490:. 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).
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570:, 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).
573:
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
221:
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
651:
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
589:
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
288:
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
523:
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.
574:
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.
478:
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.
275:
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
263:
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
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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.
168:
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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.
433:
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
139:
Comparison of labeling schemes for the 12 possible pentomino shapes. The first naming convention is the one used in this article. The second method is Conway's.
296:
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:
559:
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.
614:, 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.
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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.
600:
Pentominoes, and similar shapes, are also the basis of a number of other tiling games, patterns and puzzles. For example, the French board game
1033:
Dana S. Scott (1958). "Programming a combinatorial puzzle". Technical Report No. 1, Department of
Electrical Engineering, Princeton University.
1085:
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501:
computer program. Variations of this puzzle allow the four holes to be placed in any position. One of the external links uses this rule.
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consider mirror reflections to be distinct, and thus use the full set of 18 one-sided pentominoes. (Tetris itself uses 4-square shapes.)
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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."
948:
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1223:
1174:
1155:
1056:
Barequet, Gill; Tal, Shahar (2010). "Solving
General Lattice Puzzles". In Lee, Der-Tsai; Chen, Danny Z.; Ying, Shi (eds.).
853:
Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes".
677:
was inspired by pentomino puzzles, although it uses four-block tetrominoes. Some Tetris clones and variants, like the game
201:, fancifully interpreting the "d-" of "domino" as if it were a form of the Greek prefix "di-" (two). Golomb named the 12
639:
527:
The pentomino set is the only free polyomino set that can be packed into a rectangle, with the exception of the trivial
1512:
1259:
An exhaustive listing of solutions to many of the classic problems showing how each solution relates to the others.
610:, each consisting of every pentomino (12), tetromino (5), triomino (2) domino (1) and monomino (1). Like the game
256:
Z can be oriented in 4 ways: 2 by rotation, and 2 more for the mirror image. It has point symmetry, also known as
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in 1996 by
Hilarie Orman. It was proved to be a first-player win by examining around 22 billion board positions.
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additional 25 squares (two more rows of 10 and one offset row of five) on each side for more than two players.
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Fairy Chess
Supplement in 1935, and further tiling problems were explored in the PFCS, and its successor, the
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A somewhat easier (more symmetrical) puzzle, the 8×8 rectangle with a 2×2 hole in the center, was solved by
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The 12 pentominoes can form 18 different shapes, with 6 of them (the chiral pentominoes) being mirrored.
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151:, published in 1907. The earliest tilings of rectangles with a complete set of pentominoes appeared in
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as far back as 1958. There are 65 solutions. Scott's algorithm was one of the first applications of a
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F, L, N, P, and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image. Their
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of skill based entirely on pentominoes. Such games are often simply called "Pentominoes".
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Sample solutions to pentacube puzzles of the stated dimensions, drawn one layer at a time.
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81:
1007:
718:. Clarke also wrote an essay in which he described the game and how he got hooked on it.
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has a number of games that use the same pentominoes, but on different game planes. Their
1219:, with information about the book Chasing Vermeer and a click-and-drag pentomino board.
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of order 2. Its symmetry group has two elements, the identity and the 180° rotation.
1046:(Postscript, 1.6 megabytes). Includes a summary of Scott's and Fletcher's articles.
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Efficient algorithms have been described to solve such problems, for instance by
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883:"The Project Gutenberg eBook of The Canterbury Puzzles, by Henry Ernest Dudeney"
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Being orientable in 2 ways, which are each other's mirror images, for example a
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in which an astronaut is playing a two-player pentomino game against the
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The earliest puzzle containing a complete set of pentominoes appeared in
1064:. Lecture Notes in Computer Science. Vol. 6213. Berlin Heidelberg:
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pentominoes. When rotations are also considered distinct, there are 63
48:
284:; adding their reflections (F′, J, N′, Q, Y′, S) brings the number of
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T, and U can be oriented in 4 ways by rotation. They have an axis of
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pentominoes. When reflections are considered distinct, there are 18
907:"Dissection Problems in PFCS/FCR: Summary of Results in Date Order"
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in 1966. Its theme is based on a deleted scene from the 1968 film
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514:
461:
134:
18:
67:
are not considered to be distinct shapes, there are 12 different
992:
1266:
512:, these pentomino puzzles can now be solved in mere seconds.
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1262:
965:"people.rit.edu - Introduction - polyomino and pentomino"
535:
sets, each of which consists only of a single rectangle.
429:
For 2D figures in general there are two more categories:
28:
840:
Planar
Tilings and the Search for an Aperiodic Prototile
162:
Pentominoes were formally defined by
American professor
215:
FILiPiNo along with the end of the alphabet (TUVWXYZ).
169:
Polyominoes: Puzzles, Patterns, Problems, and
Packings
127:
pentomino can tile the plane without being reflected.
1006:
C. B. Haselgrove; Jenifer
Haselgrove (October 1960).
943:. The University of Chicago Press. pp. 124–140.
708:
Pentominoes were featured in a prominent subplot of
633:
released a multi-player pentomino board game called
1420:
1379:
1358:
1300:
1186:Buckley, Mike (June 27, 2012). Shortz, Will (ed.).
731:, which was published in 2003 and illustrated by
649:a scene with a different astronaut playing chess
172:. They were introduced to the general public by
940:Hexaflexagons and other mathematical diversions
184:. Golomb coined the term "pentomino" from the
1278:
819:
817:
444:. This type of symmetry requires at least an
115:Each of the twelve pentominoes satisfies the
8:
1146:, September 14, 1975; reprinted in Clarke's
166:starting in 1953 and later in his 1965 book
1169:, by Blue Balliett, Scholastic Paperbacks,
1148:Ascent to Orbit: A Scientific Autobiography
699:uses pentomino puzzles throughout the game.
1285:
1271:
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482:The 6×10 case was first solved in 1960 by
1150:, New York: John Wiley & Sons, 1984.
1100:
280:The F, L, N, P, Y, and Z pentominoes are
16:Geometric shape formed from five squares
842:. PhD dissertation, Rutgers University.
813:
590:Pentominoes is called "Golomb's Game".
363:
298:
119:; hence, every pentomino is capable of
1257:Pentomino configurations and solutions
987:. New York: Charles Scribner's Sons.
7:
1008:"A Computer Program for Pentominoes"
1479:
869:10.1038/scientificamerican0775-112
14:
606:is played with 4 colored sets of
222:used by convention in discussing
207:pentominoes after letters of the
27:Derived from the Greek word for '
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825:"Eric Harshbarger - Pentominoes"
593:The two-player version has been
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1114:Pentominoes: A First Player Win
1066:Springer Science+Business Media
268:of order 2, also known as the
211:that they resemble, using the
1:
59:connected edge to edge. When
1140:Could you solve Pentominoes?
1078:10.1007/978-3-642-14553-7_14
981:; Lushbaugh, Warren (1965).
197:, "five", and the -omino of
721:They were also featured in
300:Eight possible orientations
1539:
735:, as well as its sequels,
365:Four possible orientations
189:
1476:
1226:(1982). "Golomb's Game".
1144:Sunday Telegraph Magazine
1060:Frontiers in Algorithmics
838:Rhoads, Glenn C. (2003).
178:Mathematical Games column
92:and games are popular in
766:Previous and Next orders
688:Magical Tetris Challenge
94:recreational mathematics
539:Box filling puzzle (3D)
47:of order 5; that is, a
578:Commercial board games
548:
520:
484:Colin Brian Haselgrove
467:
149:The Canterbury Puzzles
140:
55:made of 5 equal-sized
24:
1142:by Arthur C. Clarke,
691:, do use pentominoes.
683:Plan 9 from Bell Labs
640:2001: A Space Odyssey
546:
518:
465:
239:consists only of the
224:Conway's Game of Life
138:
22:
508:. Running on modern
176:in his October 1965
911:www.mayhematics.com
856:Scientific American
519:Unsolvable patterns
258:rotational symmetry
182:Scientific American
1513:Mathematical games
1234:. pp. 83–85.
1112:Hilarie K. Orman.
979:Golomb, Solomon W.
753:The New York Times
655:Game manufacturer
549:
521:
488:Jenifer Haselgrove
468:
458:Tiling puzzle (2D)
219:John Horton Conway
157:Fairy Chess Review
141:
25:
1500:
1499:
1359:Higher dimensions
1232:Penguin Books Ltd
1087:978-3-642-14552-0
1042:Donald E. Knuth.
935:"13: Polyominoes"
887:www.gutenberg.org
802:Solomon W. Golomb
645:HAL 9000 computer
623:is also based on
164:Solomon W. Golomb
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1407:Pseudo-polyomino
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174:Martin Gardner
153:the Problemist
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738:The Wright 3
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1484:WikiProject
1392:Polydrafter
1366:Polyominoid
1302:Polyominoes
1228:Brain Games
1068:. pp.
984:Polyominoes
667:Video games
625:polyominoes
612:Pentominoes
608:polyominoes
584:board games
470:A standard
424:Z-pentomino
412:W-pentomino
400:V-pentomino
388:U-pentomino
376:T-pentomino
359:Y-pentomino
347:P-pentomino
335:N-pentomino
323:L-pentomino
311:F-pentomino
98:video games
96:. Usually,
65:reflections
1507:Categories
1444:Snake cube
1402:Polyiamond
1210:References
1175:0439372976
1156:047187910X
916:2022-03-26
892:2022-03-26
798:board game
704:Literature
582:There are
495:Dana Scott
248:reflection
88:Pentomino
1518:Polyforms
1439:Soma cube
1412:Polystick
1387:Polyabolo
1335:Heptomino
1325:Pentomino
1320:Tetromino
1294:Polyforms
795:Cathedral
772:Tetromino
620:Cathedral
553:pentacube
435:heptomino
286:one-sided
276:elements.
147:'s book,
76:one-sided
61:rotations
45:polyomino
37:pentomino
1454:Hexastix
1371:Polycube
1350:Decomino
1345:Nonomino
1340:Octomino
1330:Hexomino
1023:: 16–18.
993:64-24805
933:(1988).
777:Hexomino
761:See also
661:101 Game
635:Universe
557:polycube
529:monomino
510:hardware
446:octomino
442:swastika
230:Symmetry
213:mnemonic
100:such as
31:', and "
1460:Tantrix
1449:Tangram
1426:puzzles
1397:Polyhex
1315:Tromino
1197:30 July
131:History
123:. Each
109:Rampart
57:squares
51:in the
49:polygon
43:) is a
41:5-omino
1494:Portal
1467:Tetris
1434:Blokus
1380:Others
1310:Domino
1238:
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1084:
1072:–135.
1016:Eureka
991:
947:
783:Others
685:, and
674:Tetris
657:Lonpos
603:Blokus
566:or 3D
533:domino
474:is to
282:chiral
199:domino
125:chiral
103:Tetris
33:domino
1422:Games
1127:"FAQ"
1011:(PDF)
808:Notes
555:is a
453:Games
291:fixed
195:pénte
190:πέντε
82:fixed
53:plane
35:", a
1424:and
1236:ISBN
1199:2020
1171:ISBN
1152:ISBN
1082:ISBN
989:LCCN
945:ISBN
741:and
531:and
486:and
476:tile
204:free
70:free
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1074:doi
1070:124
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750:In
725:'s
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