You may also like

problem icon

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?

problem icon

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?

problem icon

3388

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

 
What can you see?
How do you think the pattern was made?
How do you think the pattern would continue?

What proportion of the total area of the square is taken up by the small square in the middle?
How do you know?

You might like to make a copy of the pattern for yourself using coloured paper. Or you could click below to show the different stages of construction of the image.




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

 

sixth image

 

first image

 
 

You may wish to take a look at Vanishing Point after working on this problem.