### Cubes

Investigate the number of faces you can see when you arrange three cubes in different ways.

### The Big Cheese

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

### Wrapping Presents

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

##### Stage: 2 Challenge Level:

Well, you have a whole collection of cubes and we're going to paint the outside that's visible once we've put them together.
A few simple rules:
• the cubes must be together;
• they must not be toppling over.
We'll need to look at how much paint is needed. One Brush Load (a kind of unit that we'll use) will paint one square face.
Well here are $5$ cubes:

Counting the faces to be painted comes to $15$, so $15$ Brush Loads (remember we're only counting visible faces, so not those that are touching the surface where the cubes are placed).

But of course we could have placed the $5$ cubes differently:

Counting the faces to be painted we now have $17$, so $17$ Brush Loads.

Now we'll need $21$ Brush Loads.
O.K. Now let's explore different numbers of cubes, say $6, 7, 8, 9,$ etc. Each time we count the number of square faces showing and so find the number of Brush Loads (BLs).
The cubes - as in the case of $5$ - can be arranged in different ways, sometimes giving different BLs.
Try first with $5$ cubes.
How about $6, 7, 8$ and $9$ cubes?
If you wish to go further, then increase the number of cubes. See if you can get all the numbers between the largest and the smallest e.g. if $15$ is the smallest number for $5$ cubes and $21$ is the largest, can you get arrangements that'll give $16, 17, 18, 19,$ and $20$ BLs?