### 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.

# Peaches in General

##### Stage: 4 Challenge Level:

There's a problem which goes like this :

A monkey had some peaches.

He ate half of them plus one more.

On the second day, he ate half of the rest plus one more.

On the third day, he ate half of the rest plus one more again.

On the fourth day, he found there was only one left.

How many did he have at the beginning?

By the time you reach Stage 4 you will start to feel that problems like this are examples or instances of something more general.

Try this problem as it is and then generalise your solution as far as you think you can.

Using a spreadsheet may be a help, but you can judge that for yourself.