### 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 $1 200. What were the assets of the players at the beginning of the evening? ### Plutarch's Boxes According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes? ### 3388 Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24. # Mathematical Swimmer ##### Age 11 to 14 Challenge Level: Every day I go to the swimming pool and swim the same number of lengths. I like to count the number of lengths I've done as I go as a fraction of the total number of lengths I'm going to do that day. If I swam ten lengths a day... • After five lengths I would say to myself, "I've managed$\frac{5}{10}$of my day's swimming, which simplifies to$\frac{1}{2}$." • After seven lengths, I would say "I've managed to swim a prime number of lengths." • After eight lengths, I would say "I've done$\frac{8}{10}$of my day's swimming, which simplifies to$\frac{4}{5}$." • After nine lengths, I'd say "I've done 9 lengths, which isn't prime, and$\frac{9}{10}$does not simplify." I don't swim ten lengths a day. In fact, the total number of lengths I swim each day is rather special... Let's call the total number of lengths I swim$n$. After the first length, for every length I swim, the total so far (let's call it$t$) is either a prime number, or the fraction$\frac{t}{n}\$ will simplify (or both).

It is, in fact, the largest number for which this is true.

Can you work out how many lengths I swim each day?