![All 18 Pentominoes](https://upload.wikimedia.org/wikipedia/commons/a/aa/All_18_Pentominoes.svg)
Derived from the Greek word for '
5', and "
domino
Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also ca ...
", a pentomino (or 5-omino) is a
polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in pop ...
of order 5, that is, a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
made of 5 equal-sized
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
s connected edge-to-edge. When
rotations and
reflections are not considered to be distinct shapes, there are 12 different ''
free'' pentominoes. When reflections are considered distinct, there are 18 ''
one-sided'' pentominoes. When rotations are also considered distinct, there are 63 ''
fixed
Fixed may refer to:
* ''Fixed'' (EP), EP by Nine Inch Nails
* ''Fixed'', an upcoming 2D adult animated film directed by Genndy Tartakovsky
* Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System
* ...
'' pentominoes.
Pentomino
tiling puzzle
Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...
s and games are popular in
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
.
Usually,
video game
Video games, also known as computer games, are electronic games that involves interaction with a user interface or input device such as a joystick, controller, keyboard, or motion sensing device to generate visual feedback. This fee ...
s such as ''
Tetris
''Tetris'' (russian: link=no, Тетрис) is a puzzle video game created by Soviet software engineer Alexey Pajitnov in 1984. It has been published by several companies for multiple platforms, most prominently during a dispute over the approp ...
'' imitations and
''Rampart'' consider mirror reflections to be distinct, and thus use the full set of 18 one-sided pentominoes.
Each of the twelve pentominoes satisfies the
Conway criterion
In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements:Will It Tile? Try ...
; hence every pentomino is capable of
tiling the plane. Each chiral pentomino can tile the plane without being reflected.
History
![Pentomino Naming Conventions](https://upload.wikimedia.org/wikipedia/commons/6/69/Pentomino_Naming_Conventions.svg)
The earliest puzzle containing a complete set of pentominoes appeared in
Henry Dudeney
Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the country's foremost creators of mathematical puzzles.
Early life
...
's book,
The Canterbury Puzzles
''The Canterbury Puzzles and Other Curious Problems'' is a 1907 mathematical puzzle book by Henry Dudeney. The first part of the book features a series of puzzles based on the characters from ''The Canterbury Tales'' by Geoffrey Chaucer.
Refere ...
, published in 1907. The earliest tilings of rectangles with a complete set of pentominoes appeared in
the Problemist
''The Problemist'' is a bimonthly chess problem magazine which has been in publication since January 1926. It originally had the subtitle "Proceedings of the British Chess Problem Society" but the words "Proceedings of" were dropped in January 1 ...
Fairy Chess Supplement in 1935, and further tiling problems were explored in the PRCS, and its successor, the
Fairy Chess Review
''Fairy Chess Review'' () was a magazine that was devoted principally to fairy chess problems, but also included extensive original results on related questions in mathematical recreations, such as knight's tours and polyominoes (under the title o ...
. Pentominoes were formally defined by American professor
Solomon W. Golomb starting in 1953 and later in his 1965 book ''
Polyominoes: Puzzles, Patterns, Problems, and Packings''.
They were introduced to the general public by
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
in his October 1965
Mathematical Games column
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, ...
in
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
. Golomb coined the term "pentomino" from the
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
/ ''pénte'', "five", and the -omino of
domino
Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also ca ...
, fancifully interpreting the "d-" of "domino" as if it were a form of the Greek prefix "di-" (two). Golomb named the 12
''free'' pentominoes after letters of the
Latin alphabet
The Latin alphabet or Roman alphabet is the collection of letters originally used by the ancient Romans to write the Latin language. Largely unaltered with the exception of extensions (such as diacritics), it used to write English and the o ...
that they resemble.
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
proposed an alternate labeling scheme for pentominoes, using O instead of I, Q instead of L, R instead of F, and S instead of N. The resemblance to the letters is more strained, especially for the O pentomino, but this scheme has the advantage of using 12 consecutive letters of the alphabet. It is used by convention in discussing
Conway's Game of Life
The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is a zero-player game, meaning that its evolution is determined by its initial state, requiring no further ...
, where, for example, one speaks of the R-pentomino instead of the F-pentomino.
Symmetry
* F, L, N, P, and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image. Their
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
consists only of the
identity mapping
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
.
* T, and U can be oriented in 4 ways by rotation. They have an axis of
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares.
* V and W also can be oriented in 4 ways by rotation. They have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection.
* Z can be oriented in 4 ways: 2 by rotation, and 2 more for the mirror image. It has point symmetry, also known as
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
of order 2. Its symmetry group has two elements, the identity and the 180° rotation.
* I can be oriented in 2 ways by rotation. It has two axes of reflection symmetry, both aligned with the gridlines. Its symmetry group has four elements, the identity, two reflections and the 180° rotation. It is the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
of order 2, also known as the
Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
.
* X can be oriented in only one way. It has four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4. Its symmetry group, the dihedral group of order 4, has eight elements.
The F, L, N, P, Y, and Z pentominoes are
chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
; adding their reflections (F′, J, N′, Q, Y′, S) brings the number of ''one-sided'' pentominoes to 18. If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories (T, U, V, W, Z) count fourfold, I counts twice, and X counts only once. This results in 5×8 + 5×4 + 2 + 1 = 63 ''fixed'' pentominoes.
For example, the eight possible orientations of the L, F, N, P, and Y pentominoes are as follows:
![Y-pentomino Symmetry](https://upload.wikimedia.org/wikipedia/commons/d/dd/Y-pentomino_Symmetry.svg)
For 2D figures in general there are two more categories:
* Being orientable in 2 ways by a rotation of 90°, with two axes of reflection symmetry, both aligned with the diagonals. This type of symmetry requires at least a
heptomino
A heptomino (or 7-omino) is a polyomino of order 7, that is, a polygon in the plane made of 7 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hept(a)-. When rotations and reflections are not ...
.
* Being orientable in 2 ways, which are each other's mirror images, for example a
swastika
The swastika (卐 or 卍) is an ancient religious and cultural symbol, predominantly in various Eurasian, as well as some African and American cultures, now also widely recognized for its appropriation by the Nazi Party and by neo-Nazis. It ...
. This type of symmetry requires at least an
octomino
An octomino (or 8-omino) is a polyomino of order 8, that is, a polygon in the plane made of 8 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 369 different ''free'' ...
.
Constructing rectangular dimensions
![Pentomino Puzzle Solutions](https://upload.wikimedia.org/wikipedia/commons/0/09/Pentomino_Puzzle_Solutions.svg)
A standard pentomino puzzle is to
tile
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or o ...
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 6×10 case was first solved in 1960 by
Colin Brian Haselgrove and
Jenifer Haselgrove. 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 (which sometimes provides an additional solution in a simple way). The 5×12 box has 1010 solutions, the 4×15 box has 368 solutions, and the 3×20 box has just 2 solutions (one is shown in the figure, and the other one can be obtained from the solution shown by rotating, as a whole, the block consisting of the L, N, F, T, W, Y, and Z pentominoes).
A somewhat easier (more symmetrical) puzzle, the 8×8 rectangle with a 2×2 hole in the center, was solved by
Dana Scott
Dana Stewart Scott (born October 11, 1932) is an American logician who 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, Ca ...
as far back as 1958. There are 65 solutions. Scott's algorithm was one of the first applications of a
backtracking
Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it de ...
computer program. 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.
Efficient algorithms have been described to solve such problems, for instance by
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
. Running on modern
hardware, these pentomino puzzles can now be solved in mere seconds.
The pentomino set is the only free polyomino set that can be packed into a rectangle, with the exception of the trivial
monomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in pop ...
and
domino
Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also ca ...
sets, each of which consists only of a single rectangle.
Filling boxes
A pentacube is a
polycube
upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
A puzzle involving arranging nine L tricubes into a 3×3 cube
A polycube is a solid figure formed by j ...
of five cubes. Of the 29 pentacubes, exactly twelve pentacubes are flat (1-layer) and 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 the 12 flat pentacubes, i.e. cover it without overlap and without gaps. Since each pentacube has a volume of 5 unit cubes, the box must have a volume of 60 units. Possible sizes are 2×3×10 (12 solutions), 2×5×6 (264 solutions) and 3×4×5 (3940 solutions). Following are one solution of each case.
Alternatively one could also consider combinations of five cubes that 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 in 145 cubes, which will not make a 3D box (as 145 can only be 29×5×1, which the non-flat pentominoes cannot fit into).
Board games
There are
board game
Board games are tabletop games that typically use . These pieces are moved or placed on a pre-marked board (playing surface) and often include elements of table, card, role-playing, and miniatures games as well.
Many board games feature a comp ...
s of skill based entirely on pentominoes. Such games are often simply called "Pentominoes".
One of the games 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. This version of Pentominoes is called "Golomb's Game".
The two-player version has been
weakly solved in 1996 by Hilarie Orman. It was proved to be a first-player win by examining around 22 billion board positions.
Pentominoes, and similar shapes, are also the basis of a number of other tiling games, patterns and puzzles. For example, the French board game ''
Blokus
Blokus ( ) is an abstract strategy board game for two to four players, where players try to score points by occupying most of the board with pieces of their colour. The board is a square regular grid and the pieces are polyominoes. It was designe ...
'' is played with 4 colored sets of
polyominoes
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in po ...
, each consisting of every pentomino (12), tetromino (5), triomino (2) domino (1) and monomino (1). Like the game ''Pentominoes'', the goal is to use all of your tiles, and a bonus is given if the monomino is played on the last move. The player with the fewest blocks remaining wins.
The game of ''
Cathedral
A cathedral is a church that contains the '' cathedra'' () of a bishop, thus serving as the central church of a diocese, conference, or episcopate. Churches with the function of "cathedral" are usually specific to those Christian denomination ...
'' is also based on
polyominoes
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in po ...
.
Parker Brothers
Parker Brothers (known by Parker outside of North America) was an American toy and game manufacturer which in 1991 became a brand of Hasbro. More than 1,800 games were published under the Parker Brothers name since 1883. Among its products wer ...
released a multi-player pentomino board game called ''Universe'' in 1966. Its theme is based on a deleted scene from the 1968 film ''
2001: A Space Odyssey'' in which an astronaut is playing a two-player pentomino game against the
HAL 9000 computer (
a scene with a different astronaut playing chess was retained). 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 four 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 five) on each side for more than two players.
Game manufacturer
Lonpos has a number of games that use the same pentominoes, but on different game planes. Their ''101 Game'' has a 5 x 11 plane. By changing the shape of the plane, thousands of puzzles can be played, although only a relatively small selection of these puzzles are available in print.
Literature
Pentominoes were featured in a prominent subplot of
Arthur C. Clarke's 1975 novel ''
Imperial Earth
''Imperial Earth'' is a science fiction novel by British writer Arthur C. Clarke, published in 1975 by Gollancz Books. The plot follows the protagonist, Duncan Makenzie, on a trip to Earth from his home on Titan, in large part as a diplo ...
''. Clarke also wrote an essay in which he described the game and how he got hooked on it.
They were also featured in
Blue Balliett
Blue Balliett (born 1955 in New York) is an American author, who lives with her husband, three children, a grandson, and a cat. She is best known for her award-winning novel for children, '' Chasing Vermeer''. She was born Elizabeth Balliett, but ...
's ''
Chasing Vermeer'', which was published in 2003 and illustrated by
Brett Helquist
Brett L. Helquist (born November 1965) is an American illustrator best known for his work in the children's book series ''A Series of Unfortunate Events''. As such, his illustrations for that series have appeared in multiple media, including the bo ...
, as well as its sequels, ''
The Wright 3'' and ''
The Calder Game
''The Calder Game'' is a children's novel written by Blue Balliett and illustrated by Brett Helquist, published in 2008. It is the sequel to '' The Wright 3'', which in turn is the sequel to '' Chasing Vermeer''. Some underlying themes includ ...
''.
In the New York Times crossword puzzle for June 27, 2012, the clue for an 11-letter word at 37 across was "Complete set of 12 shapes formed by this puzzle's black squares."
Video games
* ''
Tetris
''Tetris'' (russian: link=no, Тетрис) is a puzzle video game created by Soviet software engineer Alexey Pajitnov in 1984. It has been published by several companies for multiple platforms, most prominently during a dispute over the approp ...
'' was inspired by pentomino puzzles, although it uses four-block tetrominoes. Some Tetris clones and variants, like the game ''5s'' included with
Plan 9 from Bell Labs
Plan 9 from Bell Labs is a distributed operating system which originated from the Computing Science Research Center (CSRC) at Bell Labs in the mid-1980s and built on UNIX concepts first developed there in the late 1960s. Since 2000, Plan 9 has be ...
, and ''
Magical Tetris Challenge'', do use pentominoes.
* ''
Daedalian Opus'' uses pentomino puzzles throughout the game.
See also
Previous and Next orders
*
Tetromino
A tetromino is a geometric shape composed of four squares, connected orthogonally (i.e. at the edges and not the corners). Tetrominoes, like dominoes and pentominoes, are a particular type of polyomino. The corresponding polycube, called a tetracu ...
*
Hexomino
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. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not c ...
Others
*
Tiling puzzle
Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...
*
''Cathedral'' board game
*
Solomon W. Golomb
Notes
References
Chasing Vermeer with information about the book Chasing Vermeer and a click-and-drag pentomino board.
*
External links
Pentomino configurations and solutionsAn exhaustive listing of solutions to many of the classic problems showing how each solution relates to the others.
{{Polyforms
Mathematical games
Polyforms
Solved games