Symmetric octominoid configurations

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Symmetric configurations with vanishing invariants not listed in table magic27.gif of Jürgen Köller [Ko] (included below) and taken from: http://www.mathematische-basteleien.de/magics.htm:

TABLE 1

011-013-101-110-112-123-211-213s

011-013-101-112-123-211-213-312s

011-013-101-103-110-112-121-211s

011-013-103-105-112-114-123-213s

011-013-101-110-112-121-211-231s

011-013-031-123-132-134-143-233s

011-013-101-110-114-123-211-213s

011-013-015-101-103-123-125-213s

112-121-123-132-211-233-312-332s

112-121-123-211-213-231-323-332s


Copy of the table magic27.gif
TABLE 2


Snapshots of real puzzle configurations

TABLE 3

011-013-101-110-112-123-211-213s

011-013-101-112-123-211-213-312s

011-013-101-103-110-112-121-211s

011-013-103-105-112-114-123-213s

011-013-101-110-112-121-211-231s

011-013-031-123-132-134-143-233s

011-013-101-110-114-123-211-213s

011-013-015-101-103-123-125-213s

112-121-123-132-211-233-312-332s

112-121-123-211-213-231-323-332s

Trivia and remarks


Octominoid code syntax

An octominoid is described by a string of the form xyz-xyz-xyz-xyz-xyz-xyz-xyz-xyz[s]. It consists of a set of 8 squares of side 2 with vertices having even integer coordinates.

Each sequence of three integers xyz indicates the position of a square by means of the coordinates of its center. The only even coordinate indicates the orientation of the square.

The presence of the (optional) final 's' indicates that the given configuration corresponds to a symmetric octominoid (with respect to any isometry of space).

Two octominoids are considered equivalent if related by an isometry of space (reflections included), the printed configuration descriptions correspond to a canonical selection among all equivalent octominoids, based on a suitable total ordering. The software code can compute the canonical representative of a given configuration.


Rubik's Magic configuration code syntax

A Rubik's Magic configurations is a circular list of the eight tiles positioned in space in such a way that consecutive tiles are connected through a common side.

Each tile can be oriented by drawing an arrow on one of its faces parallel to a side as explained in [Pao]. A valid Rubik's Magic configuration admits a consistent orientation of the tiles (one tile is oriented in any way, then all the others uniquely inherit their orientation). As explained in [Pao] the tile orientation is inherited by reflecting the arrow drawn on an adjacent tile.

In the octominoid configurations, consecutive tiles form an angle with three possible values:

Now a configuration description consists of a sequence of eight directions that indicate the relative position of a tile with respect to the previous one. A direction consists of a capital letter among R, L, U, D, standing repectively for right, left, up, down. The capital letter is followed by m in case of a mountain fold, by v in case of a valley fold.

For example the sequence RRRURRRU is the flat starting configuration of the puzzle.

A sequence can be prefixed by a \ if the starting tile is of backslash type (the type of a tile, slash or backslash, is decided once we chose its orientation and depends on the direction of the nylon strings). Two configurations are considered equivalent if they are the same up to choice of the first tile of the circular list, the traversal direction of circular list and the choice of orientation (slash and backslash tiles must be considered as distinct).


Web References

[Ko]
http://www.mathematische-basteleien.de/magics.htm
[Pao]
http://arxiv.org/abs/1401.3699
[Ver]
Tom Verhoeff (1987). Magic and Is Nho Magic, Cubism For Fun (15): 24-31. Retrieved 2014-08-28.