "chocolate" squares), while the dark chocolate version has 11 pieces, each a hexomino (6 squares). They aren't available for retail online, it seems, but you may be able to find it at your ...
Mon, 26 May 2014 20:01:30 -0700

「94-14 REMIX」はTOWA TEIのデビュー20周年記念盤。「Luv Connection」のMasters At Workミックス、「HAPPY」のフランキー・ナックルズミックス、「Sound Museum」の細野晴臣ミックス、「Radio」の砂原良徳ミックスや、「Technova」をサンプリングしたA Tribe Called ...
Wed, 15 Jan 2014 19:02:11 -0800

吉木りさが初のオリジナルアルバム「ペントミノ」を3月5日にリリースすることが明らかになった。 本作はヒャダインこと前山田健一プロデュースのシングル曲「ボカロがライバル☆」「世界は教室だけじゃない」、吉木りさ&ヒャダイン名義で発表されたクリスタルキング「愛をとりもどせ!

Oops, we seem to be having trouble contacting Twitter

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!

Searchlight Group

Digplanet also receives support from Searchlight Group. Visit Searchlight