### 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 $1 200. What were the assets of the players at the beginning of the evening? ### Plutarch's Boxes According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes? ### 3388 Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24. # Farey Sequences ##### Age 11 to 14 Challenge Level: If I gave you a list of decimals, you might find it quite straightforward to put them in order of size. But what about ordering fractions? A man called John Farey investigated sequences of fractions in order of size - they are called Farey Sequences. The third Farey Sequence,$F_3$, looks like this: $$\frac01 \qquad \frac13 \qquad \frac12 \qquad \frac23 \qquad \frac11$$ 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.