### 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:

See the Hint section for detailed support for students, but the main aims of this problem are :

1. If a calculation approach is adopted, to see the value of using a spreadsheet . Thinking through the calculation required and the spreadsheet commands (formulae) necessary to achieve that are prime Stage 4 activities.

2. To account for the general result that emerges, in such a way that the 'solution' then becomes obvious, adds an additional visualisation to the student's repertoire of possibilities.