### Smaller and Smaller

Can you predict, without drawing, what the perimeter of the next shape in this pattern will be if we continue drawing them in the same way?

### Farey Sequences

There are lots of ideas to explore in these sequences of ordered fractions.

### Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

# Twisting and Turning

##### Age 11 to 14Challenge 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.