Phil arrived at the Broadway theater. Mr. Smith, the director, had called him in for an emergency.
Phil: What is the problem, David?
David: Thank god you've arrived. We have a serious issue for tonight's show. Our electrical system has gone haywire and we need your help.
Phil: Electrical system? I'm not an electrical engineer. I don't know how I could possibly help.
David: Here's the problem: we managed to fix most of our lighting, but our biggest spotlights still have a serious issue. We have no idea what causes it, but it only started an hour ago, and the electrician said we would have to dismantle a large part of our rigging to fix it, which we don't have time for.
Phil: I still don't see where I come in...
David: Here's the problem:
Our 3 main spotlights that we don't use except for the finale are broken. They will work only for 6 seconds each before the fuse blows. The off-switches are broken, and the fastest we can replace the fuse when it blows is 10 seconds, and we need to start from the moment the light goes on, but your assistant is confident that it can always be done that fast. However, the rigging cannot support more than 1 person to replace a fuse at once, so they need to be replaced sequentially.
Our finale lasts 1 minute, and we need at least one light to be active during that entire period. If that's not possible, we need to know in advance how long we can keep those lights going before all the lights are without fuses. In theory, we can make the finale anywhere between 30 and 60 seconds long.
Phil: So, to summarize:
you only have 6 seconds per light, a fuse takes 10 seconds to replace, and the rigging can only support 1 person to replace fuses at a time. You wonder whether you can let the finale last the full 60 seconds or whether it needs to be shortened.
David: Correct. You're the best lighting expert we have. What's your verdict?
Can David still do the full finale? Or does he need to shorten it? And if so, how long can the finale last? Obviously, since this is a math problem, you need to "show your work".
Bonus: Friday the 12th of June, I will hand out a bounty of 50 reputation to the answer with the highest score that continues the dialog and gives the right answer.
Explanation for those that want the story behind this question
This core of this question is not about lighting. The original question I thought of was based around a mechanic from the video game World of Warcraft, more precisely the Priest's Angelic Feather ability:
Angelic Feather 40 yd range Instant 10 sec recharge 3 Charges Requires Priest Requires level 30 Place a feather at the target location, granting the first ally to walk through it 60% increased movement speed for 6 sec. Max 3 charges. Only 3 feathers can be placed at one time.
As you can see, one charge of the feather is equal in functionality to 1 light: once you activate it, it lasts 6 seconds, each charge has a separate 10 second cool-down period (called "recharge" here) and there are 3 charges. Charges in World of Warcraft have sequential cool-downs: they don't cool down at the same time, but one after another.
The reason I translated this to lighting instead of keeping it about the spell is because I assumed that people (especially those that don't play World of Warcraft) would understand the problem easier if I used more known terms like fuses and lighting than recharge, spells and duration.