### Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square. Three of the numbers that he found are a = 18530, b=65570, c=45986. Find the fourth number, x. You could do this by trial and error, and a spreadsheet would be a good tool for such work. Write down a+x = P^2, b+x = Q^2, c+x = R^2, and then focus on Q^2-R^2=b-c which is known. Moreover you know that Q > sqrtb and R > sqrtc . Use this to show that Q-R is less than or equal to 41 . Use a spreadsheet to calculate values of Q+R , Q and x for values of Q-R from 1 to 41 , and hence to find the value of x for which a+x 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.

### Substitution Cipher

Find the frequency distribution for ordinary English, and use it to help you crack the code.

# 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.