# Number - September 2007, All Stages

## Problems

### Caterpillars

##### Stage: 1 Challenge Level:

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

### Carroll Diagrams

##### Stage: 1 Challenge Level:

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

### Venn Diagrams

##### Stage: 1 and 2 Challenge Level:

Use the interactivities to complete these Venn diagrams.

### Mobile Numbers

##### Stage: 1 and 2 Challenge Level:

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

### More Carroll Diagrams

##### Stage: 2 Challenge Level:

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

### Various Venns

##### Stage: 2 Challenge Level:

Use the interactivities to complete these Venn diagrams.

### Alien Counting

##### Stage: 2 Challenge Level:

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

### Factors and Multiples Puzzle

##### Stage: 3 Challenge Level:

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

### The Cantor Set

##### Stage: 3 Challenge Level:

Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you picture it?

### How Long Is the Cantor Set?

##### Stage: 3 Challenge Level:

Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you find its length?

### Napier's Location Arithmetic

##### Stage: 4 Challenge Level:

Have you seen this way of doing multiplication ?

### Expenses

##### Stage: 4 Challenge Level:

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

### Interactive Number Patterns

##### Stage: 4 Challenge Level:

How good are you at finding the formula for a number pattern ?

### Two and Four Dimensional Numbers

##### Stage: 5 Challenge Level:

Investigate matrix models for complex numbers and quaternions.

### Quaternions and Rotations

##### Stage: 5 Challenge Level:

Find out how the quaternion function G(v) = qvq^-1 gives a simple algebraic method for working with rotations in 3-space.

### Quaternions and Reflections

##### Stage: 5 Challenge Level:

See how 4 dimensional quaternions involve vectors in 3-space and how the quaternion function F(v) = nvn gives a simple algebraic method of working with reflections in planes in 3-space.