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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)