### 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?

### N000ughty Thoughts

How many noughts are at the end of these giant numbers?

##### Age 14 to 16 Challenge Level:

A walk is made up of diagonal steps, starting at the bottom left and ending up back on the bottom line (x-axis). You can move diagonally up and down towards the right but you cannot move towards the left.

A diagonal must go from a left-hand corner of a square to the opposite right-hand corner of the same square.

The examples above all show 10-step walks. So let's look at a 2-step walk in more detail:

There is only one way to make a 2-step walk - from A up to x and down to B:

Here are two 4-step walks:

Are there any more 4-step walks?

How do you know you have them all?

How many 6-step, 8-step walks are there?

Can you find a general rule?