### 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?

# Diminishing Returns

##### Stage: 3 Challenge Level:

Take a look at the image below:

Work out what proportion of the image is coloured blue.

Imagine continuing the pattern towards the centre of the square:

If this process could be continued forever, what proportion of the image would be coloured blue?

Below is a collection of images created using repeating processes.

Choose a few of the images below, and work out what fraction of the total is taken up by the five largest blue shapes.
Add your fractions to estimate the proportion of the total that is coloured blue.

If each process could be continued for ever, what proportion of the whole image would be coloured blue?