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.
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.