Pentomino
![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/aa/All_18_Pentominoes.svg/370px-All_18_Pentominoes.svg.png)
Derived from the Greek word for '5', and "domino",apentomino(or5-omino) is apolyominoof order 5; that is, apolygonin theplanemade of 5 equal-sizedsquaresconnected edge to edge. Whenrotationsandreflectionsare not considered to be distinct shapes, there are 12 differentfreepentominoes. When reflections are considered distinct, there are 18one-sidedpentominoes. When rotations are also considered distinct, there are 63fixedpentominoes.
Pentominotiling puzzlesand games are popular inrecreational mathematics.[1]Usually,video gamessuch asTetrisimitations andRampartconsider mirror reflections to be distinct, and thus use the full set of 18 one-sided pentominoes. (Tetris itself uses 4-square shapes.)
Each of the twelve pentominoes satisfies theConway criterion;hence, every pentomino is capable oftiling the plane.[2]Eachchiralpentomino can tile the plane without being reflected.[3]
History
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Pentomino_Naming_Conventions.svg/330px-Pentomino_Naming_Conventions.svg.png)
The earliest puzzle containing a complete set of pentominoes appeared inHenry Dudeney's book,The Canterbury Puzzles,published in 1907.[4]The earliest tilings of rectangles with a complete set of pentominoes appeared inthe ProblemistFairy Chess Supplement in 1935, and further tiling problems were explored in the PFCS, and its successor, theFairy Chess Review.[5][6]: 127
Pentominoes were formally defined by American professorSolomon W. Golombstarting in 1953 and later in his 1965 bookPolyominoes: Puzzles, Patterns, Problems, and Packings.[1][7]They were introduced to the general public byMartin Gardnerin his October 1965Mathematical Games columninScientific American.Golomb coined the term "pentomino" from theAncient Greekπέντε/pénte,"five", and the -omino ofdomino,fancifully interpreting the "d-" of "domino" as if it were a form of the Greek prefix "di-" (two). Golomb named the 12freepentominoes after letters of theLatin alphabetthat they resemble, using themnemonicFILiPiNo along with the end of the alphabet (TUVWXYZ).[8]: 23
John Horton Conwayproposed 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 discussingConway's Game of Life,where, for example, one speaks of the R-pentomino instead of the F-pentomino.
Symmetry
[edit]- F, L, N, P, and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image. Theirsymmetry groupconsists only of theidentity mapping.
- T, and U can be oriented in 4 ways by rotation. They have an axis ofreflectionaligned 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 asrotational symmetryof 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 thedihedral groupof order 2, also known as theKlein four-group.
- 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 arechiral;adding their reflections (F′, J, N′, Q, Y′, S) brings the number ofone-sidedpentominoes 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 = 63fixedpentominoes.
The eight possible orientations of the F, L, N, P, and Y pentominoes, and the four possible orientations of the T, U, V, W, and Z pentominoes are illustrated:
- Eight possible orientations
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F-pentomino
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L-pentomino
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N-pentomino
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P-pentomino
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Y-pentomino
- Four possible orientations
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T-pentomino
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U-pentomino
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V-pentomino
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W-pentomino
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Z-pentomino
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 aheptomino.
- Being orientable in 2 ways, which are each other's mirror images, for example aswastika.This type of symmetry requires at least anoctomino.
Games
[edit]Tiling puzzle (2D)
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/09/Pentomino_Puzzle_Solutions.svg/440px-Pentomino_Puzzle_Solutions.svg.png)
A standardpentomino puzzleis totilea 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 byColin Brian HaselgroveandJenifer Haselgrove.[9]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 byDana Scottas far back as 1958.[10]There are 65 solutions. Scott's algorithm was one of the first applications of abacktrackingcomputer program. Variations of this puzzle allow the four holes to be placed in any position. One of the external links uses this rule.
Efficient algorithms have been described to solve such problems, for instance byDonald Knuth.[11]Running on modernhardware,these pentomino puzzles can now be solved in mere seconds.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Pentomino_unsolvable.svg/330px-Pentomino_unsolvable.svg.png)
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.
The pentomino set is the only free polyomino set that can be packed into a rectangle, with the exception of the trivialmonominoanddominosets, each of which consists only of a single rectangle.
Box filling puzzle (3D)
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/47/Pentomino_Cube_Solutions.svg/440px-Pentomino_Cube_Solutions.svg.png)
Apentacubeis apolycubeof 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.
Apentacube puzzleor 3Dpentomino 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).[12]
Alternatively one could also consider combinations of five cubes that are themselves 3D, i.e., those which include more than just the 12 "flat" single-layer thick combinations of cubes. However, in addition to the 12 "flat" pentacubes formed by extruding the pentominoes, there are 6 sets of chiral pairs and 5 additional pieces, forming a total of 29 potential pentacube pieces, which gives 145 cubes in total (=29×5); as 145 can only be packed into a box measuring 29×5×1, it cannot be formed by including the non-flat pentominoes.
Commercial board games
[edit]There areboard gamesof 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".[13]
The two-player version has beenweakly solvedin 1996 by Hilarie Orman. It was proved to be a first-player win by examining around 22 billion board positions.[14]
Pentominoes, and similar shapes, are also the basis of a number of other tiling games, patterns and puzzles. For example, the French board gameBlokusis played with 4 colored sets ofpolyominoes,each consisting of every pentomino (12), tetromino (5), triomino (2) domino (1) and monomino (1). Like the gamePentominoes,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 ofCathedralis also based onpolyominoes.[15]
Parker Brothersreleased a multi-player pentomino board game calledUniversein 1966. Its theme is based on a deleted scene from the 1968 film2001: A Space Odysseyin which an astronaut is playing a two-player pentomino game against theHAL 9000 computer(a scene with a different astronaut playing chesswas 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 manufacturerLonposhas a number of games that use the same pentominoes, but on different game planes. Their101 Gamehas 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.
Video games
[edit]- Tetriswas inspired by pentomino puzzles, although it uses four-block tetrominoes. Some Tetris clones and variants, like the game5sincluded withPlan 9 from Bell Labs,andMagical Tetris Challenge,do use pentominoes.
- Daedalian Opususes pentomino puzzles throughout the game.
Literature
[edit]Pentominoes were featured in a prominent subplot ofArthur C. Clarke's 1975 novelImperial Earth.Clarke also wrote an essay in which he described the game and how he got hooked on it.[16]
They were also featured inBlue Balliett'sChasing Vermeer,which was published in 2003 and illustrated byBrett Helquist,as well as its sequels,The Wright 3andThe Calder Game.[17]
InThe New York Timescrossword puzzlefor 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."[18]
See also
[edit]Previous and Next orders
[edit]Others
[edit]![](https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
- Tiling puzzle
- Cathedralboard game
- Solomon W. Golomb
Notes
[edit]- ^ab"Eric Harshbarger - Pentominoes".
- ^Rhoads, Glenn C. (2003).Planar Tilings and the Search for an Aperiodic Prototile.PhD dissertation, Rutgers University.
- ^Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes".Scientific American.233(2): 112–115.doi:10.1038/scientificamerican0775-112.
- ^"The Project Gutenberg eBook of The Canterbury Puzzles, by Henry Ernest Dudeney".www.gutenberg.org.Retrieved2022-03-26.
- ^"Dissection Problems in PFCS/FCR: Summary of Results in Date Order".www.mayhematics.com.Retrieved2022-03-26.
- ^Gardner, Martin(1988)."13: Polyominoes".Hexaflexagons and other mathematical diversions.The University of Chicago Press. pp. 124–140.ISBN0-226-28254-6.
- ^"people.rit.edu - Introduction - polyomino and pentomino".
- ^Golomb, Solomon W.;Lushbaugh, Warren (1965).Polyominoes.New York: Charles Scribner's Sons.LCCN64-24805.
- ^C. B. Haselgrove; Jenifer Haselgrove (October 1960)."A Computer Program for Pentominoes"(PDF).Eureka.23:16–18.
- ^Dana S. Scott (1958). "Programming a combinatorial puzzle". Technical Report No. 1, Department of Electrical Engineering, Princeton University.
- ^Donald E. Knuth."Dancing links"(Postscript, 1.6 megabytes). Includes a summary of Scott's and Fletcher's articles.
- ^Barequet, Gill; Tal, Shahar (2010). "Solving General Lattice Puzzles". In Lee, Der-Tsai; Chen, Danny Z.; Ying, Shi (eds.).Frontiers in Algorithmics.Lecture Notes in Computer Science. Vol. 6213. Berlin Heidelberg:Springer Science+Business Media.pp.124–135.doi:10.1007/978-3-642-14553-7_14.ISBN978-3-642-14552-0.
- ^Pritchard (1982),p. 83.
- ^Hilarie K. Orman.Pentominoes: A First Player Win(Pdf).
- ^"FAQ".
- ^Could you solve Pentominoes?by Arthur C. Clarke,Sunday Telegraph Magazine,September 14, 1975; reprinted in Clarke'sAscent to Orbit: A Scientific Autobiography,New York: John Wiley & Sons, 1984.ISBN047187910X
- ^Chasing Vermeer,by Blue Balliett, Scholastic Paperbacks,ISBN0439372976
- ^Buckley, Mike (June 27, 2012). Shortz, Will (ed.)."The Crossword".New York Times.Retrieved30 July2020.
References
[edit]- Chasing Vermeer,with information about the book Chasing Vermeer and a click-and-drag pentomino board.
- Pritchard, D. B.(1982). "Golomb's Game".Brain Games.Penguin Books Ltd.pp. 83–85.ISBN0-14-00-5682-3.
External links
[edit]- Pentomino configurations and solutionsAn exhaustive listing of solutions to many of the classic problems showing how each solution relates to the others.