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Twisting and Turning

Age 11 to 14 Challenge Level:

Ben from the UK, Arkadiusz from the Costello School in the UK and Ved from WBGS in the UK worked out how the rope ends up after the series of moves
Twist, twist, twist, turn, twist, twist, twist, turn, twist, twist, twist, turn
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.