# Your Solutions - Stage 4 & 5

### Tens

##### Stage: 5 Challenge Level:

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

### Giants

##### Stage: 5 Challenge Level:

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

### Rhombuses from Diagonals

##### Stage: 3 and 4

Take a look at the video showing rhombuses and their diagonals...

### Getting Round the City

##### Stage: 3 and 4

In a city with a grid system of roads, how do you get from A to B?

### Garden Shed

##### Stage: 4 Challenge Level:

Can you minimise the amount of wood needed to build the roof of my garden shed?

##### Stage: 5

This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

### Summing to One

##### Stage: 5 Challenge Level:

This problem is a nice introduction that will give you a feeling for how logs work and what that button on your calculator might be doing.

### Proving the Laws of Logarithms

##### Stage: 5 Challenge Level:

Here you have an opportunity to explore the proofs of the laws of logarithms.

### Can You Find ... Cubic Edition

##### Stage: 5 Challenge Level:

Can you find cubic functions which satisfy each condition?

### Can You Find ... Asymptote Edition

##### Stage: 5 Challenge Level:

Given a sketch of a curve with asymptotes, can you find an appropriate function?

### A Tangent Is ...

##### Stage: 5 Challenge Level:

What do we REALLY mean when we talk about a tangent to a curve?

### Transformation ... or Not?

##### Stage: 5 Challenge Level:

This graph looks like a transformation of a familiar function...