Ben wrote:
Having explored the use of twisting and turning I used the example given to
help me solve the first problem. I found an answer of $-\frac8{21}$ having gone through the sequence of
$1,2,3,-\frac13,\frac23,1 \frac23,2 \frac23,-\frac38,\frac58,1 \frac58,2
\frac58$ to finally get to $-\frac8{21}$
Arkadiusz and Ved also worked out how to get back to $0$. This is Arkadiusz' work:
Mohit from Burnt Mill Academy, Harlow in the UK untangled ropes starting from a different state:
TANGLE :
$-\frac{11}{30}$
UNTANGLE:
$-\frac{11}{30}+1=\frac{19}{30}, \\
\frac{19}{30} \rightarrow -\frac1x=-\frac{30}{19},\\
-\frac{30}{19}+1=-\frac{11}{19},\\
-\frac{11}{19}+1=\frac8{19},\\
\frac8{19}\rightarrow -\frac1x=-\frac{19}8, \\
-\frac{19}8+1=-\frac{11}8, \\
-\frac{11}8+1=-\frac38, \\
-\frac38+1=\frac58,\\
\frac58\rightarrow-\frac1x= -\frac85,\\
-\frac85+1=-\frac35, \\
-\frac35+1=\frac25,
\frac25\rightarrow-\frac1x=-\frac52, \\
-\frac52+1=-\frac32,\\
-\frac32+1=-\frac12,\\
-\frac12+1=\frac12, \\
\frac12\rightarrow-\frac1x=-2, \\
-2+1=-1, \\
-1=1=0$
T,R,T,T,R,T,T,T,R,T,T,R,T,T,T,R (T,T)
Arkadiusz described a method to untangle ropes starting from any position:
We keep twisting until we get a positive value, then we turn and repeat over and over until we get zero.