### 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. # More Twisting and Turning ##### Age 11 to 14 Challenge Level: This problem follows on from Twisting and Turning in which twisting has the effect of adding$1$and turning transforms any number into the negative of its reciprocal. It would be nice to have a strategy for disentangling any tangled ropes... I wonder if it is always possible to disentangle them... Before reading on, select a few fractions and try to get back to$0$. You could consider ropes that have been tangled up and have left you with a negative fraction containing a$2$as the denominator. e.g:$-\frac{5}{2}$or$-\frac{17}{2}$or$-\frac{23}{2}$How would you disentangle them? Try to describe an efficient strategy for disentangling any fraction of the form $$-\frac{n}{2}$$ Can this help you disentangle any positive fraction containing a 2 as the numerator? eg:$\frac{2}{7}$or$\frac{2}{15}$or$\frac{2}{32}$Next, you could consider ropes that have been tangled up and have left you with a negative fraction containing a$3$as the denominator e.g:$-\frac{5}{3}$or$-\frac{17}{3}$or$-\frac{23}{3}$Try to describe an efficient strategy for disentangling any fraction of the form $$-\frac{n}{3}$$ and use this to suggest a strategy for disentangling any fraction of the form $$\frac{3}{n}$$ Next, you could consider ropes that have been tangled up and have left you with negative fractions containing$4, 5, 6 \ldots$as the denominator, or positive fractions containing$4, 5, 6 \ldots\$ as the numerator.

Can you develop a strategy for disentangling any tangled ropes, irrespective of the fraction you have ended up with?

You may want to take a look at All Tangled Up after this.