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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?

Farey Sequences

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

This problem concerns a sequence of numbers named after John Farey, an English geologist. He wrote about these sequences in an article called "On a curious property of vulgar fractions", published in 1816. Back in those days, "vulgar fractions" were what we now call "improper fractions"; "vulgar" meant (and still does mean) commonplace, rather than obscene or uncouth! The "curious property" that Farey noticed is something that you are asked to explore in this problem, so more on this later...

Holly, Shannon and Jasper from Litcham School created graphical representations of $F_5$, $F_6$ and $F_7$:

Kieran, from Oakwood Park Grammar School worked out each sequence by starting with the previous one, using it as a framework. From this, he then slotted in the new fractions, rather than creating each new sequence from scratch.

Weida, from Collaton St Mary Primary School, submitted the correct sequence for $F_5$:

F5 = $\frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1}$

The extra fractions in $F_5$ are $\frac{1}{5}, \frac{2}{5}, \frac{3}{5}$ and $\frac{4}{5}$.

Well done also to Stephen from Tudhoe Grange and Adam from Wilson's School. Weida, Adam and Stephen all submitted correct sequences for $F_6$, and also $F_7$. Adam wrote:

$F_6$ goes $\frac{0}{1}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{1}{1}$.

$F_7$ goes $\frac{0}{1}, \frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{2}{7}, \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{2}{3}, \frac{5}{7}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{1}{1}$.

Weida was very thorough, and submitted $F_8$ as well. Can you work this one out?

As you can see from these sequences, the number of "extra" fractions added to the next Farey sequence varies. Why is this the case? In what situations are lots of extra fractions added, and when are only a few?

Looking at $F_{10}$, $F_{11}$, and $F_{12}$ is a good starting point, as there are ten extra fractions going from $F_{10}$ to $F_{11}$, but only four going from $F_{11}$ to $F_{12}$. Why does this happen?

Stephen, from Tudhoe Grange explained:

There are a lot more extra fractions in $F_{11}$ than in $F_{10}$ because $11$ is a prime number. The new fractions that would go into $F_{11}$ are $\frac{1}{11}, \frac{2}{11}, \frac{3}{11}$ and so on. None of the new fractions in $F_{11}$ are equivalent to other fractions i.e. can be simplified (except $\frac{11}{11}=\frac{1}{1}$) in the $F_{11}$ sequence.

$10$ has factors of $1$, $2$, $5$ and $10$.
In the $F_{10}$ sequence, the new fractions are $\frac{1}{10}, \frac{2}{10}, \frac{3}{10}$ and so on.
$\frac{2}{10}=\frac{1}{5}, \frac{4}{10}=\frac{2}{5}$... , so $\frac{2}{10}, \frac{4}{10}, \frac{6}{10}, \frac{8}{10}$ won't be in the $F_{10}$ sequence - they will stay as $\frac{1}{5}, \frac{2}{5}, \frac{3}{5}$ and $\frac{4}{5}$. That is why there are less extra fractions in $F_{10}$ but there are a lot of extra fractions in $F_{11}$.

Heather from GHS and Adam and Weida also came to the same conclusion. Weida summarised more generally:

I will need lots of extra fractions to get to the next Farey sequence when the next sequence's number is a prime number. Every Farey sequence will be longer than the one before because each one will have at least two new fractions:

$\frac{1}{F_{n}}$ & $\frac{n-1}{F_{n}}$

Adam agreed with this. He went on to discuss Farey sequences containing odd and even numbers of fractions:

The only sequence with an even number of fractions is the $F_1$ sequence. All other sequences will contain an odd number of fractions since $F_2$ contains $3$ fractions and you will always need an even number of additional fractions after that.

We have seen, and begun to explain, some properties of Farey sequences. What about the "curious property" that fascinated Farey? The problem hints at this by asking you to focus on $F_4$: $\frac{3}{4}$ slotted in between $\frac{2}{3}$ and $\frac{1}{1}$. It asks what you notice about the fractions on either side when you slot in a new fraction. Weida made this observation:

I noticed that the numerators of the fractions on either side of the new fraction added together make the numerator of the new fraction; and that the denominators of the fractions on either side of the new fraction added together make the denominator of the new fraction.

She then continued to look at other consecutive fractions in a sequence, to see if this relationhip between two surrounding fractions and the fraction in the middle is a general property of Farey sequences or a "one-off":

Three consecutive fractions from a Farey sequence:

$\frac{2}{5}$, $\frac{3}{7}$, $\frac{1}{2}$.

Total of numerators from outer fractions = $3$ = numerator of middle fraction
Total of denominators from outer fractions = $7$ = denominator of middle fraction
When I put the new numerator and denominator together, I got $\frac{3}{7}$. This works with all other consecutive fractions from Farey sequences when the middle fraction is new to that sequence.

This is indeed the "curious property" that Farey described. Adam and Stephen noticed this too, so well done. Great minds think alike!

When Farey sent his letter to the Philosophical Magazine to describe his findings, he wrote:

"I am not acquainted whether this curious property of vulgar fractions has been before pointed out; or whether it may admit of some easy or general demonstration; which are points on which I should be glad to learn the sentiments of some of your mathematical readers..."

A "reader", Augustin Cauchy, gave the necessary proof. In addition, Farey's observations had indeed been seen before; it was noted by C. Haros in 1802. Nevertheless, the results still bear Farey's name. Indeed, many theorems are misnamed (depending on who you "side" with)...

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.