the maximum number of bounded regions created by n line segments w/o lifing pencil?
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Dr.Peterson
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The picture seems to be unreadable, but I can easily imagine it.
But what thoughts do you have? This sort of exercise is meant to give you a chance to practice thinking for yourself, not just to pass the work on to someone else. And there are a number of good ways you can think about it.
maximum number of bounded regions created by n line segments without lifing pencil?
i draw it with 6 line segments. i find 8 bounded regions. Thus
3 line segments ----- 1 bounded region
4 ------ 2
5 ------ 6
6 ---- 8
i cant draw it with 7 line segments...
i draw it with 6 line segments. i find 8 bounded regions. Thus
3 line segments ----- 1 bounded region
4 ------ 2
5 ------ 6
6 ---- 8
i cant draw it with 7 line segments...
Dr.Peterson
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I see your question at https://math.stackexchange.com/questions/2843673/regions-bounded-by-n-line-segments .
I played with the problem, finding what appear to be optimal polygons for n=3 to 8 (they are rather pretty!); but the sequence I get doesn't show up with such a description in OEIS.org (where I would expect it to be found), so either I have a number wrong, or no formula has been discovered. (There is a very simple formula for the sequence I got, but it is entirely possible that when I get to n=9, that formula will no longer apply.)
Have you made any progress, either in researching the question, or in finding numbers that we can compare to mine?
And can you tell us where the question originated? It is an obvious one to ask, but I have not found it discussed anywhere.
I played with the problem, finding what appear to be optimal polygons for n=3 to 8 (they are rather pretty!); but the sequence I get doesn't show up with such a description in OEIS.org (where I would expect it to be found), so either I have a number wrong, or no formula has been discovered. (There is a very simple formula for the sequence I got, but it is entirely possible that when I get to n=9, that formula will no longer apply.)
Have you made any progress, either in researching the question, or in finding numbers that we can compare to mine?
And can you tell us where the question originated? It is an obvious one to ask, but I have not found it discussed anywhere.
Dr.Peterson
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- Nov 12, 2017
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Any more progress?
I've extended my sequence to n=10 and have found a nice pattern such that I can probably make the maximum for any n, by starting with a regular polygon and taking some of its diagonals, then making adjustments to avoid multiple intersections. The method alternates between even and odd n, and the formula for the number of regions reflects that, though it can be written as a single formula using the ceil function. All I lack is proof that my method really accomplishes the goal (as if that were trivial).
I'm eager to help if you have been working on it. Until then, all I have are these hints.
I've extended my sequence to n=10 and have found a nice pattern such that I can probably make the maximum for any n, by starting with a regular polygon and taking some of its diagonals, then making adjustments to avoid multiple intersections. The method alternates between even and odd n, and the formula for the number of regions reflects that, though it can be written as a single formula using the ceil function. All I lack is proof that my method really accomplishes the goal (as if that were trivial).
I'm eager to help if you have been working on it. Until then, all I have are these hints.