### 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. # All Tangled Up ##### Age 11 to 14 Challenge Level: This problem follows on from Twisting and Turning and More Twisting and Turning in which twisting has the effect of adding 1 and turning transforms any number into the negative of its reciprocal. We can start at 0 and end up at any fraction of the form $$\frac{n}{n+1}$$ by following the sequence: twist, twist, twist, ... , twist, twist, turn, twist eg. to end up at$\frac{4}{5}$: twist, twist, twist, twist, twist, turn, twist to produce:$0, 1, 2, 3, 4, 5, \frac{-1}{5}, \frac{4}{5}$Check you can reach$\frac{9}{10}$The sequence twist, twist, turn, twist, twist, turn, twist, twist, turn, ... , twist, twist, turn, twistwill lead us from 0 to all the fractions of the form $$\frac{1}{n}$$ eg. to end up at$\frac{1}{5}$(and$\frac{1}{2}$,$\frac{1}{3}$and$\frac{1}{4}$along the way): twist, twist, turn, twist, twist, turn, twist, twist, turn, twist, twist, turn, twist to produce: 0, 1, 2,$\frac{-1}{2}$,$\frac{1}{2}$,$\frac{3}{2}$,$\frac{-2}{3}$,$\frac{1}{3}$,$\frac{4}{3}$,$\frac{-3}{4}$,$\frac{1}{4}$,$\frac{5}{4}$,$\frac{-4}{5}$,$\frac{1}{5}$Check you can reach$\frac{1}{10}\$
Can you find other sequences of twists and turns that lead to special fractions?

Is it possible to start at 0 and end up at any fraction?