A pentomino is a polyomino composed of five (Greek πέντε / pente) congruent squares, connected orthogonally. Image File history File links Pentominos. ... The 35 possible hexominoes. ... As an abstract term, congruence means similarity between objects. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

There are twelve different pentominoes, and they are named after the letters of the Latin alphabet that they resemble. Ordinarily, the reflection symmetry and rotation symmetry of a pentomino does not count as a different pentomino. Abcdefghijklmnopqrstuvwxyz redirects here. ... Figures with the axes of symmetry drawn in. ... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ...

F, L, N, P, Y, and Z pentominoes are chiral in two dimensions; adding their reflections (F', J, N', Q, Y', S) brings the number of "one-sided" pentominoes to 18. The others, lettered I, T, U, V, W, and X, are equivalent to some rotation of their mirror images. This matters in some computer games, where mirror image moves are not allowed, such as Tetris-clones and Rampart. In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ... Tetris (Russian: ) is a falling-blocks puzzle video game, released on a large spectrum of platforms. ... This page is a candidate for speedy deletion. ...

Each of the twelve pentominoes can be tiled to fill the plane. In addition, each chiral pentomino can be tiled without using its reflection. A tessellated plane seen in street pavement. ... This article is about the mathematical construct. ...

John Horton Conway proposed an alternate labeling scheme -- he uses O instead of I, Q instead of L, R instead of F, and S instead of N. The resemblance to the letters is a bit more strained, but this scheme has the advantage that it uses 12 consecutive letters of the alphabet. In reference to Conway's Game of Life, this scheme is used, so it talks about the R-pentomino instead of the F-pentomino. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Gospers Glider Gun creating gliders. The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. ...

Considering rotations of multiples of 90 degrees only, there are the following symmetry categories: Sphere symmetry group o. ...

• L, N, P, F and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image.
• Z can be oriented in 4 ways: 2 by rotation, and 2 more for the mirror image.
• T, V, U and W can be oriented in 4 ways by rotation.
• I can be oriented in 2 ways by rotation.
• X can be oriented in only one way.

For 2D figures in general there is one more category: being orientable in 2 ways, which are each other's mirror image, for example a swastika. There is no pentomino in this category (this type of symmetry requires at least an octomino). This article is about the symbol. ... The 35 possible hexominoes. ...

For example, the eight possible orientations of the Y pentomino are as follows:

Tiling rectangles

A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box must have an area of 60 units. Possible sizes are 6×10, 5×12, 4×15 and 3×20. The avid puzzler can probably solve these problems by hand within a few hours. A more challenging task, typically requiring a computer search, is to count the total number of solutions in each case. Image File history File links Pentomino_sol. ... A tessellated plane seen in street pavement. ... A puzzle undone, which forms a cube Puzzle cube; a type of puzzle For other uses, see Puzzle (disambiguation). ... In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions. ...

The 6×10 case was first solved in 1960 by C. B. Haselgrove and Jenifer Haselgrove.^[1] There are exactly 2339 solutions, excluding trivial variations obtained by rotation and reflection of the whole rectangle, but including rotation and reflection of a subset of pentominoes (sometimes this is possible and provides in a simple way an additional solution; e.g., with the 3×20 solution shown, the other one is obtained by rotating a set of seven pentominoes, or put differently, by rotating the four leftmost and the rightmost to the other side).

The 5×12 box has 1010 solutions, the 4×15 box has 368 solutions, and the 3×20 box has just 2 solutions.

A somewhat easier (more symmetrical) puzzle, the 8×8 rectangle with a 2×2 hole in the center, was solved by Dana Scott as far back as 1958^[2]. There are 65 solutions. Scott's algorithm was one of the first applications of a backtracking computer program ever. Variations of this puzzle allow the four holes to be placed in any position. One of the external links uses this rule. Most such patterns are solvable, with the exceptions of placing each pair of holes near two corners of the board in such a way that both corners could only be fitted by a P-pentomino, or forcing a T-pentomino or U-pentomino in a corner such that another hole is created. Dana Stewart Scott (born 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. ... Backtracking is a type of algorithm that is a refinement of brute force search. ...

Efficient algorithms have been described to solve such problems, for instance by Donald Knuth^[3]. Running on modern hardware, these pentomino puzzles can now be solved in mere seconds. Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ... A stylised illustration of a modern personal computer A personal computer (PC) is a computer whose original sales price, size, and capabilities make it useful for individuals. ...

Filling boxes

A pentacube is a polycube of five cubes. Twelve of the 29 pentacubes correspond to the twelve pentominoes extruded to a depth of one square. A pentacube puzzle or 3D pentomino puzzle, amounts to filling a 3-dimensional box with these 1-layer pentacubes, i.e. cover it without overlap and without gaps. Each of the 12 pentacubes consists of 5 unit cubes, and are like 2D pentominoes but with unit thickness. Clearly the box must have a volume of 60 units. Possible sizes are 2×3×10, 2×5×6 and 3×4×5. Following are several solutions.
In recreational mathematics, a polycube is a polyform with a cube as the base form. ...

 2 x 3 x 10 box P P F N N W T U X U V V V Z N N N U U U P P F F W W T X X X V Z Z Z Y I I I I I P F F W W T T T X L V Z Y Y Y Y L L L L 2 x 5 x 6 box P P P N N N P P L L L L Y W N N X U F F L Z Z U Y W W X X X V F F Z T U Y Y W W X U V F Z Z T U Y I I I I I V V V T T T 3 x 4 x 5 box F F V V V X F F P T U F U P P U U U P P X N N N V X L T T T X L L L L I I I I I N N Z Z V X W W Z T W W Y Z Z W Y Y Y Y 

Alternatively one could also consider combinations of five cubes which are themselves 3D, i.e., are not part of one layer of cubes. However, in addition to the 12 extruded pentominoes, 6 sets of chiral pairs and 5 pieces make total 29 pieces, resulting 145 cubes, which will not make 3D box. Furthermore, it will be a computer puzzle.

Trivia

Pentominoes are a key part of the 2003 children's novel Chasing Vermeer by Blue Balliett and its 2005 sequel, The Wright 3. The plot revolves around pentominoes and there is a puzzle for readers to solve throughout the book that is based on pentominoes. The books are based in Johannes Vermeer's painting A lady writing a letter. Image File history File links Broom_icon. ... Chasing Vermeer is a childrens book by Blue Balliett and illustrated by Brett Helquist, illustrator of A Series of Unfortunate Events. ... Elizabeth Blue Balliett Klein is an American author, best known for her award-winning novel for children, Chasing Vermeer. ... The Wright 3 is the sequel to Chasing Vermeer, by Blue Balliett and illustrated by Brett Helquist. ...

Pentominoes are prominently featured in a subplot of the novel Imperial Earth by Arthur C. Clarke. Imperial Earth (ISBN 0-15-144233-9) is a novel written by Arthur C. Clarke, and published in time for the U.S. bicentennial in 1976 by Ballantine Books. ... Wikiquote has a collection of quotations related to: Arthur C. Clarke Sir Arthur Charles Clarke, CBE (born 16 December 1917) is a British science-fiction author and inventor, most famous for his novel 2001: A Space Odyssey, and for collaborating with director Stanley Kubrick on the film of the same...

"Pentominoes" was registered as a trademark by Solomon W. Golomb (#1008964 USPTO 1975 April 15), but this trademark is no longer in effect as of 1982. “(TM)” redirects here. ... Solomon W. Golomb Solomon Wolf Golomb (b. ... PTO headquarters in Alexandria The United States Patent and Trademark Office (PTO or USPTO) is an agency in the United States Department of Commerce that provides patent and trademark protection to inventors and businesses for their inventions and corporate and product identification. ... Year 1975 (MCMLXXV) was a common year starting on Wednesday (link will display full calendar) of the Gregorian calendar. ... is the 105th day of the year (106th in leap years) in the Gregorian calendar. ...

Other patterns that 9 of the 12 Pentomino pieces will solve, are the shapes of each of the 12 pieces enlarged x3. The German Wikipedia article mentions this too, giving examples in images.

A puzzle similar to Pentomino, also with 12 pieces, but each made of 6 equilateral triangles, exists under the name Hexiamond. The pieces must cover a 6x6 parallelogram divided in 72 equilateral triangles. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A polyiamond is a counterpart to a polyomino where the polygon used as the building block is an equilateral triangle rather than a square. ...

Gabriel Industries of Hagerstown, Maryland US sold a pentomino set as a puzzle under the trade name Hexed, along with a line of other geometry puzzles, including a tangram set called Pythagoras and Crazy Quilt (which was based on irregular shapes). A typical tangram construction Tangram (Chinese: ; Pinyin: ; literally seven boards of cunning) is a Dissection puzzle. ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Crazy Quilt was a villian in the Batman comic books. ...

Board game

There is a board game of skill based entirely on pentominoes, called pentominoes. A board game is a game played with counters or pieces that are placed on, removed from, or moved across a board (a premarked surface, usually specific to that game). ...

The game is played on an 8×8 grid by two or three players. Players take turns in placing pentominoes on the board so that they do not overlap with existing tiles and no tile is used more than once. The objective is to be the last player to place a tile on the board.

The two-player version has been weakly solved; it is a first-player win. A two-player game can be solved on several levels. ...

Pentominoes, and similar shapes, are also the basis of a number of other tiling games, patterns and puzzles. For example, a French board game called Blokus is played with 4 opposing color sets of polyominoes. In Blokus, each color begins with every pentomino (12), as well as every tetromino (5), every tromino (2), every domino (1) , and every monomino (1). Like the game Pentominoes, the goal is to use all of your tiles, and a bonus is given if the monomino is saved for the very last move. The player with the fewest blocks remaining wins. Blokus is an abstract strategy board game for two to four players, invented by Bernard Tavitian and published in 2000 by the Sekkoia company. ...

Parker Brothers released a multi-player pentomino board game called Universe in 1966. Its theme is based on an outtake from the movie 2001: A Space Odyssey in which the astronaut (seen playing chess in the final version) is playing a two-player pentomino game against a computer. The front of the board game box features scenes from the movie as well as a caption describing it as the "game of the future". The game comes with 4 sets of pentominoes (in red, yellow, blue, and white). The board has two playable areas: a base 10x10 area for two players with an additional 25 squares (two more rows of 10 and one offset row of 5) on each side for more than two players. The Parker Brothers logo. ...

Pentominoes featured in an episode of The A-Team where BA Barracus had to solve the pentominoes puzzle to save his friends and the world. For the 2008 movie, see The A-Team (film). ...

Video games

• Tetris was inspired by pentomino puzzles, although it uses four-block tetrominoes.
• Daedalian Opus uses pentomino puzzles throughout the game.
• Yohoho! Puzzle Pirates carpentry minigame is based on pentomino puzzles.

Tetris (Russian: ) is a falling-blocks puzzle video game, released on a large spectrum of platforms. ... Daedalian Opus was a puzzle game for the Game Boy released in June of 1990. ... Yohoho! Puzzle Pirates is a massively multiplayer online game by Three Rings Design. ...

See also

• Tiling puzzle
• Sudoku variants

A tessellated plane seen in street pavement. ... This article is about the logic puzzle. ...

References

Eureka is a publication of the Cambridge University Mathematical Society (The Archimedians). Its focus is on articles of interest to the society, including problems and puzzles, discussions and humour. ...

External links

• Pentamino for Pocket PC is a GPLed implementation of the puzzle with a few fun patterns besides ordinary boxes.
• Pentas a small puzzle game and solver.