# Generic examples - April 2012, All Stages

Sometimes it is possible to state what is general by perceiving the structure in just one example. Vivid proofs of generalisations can be located in a particular, well-chosen example.

## Problems

### Walking Round a Triangle

##### Stage: 1 Challenge Level:

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?

### Two Numbers Under the Microscope

##### Stage: 1 Challenge Level:

This investigates one particular property of number by looking closely at an example of adding two odd numbers together.

### Odd Times Even

##### Stage: 1 Challenge Level:

This problem looks at how one example of your choice can show something about the general structure of multiplication.

### Round a Hexagon

##### Stage: 2 Challenge Level:

This problem shows that the external angles of an irregular hexagon add to a circle.

### Take Three Numbers

##### Stage: 2 Challenge Level:

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

### Three Neighbours

##### Stage: 2 Challenge Level:

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

### Square Subtraction

##### Stage: 2 Challenge Level:

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

### Magic Letters

##### Stage: 3 Challenge Level:

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

### Coordinate Patterns

##### Stage: 3 Challenge Level:

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

### Route to Infinity

##### Stage: 3 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next?

### Seven Squares

##### Stage: 3 Challenge Level:

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

### Double Trouble

##### Stage: 4 Challenge Level:

Simple additions can lead to intriguing results...

### Slick Summing

##### Stage: 4 Challenge Level:

Watch the video to see how Charlie works out the sum. Can you adapt his method?

### Speedy Summations

##### Stage: 5 Challenge Level:

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

### Polynomial Interpolation

##### Stage: 5 Challenge Level:

Can you fit polynomials through these points?

### Particularly General

##### Stage: 5 Challenge Level:

By proving these particular identities, prove the existence of general cases.