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Do Unto Caesar

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Diminishing Returns

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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