Imagine four people standing at the four corners of a square. Let's call the corners after the compass directions NW, NE, SE and SW. To begin, NE and NW each hold an end of one rope, and SE and SW each hold the end of another rope.

There are two operations that the people can perform,

In a

In a

At every stage, the 'tangle' in the ropes can be represented by a number. The initial untangled state is represented by 0.

Take a look at this video to see an example of the ropes being tangled and untangled:

The video begins with the two ropes untangled, representing the number zero.

The operations are performed in the following sequence, resulting in a new fraction at each stage:

With the ropes now tangled, the team begin the process of untangling them again!

At the end of the video, once again the ropes are untangled. Hurrah!

The operations are performed in the following sequence, resulting in a new fraction at each stage:

Operation | Fraction |
---|---|

Twist | $1$ |

Twist | $2$ |

Turn | $-\frac12$ |

Twist | $\frac12$ |

Twist | $1\frac12$, or $\frac32$ |

Twist | $2\frac12$, or $\frac52$ |

Turn | $-\frac25$ |

Twist | $\frac35$ |

Twist | $1\frac35$, or $\frac85$ |

Twist | $2\frac35$, or $\frac{13}5$ |

Turn | $-\frac5{13}$ |

With the ropes now tangled, the team begin the process of untangling them again!

Operation | Fraction |
---|---|

Twist | $\frac8{13}$ |

Turn | $-\frac{13}8$ or $-1\frac58$ |

Twist | $-\frac58$ |

Twist | $\frac38$ |

Turn | $-\frac83$ or $-2\frac23$ |

Twist | $-\frac53$ or $-1\frac23$ |

Twist | $-\frac23$ |

Twist | $\frac13$ |

Turn | $-3$ |

Twist | $-2$ |

Twist | $-1$ |

Twist | $0$ |

At the end of the video, once again the ropes are untangled. Hurrah!

Starting at zero (with both ropes parallel), what would you end with after following this sequence of moves?

Twist, twist, twist, turn, twist, twist, twist, turn, twist, twist, twist, turn

More Twisting and Turning follows on from this problem.