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Can you develop a strategy for untangling any tangled ropes, irrespective of the fraction you have ended up with?
Is it possible to start at 0 and end up at any fraction?
Tangling and Untangling
The mathematician John Conway developed an interesting problem about tangling ropes. He discovered that a pair of "tangling operations", twisting and turning, can be represented by a pair of simple arithmetical operations.
Twisting can be represented by adding 1:
$$x \rightarrow x+1$$
To represent turning, transform a number into the negative of its reciprocal:
$$x \rightarrow -\frac{1}{x}$$
Take a look at this video:
If you can't see the video read the description below:
This is how the ropes got tangled:
Twist, twist, turn, twist, twist, twist, turn, twist, twist, twist, turn.
This is the sequence of numbers it produced:
0, 1, 2, -1/2, 1/2, 3/2, 5/2, -2/5, 3/5, 8/5, 13/5, -5/13...
and this is how they got untangled:
Twist, turn. twist, twist, turn. twist, twist, twist, turn, twist, twist, twist,
generating these numbers:
...8/13, -13/8, -5/8, 3/8, -8/3, -5/3, -2/3, 1/3, -3, -2, -1, 0.
Twist, twist, turn, twist, twist, twist, turn, twist, twist, twist, turn.
This is the sequence of numbers it produced:
0, 1, 2, -1/2, 1/2, 3/2, 5/2, -2/5, 3/5, 8/5, 13/5, -5/13...
and this is how they got untangled:
Twist, turn. twist, twist, turn. twist, twist, twist, turn, twist, twist, twist,
generating these numbers:
...8/13, -13/8, -5/8, 3/8, -8/3, -5/3, -2/3, 1/3, -3, -2, -1, 0.
Investigate tangles for yourself, with skipping ropes or string, or by creating sequences of fractions using the two operations.
There are lots of interesting mathematical questions to explore. Choose one of your own to work on or click below for some ideas:
Can you develop a strategy for untangling any tangled ropes, irrespective of the fraction you have ended up with?
Is it possible to start at 0 and end up at any fraction?