Pentomino

The 12 pentominos

Pentomino (also pentamino ) is a polyomino of the 5th order, i.e. H. a (flat) polygon that results from placing 5 squares of the same size next to each other so that the individual squares are edge to edge.

In addition, Pentomino also denotes a puzzle for one person. Under the name Pentominos it was published as a two-person version by Hallmark Verlag .

The game pieces: Pentominos

There are - apart from symmetry - 12 different pentominos. The word pentomino was invented by mathematician Solomon W. Golomb and first used in 1954 in an article in the American Mathematical Monthly . Polyominos as a superordinate group were first discussed in detail in Scientific American in 1957.

For a better understanding, the 12 pentominos (or game pieces) have been designated with letters that correspond to the approximate shape of the piece. The chiral (oriented) pieces L, Y, N, P, Z and F do not match their mirror image; However, if you turn one of these stones, you get its reflection. In this way, the 12 Pentaminos lead to a total of 18 forms in the Pentomino games.

The level can be tiled with every single one of the pentominos , with the chiral pentominos even without turning them around.

Solved pentominos

Puzzle for one person

In the pentomino puzzle , also known as the pentomino puzzle, the task is to place certain figures from the twelve pentominos as pieces (also: plates) - similar to tangram :

• Place all pentominos in a rectangle so that all 12 tiles are used and every square of the rectangle is occupied. So the rectangle always consists of small squares. ${\displaystyle 12\cdot 5=60}$

3 × 20	4 × 15	5 × 12	6 × 10
2 solutions	368 solutions	1010 solutions	2339 solutions

• Place square fields with cutouts
  • The larger 8 × 8 variants (12 plates):

2170 solutions	188 solutions	65 solutions	21 solutions	74 solutions

• The smaller 7 × 7 variants (9 plates):

• Fill in a space that is three times the size of a single tile with 9 of the 11 other tiles:

Number of possible solutions (without reflections):

Pentomino	F.	I.	L.	N	P	T	U	V	W.	X	Y	Z
solutions	125	19th	113	68	497	106	48	63	91	15th	86	131

• All pentominos can be composed of pentominos:

• Other forms:

    X               X
   XXX             XXX
  XXXXX           XXXXX
 XXXXXXX         XXXXXXX
XXXXXXXXX         XXXXX
                   XXX
                    X

There are other puzzle games, such as the birthday puzzle, in which twelve game pieces have to be placed on an 8 × 8 playing field in such a way that the four remaining squares indicate a specific date.

Game for two people

A possible set of rules for a strategy game with two (or more) people would be the following:

• All parts are split up by each player taking turns taking a part.
• When all parts are taken, the game begins. The players take turns placing one of their game pieces on a previously selected playing field (for example a chess board).
• The player who cannot place any more pieces on the playing field first has lost.

In addition, it must be determined whether parts can be turned (which corresponds to an axis mirroring).

There are several commercially available multi-person games that work with pentomino or polyomino tokens.

An (incomplete) list:

• Pentominos (published by Hallmark in 1973, based on the ideas of Golomb ). The 12 pieces are placed alternately on a chessboard until no more moves are possible - the player with the last move wins. There is a winning strategy for the player who starts.
• Blokus (flat stones, probably the best known)
• Duopento (flat stones, rule similar to the one above)
• Cathedral (The tokens represent buildings in a medieval city that occupy 1–6 fields in different formations on a 10 × 10 playing field, depending on the building. Rules similar to the above.)
• Rumis (3D stones)
• Tower builder game (3D bricks)
• Ubongo (flat stones)
• Ubongo 3D (3D bricks)

In the Schlag-den-Star issue of December 10, 2016, the game was named Katamino and played with a shared pool of 12 parts on an 8 × 8 field without the possibility of turning .

3D pentomino

Instead of squares, the game pieces can also be made from cubes (they are then also called pentacubes ). From these pieces, just like from the Soma cube, many different three-dimensional objects can be placed, for example cuboids with the following dimensions:

• 5 × 4 × 3: 3940 solutions
• 6 × 5 × 2: 264 solutions
• 10 × 3 × 2: 12 solutions

You can also enlarge some of the pieces yourself. Each cube in the stone to be reproduced is reproduced using a 2x2x3 block.

The following game pieces can be recreated: F with 1, P with 1082, U with 10, Z with 24, T with 3, V with 21, N with 51, Y with 7 and L with 99 solutions.

Pentomino as a computer game

In addition to the form of the game to touch, Pentomino was (and is) often implemented as tinkering on the computer . Pentomino inspired Alexei Paschitnow to create Tetris .

Animation of the F-Pentomino in Conway's Game of Life .

In the " Game of Life ", a two-dimensional cellular automaton designed by the English mathematician John Horton Conway , the relatively simple starting figure of the F-pentomino shows completely chaotic behavior before it forms an oscillating structure from the 1103rd step on.

variants

Instead of plates with 5 squares, the game is also available with plates made up of 6 squares. This variant is called Hexamino and has 35 different plates. Heptamino has 108 and Oktamino 369 different plates.

The 5 different Tetramino plates (with 7 shapes that cannot be reached by turning), made up of 4 squares, have found their way into the computer game Tetris .

Instead of squares, other geometric figures can also be chosen: equilateral triangles, hexagons, rectangles, even groups of two or more different figures. You don't have to let the figures butt against each other with the full edge, but can shift them by half, for example. The possibilities for variation are enormous.

The L game for two people is also a variant, but here only one character (per player) is played.

"Parallel polarized" game pieces

The so-called "polarized" game pieces are an interesting variant. If one imagines that the plane is traversed by vertical or horizontal (parallel) "polarization fields", one can distinguish two versions of most game pieces, ie practically a "horizontal" and a "vertical" version. Only the pieces W, X and V are, so to speak, polarized “in themselves” and therefore only appear simply. Of course, when constructing puzzles, it is important to ensure that the parts can only be used in one "polarization direction". The following 21 parts result under the stated conditions:

Parallel polarized game pieces

Solution examples

Periodic pattern

If the twelve Pentomino tiles are bendable, they can be used to cover the jacket of a suitably dimensioned straight circular cylinder completely and without overlapping (picture on the left). While the laying of the tiles in the rectangle is limited on four sides, it is limited in only two directions on the cylinder; the surface lines are not an obstacle for the tiles.

The unrestricted unrolling of the cylinder jacket in a plane provides a flat, infinitely long strip with the periodic repetition of a pattern, which is composed of one of the twelve Pentomino tiles each (illustration on the right).

The idea of periodicity can be extended by the entire plane so with a different twelve Pentomino plates composite pattern parquetted that through translations is displayed in two different directions in coming. The figure on the left shows the simply connected fundamental domain of a tiling with periods 10 and 6.

In contrast to the interrelationship between stripes and cylinders, in the case of double periodicity there is no closed surface on which the framed fundamental area can be mapped with true length . In the meantime, a torus , the surface of which is divided into sixty quadrilaterals by equidistant circles of longitude and latitude, can be painted according to the pattern given by the fundamental area (left here) (figure on the right).

Number of periodic solutions compared to a rectangle :

Rectangle : sides a and b; Cylinder : circumference a, height b; Level : periods a and b.

a × b	3 × 20	4 × 15	5 × 12	6 × 10	10 × 6	12 × 5	15 × 4	20 × 3
rectangle	2	368	1010	2339	2339	1010	368	2
cylinder	281728	628610	1844817	576619	28996	8272	901	2
level	160768	672778	1315356	1329411	1329411	1315356	672778	160768

literature

• Günter Albrecht-Bühler: The Pentomino workshop. A cookbook of new geometric patterns for logical thinkers and puzzle enthusiasts . Fischer-Taschenbuch-Verlag, Frankfurt am Main 1992, ISBN 3-596-10487-4 , ( Fischer 10487 Fischer logo ).
• Blue Balliet: The Pentomino Oracle .
• Jack Botermanns, Jerry Slocum: The world's puzzles. How to build them and how to solve them . Hugendubel, Munich 1987, ISBN 3-88034-336-5 .
• Pieter van Delft, Jack Botermanns: Mind games of the world. Puzzles, riddles, games of skill, puzzles . German adaptation by Eugen Oker . 2nd Edition. Hugendubel, Munich 1981, ISBN 3-88034-087-0 .
• Solomon W. Golomb : Polyominoes. Puzzles, Patterns, Problems and Packings . Princeton University Press, Princeton NJ 1994, ISBN 0-691-08573-0 .
• Maria Koth, Notburga Grosser: The Pentomino Book. Thinking game fun for children from 9 to 99. Maths for copying . Aulis-Verlag Deubner, Cologne 2004, ISBN 3-7614-2543-0 .

Web links

Commons : Pentomino  - collection of images, videos and audio files

• Pentomino at mathematische-basteleien.de
• Gerard's Universal Polyomino Solver (English)
• David Eck's Pentominos Puzzle Solver an excellent solver with all possible options
• Pentominoes (Dutch, English, French)
• Online games inspired by Pentomino. Drag, rotate and tilt, assemble Pentomino tiles, solve picture or number puzzles, Tetromino, Solitaire, ... (English)

Individual evidence

1. ↑ The game is marketed under the name Pentomino Puzzle . from Logoplay wooden games, Pentomino . from Bartl GmbH or Pentominos . at edumero
2. ^ Pentominos in the Luding game database
3. ↑ The Pentomino workshop ( Memento of the original from September 27, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. at spectrum of science 
4. ^ Glenn C. Rhoads: Planar Tilings and the Search for an Aperiodic Prototile. PhD dissertation, Rutgers University, 2003.
5. ↑ Martin Gardner: More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes. Scientific American 233 (2), 1975, 112-115.
6. ↑ See Gerard's Polyomino Solution Page , Nos. 37.1-37.12 (Triplications).
7. ↑ Hilarie K. Orman: Pentominoes: A First Player Win (PDF; 131 kB) . In: Richard J. Nowakowski (Ed.): Games of no chance: combinatorial games at MSRI, 1994 . Cambridge University Press, Cambridge 1996, ISBN 0-521-57411-0