**From New Scientist #2937, 5th October 2013** [link]

I have drawn a number of straight lines across a large sheet of paper, each extending from edge to edge on the paper, so that each line crosses all the other lines. One of the intersections is between three lines, all the others are between just two lines, and none of them are on the edge of the paper. I counted the number of non-overlapping areas formed that did not touch the edge of the paper and found that this was exactly three times the number of non-overlapping areas that did touch the edge of the paper.

How many lines did I draw?

[enigma1769]

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If my reasoning is correct (and I think it is), this is fairly straightforward. This Python program runs in 38ms.

Solution:You drew 15 lines.This diagram shows how the number of regions grows by

nwhen thenth line is added. (n– 2) internal regions (coloured), and 2 external regions. This gives a formulae of I(n) = T(n – 2) – 1, E(n) = 2n.In the final diagram n = 7, I(n) = E(n) = 14.

So the solution to the problem is when I(n) = 3E(n):

(n – 2)(n – 1)/2 – 1 = 6n

(n² – 3n + 2) – 2 = 12n

n² = 15n

n = 15 (n ≠ 0).

Using induction, for n lines the total number of areas is n(n+1)/2, the number of areas touching the edge is 2n, so we are looking for n satisfying n(n+1)/2 -2n = 3(2n)