# Visualising - October 2009, Stage 4&5

This month, our problems involve visualisation, inviting you to reflect on how you "see" mathematics. Everyone imagines a problem in a different way. By sharing our personal visualisations it can deepen our own understanding of the mathematics within a problem, and help us to make sense of someone else's route to a solution.

## Problems

### Picture Story

##### Stage: 3 and 4 Challenge Level:

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

### Speeding Boats

##### Stage: 4 Challenge Level:

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

### Mystic Rose

##### Stage: 4 Challenge Level:

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

### Summing Squares

##### Stage: 4 Challenge Level:

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

### Platonic Planet

##### Stage: 4 Challenge Level:

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

### Coordinated Crystals

##### Stage: 5 Challenge Level:

Explore the lattice and vector structure of this crystal.

### Classical Means

##### Stage: 5 Challenge Level:

Use the diagram to investigate the classical Pythagorean means.

### Farey Neighbours

##### Stage: 5 Challenge Level:

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?