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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.

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    $\begingroup$ 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. $\endgroup$ Commented Oct 15, 2010 at 13:43

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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.

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  • $\begingroup$ Thanks for your answer Robin, but Bill gives me more insight into the problem :-). $\endgroup$ Commented Oct 17, 2010 at 14:11
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This was a very old Monthly problem - see below. For an excellent introduction to the general topic see Richard Stanley's paper An Introduction to Hyperplane Arrangements (2004) and also Renteln's lecture slides It All Depends on How You Slice It: An Introduction to Hyperplane Arrangements, 2008 alt text

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This is the lazy caterer's sequence. As others have mentioned, arbitrary dimensional analogues are called hyperplane arrangements.

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