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Lines, cross overs and regions
By Vicky Turnage (P2500) on Tuesday, May 16, 2000 - 06:49 pm:

hi please could you help me with my maths problem. I have to investigate the relationships between the number of lines, the maximum number of cross over points and the maximum number of regions. i have drawn up a table and i have come up with a formula, but i don't know how to explain how i got it. i used sequences and differences (1st 2nd etc). the formulas are
n2/2 - n/2
n2/2 - n/2 - (n-1)
and n×(n+1)/2 + 1

By David Seery (Djs61) on Tuesday, May 16, 2000 - 10:04 pm:

Hi Vicky,
Write P(n) for the number of crossing points and R(n) for the number of regions. Then write an equation relating P(n) and P(n-1) (and another for R(n) and R(n-1)) and show that your solution satisfies the equation. This is plenty good enough, because actually constructing your solutions out of the difference equation is a fairly difficult proposition. (I haven't done the working because I thought you might like to work through this yourself, but I can walk you through it if you like.)

Let me know if this isn't clear, or you'd like some more detail. Incidentally (although I've only looked at this briefly) I'm not sure what your second equation is about: the first seems correct for the number of points and the third for number of regions. From memory, this problem was first solved in 1826 by the Swiss mathematician Jakob Steiner, if that's any help.

Hope this is useful,

David