The toy consists of a code wheel, that can be rotated as whole around the sphere, as well as paired sectors can be flipped.

as negative. Without restricting any solutions, be the shifts within -6 and 7.

A rotation is a half turn on the axis, while holding the center piece. Be the

the axis horizontal and the movable part with 4 numbers to the left. Counting the

positions starts on the center piece top left, clockwise.

Be an operation defined as a shift followed by a rotation.

Be a sequence a number of operations followed by a shift.

Be a setting of numbers a vector, the startup vector is 1,2,3,4,..13, yellow.

I found several cycles, paths giving the identity. I also found several paths leading to

the same combination.

Hence the questions :

- from a given vector, are all possible vectors reachable ?
- how long is the longest of the shortest paths ?

with

- 4 operations, there are 28k choices
- 6 operations, there are 4.8M choices
- 8 operations, there are 815M choices
- 10 operations, there are 138G choices

operations leading to the start vector.

operation(s) | times for identity |
---|---|

+1/ ( or -1/ symmetry) | 8 |

+2/ | 12 |

+3/ | 4 |

+4/ | 12 |

+5/ | 6 |

+6/ | 14 |

+7/ | 14 |

+1/-1/ | 20 |

+2/-2/ | 12 |

+1/+2/ ( or +2/+1/ commute) | 13 |

+1/+2/+3 or any permutation | 12 |

+2/+2/ as expected | 6 |

+3/+3/ as expected | 2 |

+3/+3/+3/ | 4 |

+3/+3/+3/+3/ as expected | 1 |

+1/+2/+3/+4/ | 16 |

+1/+3/+2/+4/ | 40 |

+2/-2/+1/ | 72 |

There is a great number of short cycles. To fill the space of paths, they have

to be avoided.

It appears strange that unsystematic tests with up to 7 operation to be repeated

showed rather short cycles.

There seem to be a greater number than assumed. Some unsystematic tests with the

number of path found depending on the search level (= how many operations)

operations | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|

+1/ | 3 | 3 | 67 | 67 | 1443 |

-1/ | 3 | 3 | 67 | 67 | 1423 |

+2/ | 3 | 3 | 67 | 67 | 1422 |

-2/ | 3 | 3 | 67 | 67 | 1417 |

+1/-1/ | 1 | 5 | 5 | 114 | NA |

-1/+1/ | 1 | 5 | 5 | 115 | 115 |

nothing.

To be continued ...

would match the end vector. Let the shifts run from -6 to +7 and

go to a predefined depth.

I may publish the software at a later stage on one of my delphi pages

simulations

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last updated subpage 27.dec.00

Copyright (99,2000) Ing.Büro R.Tschaggelar