### 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? # Chocolate ##### Stage: 2 and 3 Challenge Level: There were a few complete solutions sent in and many who showed us what the final result at the tables would be, i.e.$5$at the$1$table,$10$at the$2$table and$15$at the$3$table. The full solution showing fractions at each stage were received from Adriel, Emily and Aswaath from the Garden International School in Maylasia. Also Daniel at Staplehurst School and Megan at Twyford School. Here is Emily's contribution. So after child$9$has sat down, there are now:$2$people at table$13$people at table$24$people at table$3$This is a list of which table child$10-30$would go to: The reason I listed down which table they would go to is to see whether there was a pattern Child$10$would go to table$3$and receive$¾$of the chocolate Child$11$would go to table$3$and receive$½$of the chocolate Child$12$would go to table$2$and receive$½$of the chocolate Child$13$would go to table$3$and receive$3/7$of the chocolate Child$14$would go to table$2$and receive$2/5$of the chocolate Child$15$would go to table$3$and receive$3/8$of the chocolate Child$16$would go to table$1$and receive$1/3$of the chocolate Child$17$would go to table$3$and receive$1/3$of the chocolate Child$18$would go to table$2$and receive$1/3$of the chocolate Child$19$would go to table$3$and receive$3/10$of the chocolate Child$20$would go to table$2$and receive$2/7$of the chocolate Child$21$would go to table$3$and receive$3/11$of the chocolate Child$22$would go to table$1$and receive$¼$of the chocolate Child$23$would go to table$2$and receive$¼$of the chocolate Child$24$would go to table$3$and receive$¼$of the chocolate Child$25$would go to table$3$and receive$3/13$of the chocolate Child$26$would go to table$2$and receive$2/9$of the chocolate Child$27$would go to table$3$and receive$3/14$of the chocolate Child$28$would go to table$1$and receive$1/5$of the chocolate Child$29$would go to table$2$and receive$1/5$of the chocolate Child$30$would go to table$3$and receive$1/5$of the chocolate So now there are:$5$people in table$110$people in table$215$people in table$3$I found out a pattern from child 10 to child$30$. When it was child$16$’s turn to decide which was the best place to sit, child$16$could just choose to randomly sit on any table because he’ll still get$1/3$of the chocolate in any of the tables. Same with child$22$and$28$. Child$22$would either ways get$¼$and child$28 1/5$. The numbers also increase by$6. 16, 22, 28$. I noticed that sometimes there were tables that shares the same amount of chocolate so you could choose randomly between$2$or all of the tables which was what I did when I tried finding out how many chocolates each child receives as they go to find the best table for them For example: Child$16$gets to choose between any of the$3$tables because he’ll still get$1/3\$ of the chocolate in any of the tables.

Martino, Guglielmo, Lorenzo and Virginia at the International School of Turin in Italy sent us a picture of their final solution for 30 children:

Yinyi who goes to the Chinese International School, sent us this picture of the table he made to work on the solution:

Finally, thank you to Sam and Jack from Orchard Junior School who sent in a table of their solution for a class of 24 children.  Sam and Jack chose to represent the fractions of chocolate as decimal numbers but unforunately, I think you made a little slip-up for the last four pupils in the class.

Well done all of you for your work on quite a difficult challenge.