### 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.

# Funnel

##### Stage: 4 Challenge Level:
You may have heard the term 'Calculus', it's a big idea in mathematics and so powerful that I find it hard to imagine mathematics without it. It's main technique, differentiation, makes problems like this fairly easy - with a little care over the algebra. So if you are thinking about continuing with mathematics beyond Stage 4, learning about differentiation would be one of the big benefits that await you.

But differentiation is not really Stage 4 mathematics, though very close, so we don't use it in solutions because that would be unfair to students who haven't seen the idea before, it would be like suddenly changing into a language you haven't had a chance to learn yet.

If that's got you interested why not take a look at Vicky Neale's article : Introduction to Differentiation

And thank-you to David and Berny from Gordonstoun and to Naren from Loughborough Grammar School who sent in great solutions using that technique .

In fact the use of differentiation did more than get an answer. It found the result that the cone needed to have its height 1.41 times bigger than its radius, like the Stage 4 result below, but found it in this more interesting form.
And the value of that is that it suggests a new direction to pursue with this problem : it seems so neat.
Why the square root of 2 ? Is there a connection with the diagonal of a square ? Or is it something else ?

That's maybe a little bit on from us at Stage 4 so here's a way to solve a problem like this using Stage 4 mathematics. It's really 'trial and improvement' but using a spreadsheet to make the calculation effortless. .

We are going to use :
• the radius, which we'll adjust to get nearer and nearer to the answer,
• the height, which will be determined by our choice of radius, so that we get the chosen target volume

• the slant length, which we'll find using r and h and Pythagoras,

• and the surface area, for which there's a great little formula.

The volume of a cone is one third the volume of a cylinder with the same base and height.

If we took the target volume to be 1 litre, 1000 ml, then the formula that connects h and r is
The slant length (s), using Pythagoras, is

And the surface area of a cone is
plus the base if you need it - here we don't.

Incidentally, if you don't know where that surface area formula comes from it may be good to take a moment to look at that. Flatten the curved surface out to get a sector (how do you know it's a sector ?). The radius will be the cone's slant length, so you can calculate the area of the whole circle. To know the proportion that the sector is of that circle compare the sector arc, which is the cone's base circumference, with the circumference of this new circle, radius s.

Back to the funnel and using as little plastic as possible. Take a look at this spreadsheet : Funnel

Can you see what each column does ? Click on a cell and check the formula.

• The first column has increasing radius values which you can control.
• The next column calculates the height, because we knew the volume was 1000 ml
• The radius and height are then used to calculate the slant length.
• And the final column uses the radius and the slant length to calculate the surface area, which we want to be as small as possible
There's even a graph so you can have some sense for how surface area varies as the radius value ranges across your chosen interval.