### 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 $1 200. What were the assets of the players at the beginning of the evening? ### Plutarch's Boxes According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes? # Mathematical Swimmer ##### Stage: 3 Challenge Level: Every day I go to the swimming pool and swim the same number of lengths. I like to count the number of lengths I've done as I go as a fraction of the total number of lengths I'm going to do that day. If I swam ten lengths a day, after five lengths I would say to myself, "I've managed$\frac{5}{10}$of my day's swimming - that's$\frac{1}{2}$!" After eight lengths, I would say I'd done$\frac{8}{10}$, which simplifies down to$\frac{4}{5}$. After nine lengths, I'd say I'd done$\frac{9}{10}\$, which does not simplify.

I don't swim ten lengths a day. In fact, the total number of lengths I swim each day is rather special.

Each number of lengths I swim will either be a prime number or the fraction it makes of the total number of lengths will simplify or indeed both.
It is, in fact, the largest number possible for which this is true.

How many lengths do I swim per day?