### 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...

# Egyptian Fractions

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

This was a tough one, well done to lots of you who sent in lots of different examples, but only Rosie gave a reason and general rule for her findings:

I used Keep It Simple first to find that all unit fractions can be expressed as the sum of two other unit fractions like this,
$\frac{1}{n}=\frac{1}{n+1}+\frac{1}{n(n+1)}$
I noticed that if $n$ was odd, then the numbers $n+1$ and $n(n+1)$ were both even. So if $n$ is odd, say for $m$ a positive integer, $n=2m+1$.
$\frac{2}{n}=\frac{2}{n+1}+\frac{2}{n(n+1)}$
$\frac{2}{n}=\frac{2}{2m+1}=\frac{2}{2m+2}+\frac{2}{(2m+1)(2m+2)}$
$\frac{2}{n}=\frac{2}{2m+1}=\frac{1}{m+1}+\frac{1}{(2m+1)(m+1)}$
Which are unit fractions.

Now if $n$ is even, then $n+1$ is odd, and so will not cancel like above. However, as $n$ is even than half of it is still a whole number. So if $n=2p$, $\frac{2}{n}$ is going to cancel to $\frac{1}{p}$, and we know that $\frac{1}{p}=\frac{1}{p+1}+\frac{1}{p(p+1)}$ from above.
So
$\frac{2}{n}=\frac{1}{p+1}+\frac{1}{p(p+1)}$
which we can write in terms of $n$
$\frac{2}{n}=\frac{2}{n+2}+\frac{4}{n(n+2)}$
These are unit fractions, and so we're done.

Great, can anyone use this to find $\frac{3}{n},\frac{4}{n},\frac{5}{n}$ and so on?