I was thinking about this problem, and have searched online to see if there has been work done on this, but have not found anything.

The question is if I wanted to divide a plane into n unique regions, what is the minimum number of lines needed? here's an illustration of what I mean

I did find the formula 0.5(n²+n+2) which gives the maximum number of spaces given a line n. While very related, it's almost the reverse of my question.

I did work on coming up with experimental values given here. There does seem to be a clear pattern, except for when n=5, which would throw the whole thing off.


This is hand-wavy but if you know that you can create at most $M$ regions with $N$ lines (let's call this $f(N) = M$), then I'd think that you could create fewer than $M$ regions with $N$ lines. (This is an assumption.)

Let's arrange these maximum numbers as a sequence: $f(1), f(2), f(3), ... \to M_1, M_2, M_3, ...$

So, for any particular number of $m$ regions, the fewest number of lines needed will be the value $j$, such that $M_{j-1} < m \leq M_j$.


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