### 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.

# Cola Can

##### Stage: 4 Challenge Level:

A millilitre is one cubic centimetre.

Check that you can calculate the volume of a cylinder.

If the diameter is 6 cm, you can calculate the base area of the can.

If you know the base area you can find a height which will give you a specific volume ( 330 ml in this case)

Now start with a height (10 cm) and work your way back to a base area for a specified volume, then find a radius for that base area, and hence a diameter for the can.

What does least aluminium require ?

The top and base of the can are circles but how do you calculate the curved surface area ?