### 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:

You do not need to be able to play chess to solve this problem.

The standard move for a knight on a chess board is $2$ steps in one direction and one step in the other direction. A knight's tour is a sequence of moves in which the knight visits every square on the board once and only once and a circuit is a tour in which the knight returns to the starting point.

Prove that a knight cannot make a tour on a $2$ by $n$ board for any value of $n$.

How many different tours can you find on a $3$ by $4$ rectangular board?

Prove that a knight cannot make a circuit on a $3$ by $4$ rectangular board.