### 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

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

# Knight Defeated

##### Stage: 4 Challenge Level:

For the $2$ by $n$ board, if there is a tour then it must pass through the corner square. Is this possible?

It might help to think of the squares as vertices of a graph. Then there is an edge joining two vertices if and only if there is a knight's move between the corresponding squares.

Eight of the vertices are of degree two (only one path in and one out of that square). To construct a tour you are forced to visit these vertices in a particular order.