### Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

### Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

### Peeling the Apple or the Cone That Lost Its Head

How much peel does an apple have?

# The Fire-fighter's Car Keys

##### Stage: 4 Challenge Level:

Start with a diagram. What measurements might define the particular arrangement you have drawn ?

From your drawing try some possible positions along the bank. Which gave the shortest overall distance ?

If you are trying to find a relationship between the best bucket-filling position and the positions of the fire and fire-fighter it might be useful to test some particular arrangements.

For example, suppose the fire is half the distance from the river that the fire-fighter is. Try that the other way around as well.

Suppose the fire is still half the fire-fighter's distance but both are now nearer the river than before.

Or further away.

Suppose the fire is still half the fire-fighter's distance but the two positions are further from each other along the bank. Closer along the bank ?

How are you finding the best position each time - by measurement ?

How about using a spreadsheet ? You could calculate the total distance for bucket-filling positions all along the bank and see which gives the smallest value. What calculation is needed ?

Do one yourself with a calculator to help you see what calculation (formula) you need to make the spreadsheet do.

For the keys problem set up a real experiment with string, pins and keys.