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Interesting facts about polyominoes

The term polyomino is attributed to the American mathematician Solomon Wolf Golomb and was first mentioned in 1954 in an article in "American Mathematical Monthly".

Polyominos are surfaces consisting of one or more squares of equal size with common edges. Depending on the number of squares (n) that make up a surface, the resulting shapes are also called n-polyomino or n-mino. In addition, the following terms are also widely used:

  • Monomino (consisting of one square)
  • Domino (consisting of two squares)
  • Tromino (consisting of three squares)
  • Tetromino (consisting of four squares)
  • Pentomino (consisting of five squares)

A closer look at these configurations reveals that a certain number of unique shapes can be created with a fixed number of squares.

Unique in this context means that congruent or chiral forms of pieces created by rotation or reflection are taken into account as one form.

To illustrate this fact, the unique shapes of the configurations of one to five squares are shown below.

Monomino

monomino.png
(n = 1): There exists one form

Domino

domino.png
(n = 2): There exists one form

Tromino

tromino.png
(n = 3): There are two forms

Tetromino

tetromino.png
(n = 4): There are five forms

Pentomino

pentomino.png
(n = 5): There are twelve forms

Application of polyominoes

Since their discovery, polyominos have been widely used in puzzles and games. The aim of these games is to fill a given area with the available tiles.

Without any claim to completeness, some variants of this type of game are presented below:

Classical pentomino

Original Pentomino Spiel

The "classic" Pentomino is the original game form and possibly also the oldest variant of this category of puzzles. The tiles are representing the full set of twelve pentominoes.

The aim of the game is to place the shapes of the twelve pentominos on a rectangular game board (mostly with the dimensions 6 x 10 fields) so that the area of the rectangle is completely filled. This can be achieved in 2339 different ways.

Tetris

Tetris differs from the classic Pentomino in the game principle. Only tetrominos are used as tiles, which can be rotated but not flipped. For this reason, seven different tile shapes are available in the game (the five Tetrominos, as well as two flipped tiles).

tetris

Tetris is about placing the offered tiles in such a way that lines of the playing field are completely filled. As soon as a line is completely filled, the area of the playing field occupied by the line is released again. The size of the playing field is therefore - at least theoretically - unlimited.

LONPOS

Original Lonpos Spiel

LONPOS 4D is a beautiful designed game with a high degree of popularity. This version of LONPOS also supports 3D tasks in which the tiles have to be arranged as a pyramid.

LONPOS offers twelve polyominos in the form of eight pentominos, three tetrominos and one tromino. The 2D playing field comprises 55 fields and has the shape of an isosceles triangle. With the available tiles the playing field can be completely covered in 25833 different variants.

pentofun

The tile shapes of pentofun are defined similar to LONPOS and consisting of twelve different polyominos. In addition to eight pentominos, pentofun also uses three tetrominos and one tromino.

The gamenboard dimensions of pentofun are 7 x 8 fields. The field in the lower right corner of the playing field cannot be occupied, thus 55 playing fields are available. Theoretically, the pentofun gameboard can be completely covered with the available tiles in 843223 different ways.

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