### F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

### Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won \$1 200. What were the assets of the players at the beginning of the evening?

### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

# Farey Sequences

##### Stage: 3 Challenge Level:

If I gave you a list of decimals, you might find it quite straightforward to put them in order of size. But what about ordering fractions?

A man called John Farey investigated sequences of fractions in order of size - they are called Farey Sequences.

The third Farey Sequence, $F_3$, looks like this:

It lists in order all the fractions between $0$ and $1$, in their simplest forms, with denominators up to and including $3$.

Here is $F_4$:

Write down $F_5$.

Which extra fractions are in $F_5$ which weren't in $F_4$?

Use $F_5$ to help you complete $F_6$ and $F_7$.

Here are some questions to consider:

There are lots of extra fractions in $F_{11}$ which are not in $F_{10}$.
There are only a few extra fractions in $F_{12}$ which are not in $F_{11}$.
Can you explain why this is the case?
When will you need lots of extra fractions to get the next Farey Sequence?

Will every Farey Sequence be longer than the one before? How do you know?

So far, all the Farey Sequences have contained an odd number of fractions. Can you find a Farey Sequence with an even number of fractions?

In $F_4$, $\frac{3}{4}$ slotted in between $\frac{2}{3}$ and $\frac{1}{1}$. What do you notice about the fractions on either side when you slot in a new fraction?

Choose any three consecutive fractions from a Farey Sequence. Can you find a way to combine the two outer fractions to make the middle one?

To see how these sequences relate to some beautiful mathematical patterns see the pictures in the problem Ford Circles. There is no need to attempt this problem at this stage - it is aimed at older students.
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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