### Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

### Four Points on a Cube

What is the surface area of the tetrahedron with one vertex at O the vertex of a unit cube and the other vertices at the centres of the faces of the cube not containing O?

### Instant Insanity

We have a set of four very innocent-looking cubes - each face coloured red, blue, green or white - and they have to be arranged in a row so that all of the four colours appear on each of the four long sides of the resulting cuboid.

# Cheese Cutting

##### Stage: 5 Challenge Level:

Visualisation of 3D objects is a very useful real-life skill used in a range of everyday situations, from packing suitcases and boxes into the back of a car to travel to university through to parking the car in a tight space upon arrival. We can practise and develop our 3D skills and intuition through the study of the following sorts of mathematical problems. Spatial problems are mathematically quite different to algebraic problems and you may find one type of problem easier or more difficult depending on how your brain works!

I have some cubes of cheese and cut them into pieces using straight cuts from a very sharp cheese wire. In between cuts I do not move the pieces from the original cube shape.

For example, with just one cut I will obviously get two smaller pieces of cheese, with two cuts I can get up to 4 pieces of cheese and with three cuts I can get up to $8$ pieces of cheese, as shown in the picture:

Suppose I now make a fourth cut. How many individual pieces of cheese can I make?

Suppose now that I am allowed more generally to cut the block $N$ times. Can you say anything about the maximum or minimum number of pieces of chesse that you will be able to create?

Although you will not be able to determine the theoretical maximum number of pieces of cheese for $N$ cuts, you can always create a systematic cutting system which will generate a pre-detemined number of pieces (for example, making $N$ parallel cuts will always result in $N+1$ pieces of cheese). Investigate developing better cutting algorithms which will provide larger numbers of pieces. Using your algorithm what is the largest number of pieces of cheese you can make for $10$, $50$ and $100$ cuts?

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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