You go against the consensus without a solid explanation. For instance, why don't the weights matter? Would they matter if they were respectively 1mg, 1kg, and 1megaton?
Fair enough. I gave the summarized version. I'll try to explain better.
I'll only take basic classic mechanics assumptions: that the rope is of constant length (ie is like a cable that doesn't compress or expand). The framework is quasi-static classical mechanics; the results of the guy pulling slowly a little bit can solve the problem or be generalized (I won't consider the situation of the guy jerking quickly the rope etc).
The guy's hand is under a calculable tension Tw that will be 60 units or whatever, it doesn't matter. The problem asks what happens when the guy pulls, so we suppose that he's not the one being pulled but he moves forward to the right. The tension Tg that he applies doesn't matter; as long as it's bigger than the one from the weights (Tg > Tw) he'll move the rope (we discard friction since he moves slowly or if you take into account friction he just needs more force, it doesn't matter). So the distribution of weights or their actual measure don't matter so far. (of course if you have a million tons and you blow the guy away weighs matter).
Now since the cable/rope has constant length (there are no slacks etc since it's moving) when the guy pulls 1mm then than length needs to be taken from somewhere in the pulley systems.
The effect of a pulley is to divide the length of rope you take in two (one has to go to the left vertical part of rope and the other one to the right one); this is why the tension in each side of a pulley is 1/2 of the total tension and you can pull with 1/2T a weight of T with a pulley. So with this we can straightforwardly calculate the tensions everywhere but we don't need that.
So the that 1mm is taken from the system and the more pulleys the rope has to go through the less is taken (because of this 1/2 I explained above), so the weight closer to the guy (with less pulleys) will raise first, then the next one in the middle, then the next one etc; weighs don't matter.
clarification update: poor conclusion wording: if weights are in the air all 3 weights go up at the same time but C will move more than B and B more than A.
When a weight is at rest, the tension in the rope must be less then or equal to half the weight. The problem asks which weight will be first to not be at rest. The answer should now be obvious.
When a weight is at rest the tension in the rope is exactly half its weight.
Problem asks which weight would be first to raise (not "not be at rest" that could be going down) so that's why I'm supposing the guy can pull the whole thing. I'm also saying the first one is C (closer to guy).
Basically if the weights are at rest on the floor then mkn's answer is the correct one. If they are at rest on the air, then my answer is the correct one.
I supposed the system was at equilibrium without making the numbers. If it's not (as it seems) then sure, my reasoning is invalid and he's right, no problem.
You had the right answer for one set of assumptions. But, at the extreme weight is important EX: if the numbers where solar masses then gravity would suck them into each other.
If they pulley’s had sufficient friction C may move first. Or if you pull a real rope fast enough C will also move first (think KM/s speeds). Etc.
