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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.

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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?

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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: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem involves calculating with fractions and also uses knowledge of factors and multiples. It is a difficult problem that requires clear and logical thinking.

Key questions

How do you think you can start on this problem?
Would it help to make a list of some prime numbers?
Why not think of numbers which have many factors and try them?
Would it help to make a list of numbers which have many factors?
Why not start with $12$? What are its factors?
Are all the numbers up to $12$ either a prime number or a fraction that will simplify?
Now do you think you can try with larger numbers?

Possible extension

Learners could on to Peaches Today, Peaches Tomorrow.

Possible support

If learners are finding coping with fractions are very difficult suggest trying this Stage 2 problem, Fractions Made Faster or even Fractional Wall.