### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

# Towers

##### Stage: 3 Challenge Level:

The first part of this problem was solved by Polly, West Flegg Middle School in Great Yarmouth; Joel, ACS in Singapore; and Luke, Dan, Nicholas and Luke, all from Clevedon Community School.

They all relized that the area of the triangles on either side of the squares are half that of the square (the base angle of triangle is 45°). Therefore, each square is half of the trapezium it is contained within, and the sum of all the squares will be half of the sum of all the trapezia that make up the triangle. The area covered by the squares is half the area covered by the triangle for the right-angled isosceles triangle.

But, what happens when the triangle is equilateral or just isosceles?

You will need to do some calculating to work out what happens then.