Twisting and Turning

'Twisting and Turning' printed from https://nrich.maths.org/

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This problem introduces a trick that can be done with two skipping ropes. It was invented by the mathematician John Conway.

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, twisting and turning.

In a twist, the person standing at NE swaps places with the person standing at SE, with NE lifting the rope over SE as they swap.

In a turn, everyone moves clockwise one place - NE to SE, SE to SW, SW to NW and NW to NE.

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

Twisting has the effect of adding 1: $$x \mapsto x+1$$
Turning transforms any number into the negative of its reciprocal: $$x \mapsto -\frac1x$$

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

If you can't access YouTube, here is a direct link to the video: Twisting and Turning.mp4
If you can't see the video, click below for a description of the process.

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:

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!

Now that you understand how the two operations work, try this:

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

Can you work out a sequence of moves that will take you back to zero?

You may find it easier to call the turning move "Rotate" so that you can abbreviate the operations as T and R.

More Twisting and Turning follows on from this problem.