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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?

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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.

Diminishing Returns

Age 11 to 14 Challenge Level:

Take a look at the image below:
 seven squares

Work out what proportion of the image is coloured blue.
Try to provide a convincing explanation that your answer is right.
Imagine continuing the pattern towards the centre of the square:
seven squares going inwards

If this process could be continued forever, what proportion of the image would be coloured blue?
Try to provide a convincing explanation that your answer is right. 

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?  
Try to provide convincing explanations that your answers are right.  

second image 2third image

3fourth image 4fifth image

5sixth image 6thirds pattern

7quarters pattern 8fifths pattern

9first image