### 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...

### Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

### Special Sums and Products

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

# Farey Sequences

##### Age 11 to 14Challenge 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:

$$\frac01 \qquad \frac13 \qquad \frac12 \qquad \frac23 \qquad \frac11$$

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

$F_4$ lists all the fractions between $0$ and $1$, in their simplest forms, with denominators up to and including $4$. Can you write it out? Click below to check once you've had a go.

$$\frac01 \qquad \frac14 \qquad \frac13 \qquad \frac12 \qquad \frac23 \qquad \frac34 \qquad \frac11$$

Now that you've got the hang of it, write $F_5$.

Which extra fractions are in $F_5$ which weren't in $F_4$?
Which extra fractions will be in $F_6$ which aren't in $F_5$?
Where will they appear in the sequence?

There are lots of questions you could explore about Farey Sequences. Here are just a few that we thought of:
• How many extra fractions are there in $F_{11}$ that aren't in $F_{10}$?
• How many extra fractions are there in $F_{12}$ that aren't in $F_{11}$?
• Is every Farey Sequence longer than the one before? How do you know?
• Is there a way of working out how many fractions there will be in the next sequence?
• So far, all the Farey Sequences except $F_{1}$ have contained an odd number of fractions. Can you find a Farey Sequence with an even number of fractions?

Send us your solutions to these and any other questions you decide to explore!

There is a curious link between the mediant of two fractions and Farey Sequences. See the problem Mediant Madness to learn more about mediants, and Farey Neighbours to apply it to Farey Sequences. Finally, the problem Ford Circles shows the relationship between these fractions and some beautiful geometrical patterns.