### 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?

### Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

# Take Three from Five

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

This problem builds on What Numbers Can We Make?

Take a look at the video below. Will Charlie always find three numbers that add up to a multiple of 3?

If you can't see the video, click below to read a description.

Charlie invites James and Caroline to give him sets of five whole numbers. Each time he chooses three of their numbers that add together to make a multiple of 3:

 TOTAL 3 6 5 7 2 18 7 17 15 8 10 39 20 15 6 11 12 33 23 16 9 21 36 48 99 57 5 72 23 228 312 97 445 452 29 861 -1 -1 0 1 1 0

Charlie challenges Caroline and James to find a set of five whole numbers that doesn't include three that add up to a multiple of 3.

Can you come up with a set of five whole numbers that don't include a subset of three numbers that add up to a multiple of 3?

If you can't find a set of five whole numbers where it's impossible to choose three that add up to a multiple of three, convince us that no such set exists.