# Visualising - October 2009, All Stages

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

### Hundred Square

##### Stage: 1 Challenge Level:

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

### Cubes Cut Into Four Pieces

##### Stage: 1 Challenge Level:

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

##### Stage: 1 Challenge Level:

Here are shadows of some 3D shapes. What shapes could have made them?

### Odd Squares

##### Stage: 2 Challenge Level:

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

### Paw Prints

##### Stage: 2 Challenge Level:

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

### Colour Wheels

##### Stage: 2 Challenge Level:

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

### A Square in a Circle

##### Stage: 2 Challenge Level:

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

### Picturing Triangle Numbers

##### Stage: 3 Challenge Level:

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

### Handshakes

##### Stage: 3 Challenge Level:

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

### 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?

### Picture Story

##### Stage: 4 Challenge Level:

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

### 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?