See for the general problem here http://www.mathlinks.ro/Forum/viewtopic.php?highlight=Kvant&t=1896

I have looked at the link that Myth provided, and I could give some hints about it: Lemma: (Proof by induction) : If we have 2 straight lines that intersect and form an angle of $\frac{k\pi}{n}$, and inside the smaller region(in the angular meaning) we have some k lighthouses, which can give light at angle $\frac{\pi}{n}$, then we can light the opposite(small) region with those lighthouses. The proof that reduces it to the induction hypothesis uses a funny arithmetic Lemma: If we have $\sum_{i=1}^n a_i=n$ and $a_1\geq 1$ and $a_n\geq 1$ then there exists $k$, so that $\sum_{i=1}^k a_i=k$. Forgot to say, $a_k$ are nonegative integers, and $k<n$! See how nice it is to prove the problem with the lemma, and the Lemma with the Lemma =))

The main part I mean is the general problem.

A friend of mine supplied a proof for this. I will attach here:

Attachments:
illuminationoftheplane_210.pdf (90kb)

DusT,I CANNOT UNDERSTAND WHAT YOU HAVE WRITTEN COULD ANYONE POST THE FULL SOLUTION,PLEASE? THANKS

Actually, Myth did it... http://www.mathlinks.ro/Forum/files/illuminationoftheplane_210_221.pdf

Omid Hatami wrote: Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened. I believe this can be solved using arc lengths compared to angles right? Or is there more complexity regarding location within the gulf not included on the edges...

Originally from russian journal Kvant: No.11, 1981. See here for the solution(in russian) of the two proposers, V. Galperin and G. Galperin. You can use this article if you want english version.