# How to Slice the Cheese

I encountered a problem recently stated as below:

How many pieces of cheese we can obtain from a single thick piece by making five straight slices? (we can't move the cheese when slicing) If we want to maximize the number of pieces which is denoted by $$P(n)$$, is there any recurrence relation for $$P(n)$$, where $$n$$ is the number of slices?

Any hints will be highly appreciated.

• The first thing is that you don't have to worry too much about the shape of the cheese. If you can divide space into k pieces with n planes, then you can cut the cheese into that many pieces just by shrinking the diagram till all the intersections fit in the cheese. – Oscar Cunningham Oct 15 '10 at 13:43

This is a special case of the problem of counting the number of regions $\mathbb{R}^n$ is divided into by $k$ hyperplanes in general position. The answer is $$\sum_{j=0}^n {k\choose j}.$$ This is mentioned in Richard Stanley's notes on hyperplane arrangements.