# Are there publications or references describing $N$ dimensional space divided by $N-1$ dimensional (hyper)planes?

As a teenager I was given this problem which took me a few years to solve. I'd like to know if this hae ever been published. When I presented my solution I was told that it was similar to one of several he had seen.

The problem:

For an $$n$$ dimensional space, develop a formula that evaluates the maximum number of $$n$$ dimensional regions when divided by $$k$$ $$n-1$$ dimensional (hyper)planes.

Example: $$A$$ line is partitioned by points: $$1$$ point, $$2$$ line segments. $$10$$ points, $$11$$ line segments, and so one.

• @Jamie: Such type of questions cannot be answered. Just google it and see it for yourself.
– anonymous
Aug 11 '10 at 18:23
• @Chandru1: Why cannot it be answered? Aug 11 '10 at 18:24
• @ShreevatsaR: Common, how does one know that this has been published or not, and if he is curious in knowing it i am sure it will be available on the internet.
– anonymous
Aug 11 '10 at 18:26
• @Chandru1: Both the problem itself and the question of whether or not it has been published somewhere before can be answered. I know the problem itself can be answered because I have seen it published in a text, as noted in my answer. Aug 11 '10 at 18:30
• The proper term should be (hyper)plane, not boundary. Any surface can be a boundary. Aug 11 '10 at 18:39

Section 3 of http://www.macalester.edu/~bressoud/pub/Art_of_Counting/Art_of_Counting.pdf discusses the problem for $n=3$ and poses the problem of generalizing to higher dimensions. This article states that the first published solution was by Jakob Steiner in 1826. Other references are cited.
I presume you are asking for the number of regions into which $n$-space is divided into by $k$ hyperplanes "in general position". See the notes of Richard Stanley on hyperplane arrangements. The answer to your problem is Proposition 2.4, and there are lots more goodies too!