### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Doodles

A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not 'triple points'. Number the vertex points in any order. Starting at any point on the doodle, trace it until you get back to where you started. Write down the numbers of the vertices as you pass through them. So you have a [not necessarily unique] list of numbers for each doodle. Prove that 1)each vertex number in a list occurs twice. [easy!] 2)between each pair of vertex numbers in a list there are an even number of other numbers [hard!]

### Russian Cubes

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted in order 'blue then red then blue then red then blue then red'. Having finished one cube, they begin to paint the next one. Prove that the girl can choose the faces she paints so as to make the second cube the same as the first.

# Archimedes and Numerical Roots

### Why do this problem?

This problem offers students the opportunity to engage with and make sense of a numerical method for finding roots.

### Possible approach

"How could I find the square root of three if I didn't have a calculator?" Collect together students' suggestions - it is likely that various methods of trial and improvement will be suggested, as well as the observation that the value will be between 1 and 2.

"Trial and improvement takes time. Here is another numerical method for finding roots." Introduce the algorithm for finding a new approximation. Give students some time to experiment with the method to get a feel for it and to observe how it converges to $\sqrt{3}$.
"Can you adapt the method to find roots of other numbers? Can you explain why it works?"
Again, give the students time to explore these two questions.
Finally, bring the class together so they can share their ideas and explanations.

### Key questions

What does the method $(\frac{(\frac{3}{n} + n)}{2})$ calculate, if $n$ is an approximation to $\sqrt{3}$?
How can you change the method to work out other square roots?
How does the equation $n = (\frac{(\frac{3}{n} + n)}{2})$ help you to make sense of why the method works?