## Help with APMO (Asian Pacific Math Olympiad) problem

Here is the link to the problem and the solution below (it is problem 4) :

http://cms.math.ca/Competitions/APMO...mo2005-sol.pdf

I've correctly come to the solution using my own method (i found the correct containment method through trial and error and then came up with a general expression for the maximum amount of saved houses using sum of arithmetic series) but I'm stuck on mathematically proving that the containment strategy used is the most effective, which is done in the link done above. I didn't put too much effort into trying to understand the proof in the link as to why the containment strategy is the most effective for saving houses, since the proof is explained very briefly (its hard to understand) but mostly because I would like to be able to come up with my own proof and actually understanding it instead of just copying the one in the link. So does anyone have any tips as to how I could come up with a proof for this?
BTW this is for a mathematics essay I'm writing for school.

Any help is appreciated!

Thank You!!