### 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...

# How Long Is the Cantor Set?

##### Age 11 to 14 Challenge Level:

In the problem The Cantor Set, we met the Cantor set, which is the limit of $C_n$ as $n$ tends to infinity.

We can talk about the length of one of our sets $C_n$.
The set $C_1$ has length 1.
The set $C_2$ has length $\frac{2}{3}$, as this is the total length of the line segments in $C_2$.

What are the lengths of $C_3$, $C_4$ and $C_5$?

Can you find a general expression for the length of $C_n$?

By considering what happens as $n$ tends to infinity, can you find the length of the Cantor set?