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:
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?
Twisting has the effect of adding 1: $$x\rightarrow x + 1$$ Turning transforms any number into the negative of its reciprocal: $$x\rightarrow -\frac{1}{x}$$ Starting at zero, the first five moves: Twist, twist, twist, turn, twist
Can you continue from there and then return to zero? You can use the reciprocal button on a calculator ($\frac{1}{x}$ ) to help with the calculations.
Edward and Thomas from Dartford Grammar School worked out that:
Starting at zero (with both ropes parallel), the sequencetwist, twist, twist, turn , twist, twist, twist, turn , twist, twist, twist, turn
takes us to:$$0, 1, 2, 3, -\frac{1}{3}, \frac{2}{3}, \frac{5}{3}, \frac{8}{3}, -\frac{3}{8}, \frac{5}{8}, \frac{13}{8}, \frac{21}{8}, -\frac{8}{21}$$
Terence from The Garden International School in Kuala Lumpur, William from Shebbear College and Akshita from Tiffin Girls' School worked out how to disentangle themselves:
The following sequence takes us back to zero:
twist, turn , twist, twist, turn , twist, twist, twist, turn , twist, twist, twist, turn , twist, twist: $$ \frac{13}{21}, -\frac{21}{13}, -\frac{8}{13}, \frac{5}{13}, -\frac{13}{5}, -\frac{8}{5}, -\frac{3}{5}, \frac{2}{5}, -\frac{5}{2}, -\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, -2, -1, 0$$
The procedure is repeated until we get to 1/n.
Now we turn again, then twist n times to get the fraction back to 0.
The full proof is given in the problem More Twisting and Turning
A chance to experiment with fractions in an intriguing context.
This is the first of three related problems. The second problem is More Twisting and Turning and the third is All Tangled Up .
To find out more about John Conway's problem take a look here .