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

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Speeding Boats

Stage: 4 Challenge Level: Challenge Level:1

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

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Summing Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Picture Story

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Platonic Planet

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Coordinated Crystals

Stage: 5 Challenge Level: Challenge Level:1

Explore the lattice and vector structure of this crystal.

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Classical Means

Stage: 5 Short Challenge Level: Challenge Level:1

Use the diagram to investigate the classical Pythagorean means.

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Farey Neighbours

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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?