Octamino is an eight cell polyomino , that is, flat shapes consisting of eight equal squares connected by sides. Octamino figures, as with all polyominoes, involve many tasks of entertaining mathematics.
Apart from different shapes that coincide when turning and mirroring, there are 369 different (“free”) forms of octamino (see figure) [1] . There are 704 types of “one-sided” octaminoes (if specular reflections are considered different figures) and 2725 types of “fixed” octaminoes (turns are also considered different) [2] .
Content
Octamino Shapes by Symmetry Properties
The 369 free figures of octamino can be divided into 8 categories according to their symmetry properties:
- The 316 octamino figures (shown in gray in the picture) are asymmetric;
- 23 octamines (shown in red) have an axis of symmetry parallel to the lines of the square grid;
- 5 oktamino (shown in green) have a diagonal axis of symmetry;
- 18 oktamino (shown in blue) have a second-order central (rotational) symmetry;
- 1 octamino (shown in yellow) have a fourth-order (rotational) symmetry;
- 4 octaminos (shown in purple) have two axes of symmetry parallel to the grid lines;
- 1 octamino (shown in orange) has two diagonal axes of symmetry.
- 1 octamin (shown in blue-green) has four axes of symmetry - two parallel to the grid lines and two diagonal.
Octamino is the smallest polymino order in which all eight possible types of symmetry are realized. The next order of polymino with this property is dodecamino (twelve cell polymino).
If the mirror reflections of the figures are considered different, then the first, fourth and fifth categories are doubled in number, which gives an additional 335 octamino, that is, a total of 704 one-sided octamino.
If turns are also seen as different figures, then
- figures of the first category can be oriented in eight different ways;
- figures from categories from the second to the fourth - four;
- figures from the categories from the fifth to the seventh - two;
- the only figure from the latter category can be oriented in a unique way.
This gives fixed octamino.
Making Octamino Figures
Among the 369 free octamins there are 6 figures with holes (“non-simply-connected”). From this it follows that the continuous coverage of a rectangle with an area squares full set of octamino impossible. However, they can be laid in some rectangles with an area of 2,958 squares with six unicellular openings. Since the number 2958 is a product of simple factors 2 × 3 × 17 × 29, it is possible to raise the question of drawing up rectangles 6 × 493, 17 × 174, 29 × 102, 34 × 87 and 51 × 58.
For the rectangle 51 × 58 there is a solution with a symmetric arrangement of the holes presented in the figure. There is also an octamino stack in three 29 × 34 rectangles, each with two holes near the center. By combining them in various ways, you can get a 34 × 87 rectangle or 29 × 102 with a symmetrical arrangement of three pairs of holes. Solutions for 6 × 493 and 17 × 174 rectangles are not yet known.
Spatial Octamines
Of the 369 spatial octamins, having the form of the usual “flat” octamins, you can assemble a parallelepiped of 8 × 9 × 41. One solution uses all the shapes, except the direct octamin, to assemble eight separate layers of 1 × 9 × 41; direct octamino passes through the centers of all eight layers [3] .
Pseudooctamino
Pseudopolimino is a generalization of polyomino, a set of fields of an infinite chessboard that a king can circumvent [1] . There are 18,770 free (bilateral) [4] , 37,196 unilateral [5] and 147,941 fixed [6] pseudo-octamines.
Notes
- ↑ 1 2 Golomb, 1975 .
- ↑ Weisstein, Eric W. Octomino (Eng.) On Wolfram MathWorld .
- ↑ Ed Pegg, Jr. material added 11 March 2001 . Three-coloring! MathPuzzle.com .
- ↑ Sequence A030222 in OEIS
- ↑ Sequence A030233 in OEIS
- ↑ Sequence A006770 in OEIS
Literature
- Golomb S.V. . Polymino = Polyominoes / Per. from English V. Firsova. Preface and ed. I. Yagloma . - M .: Mir , 1975. - 207 p.