Unless I completely misunderstand how this works, I think 5 is the only one that will fill up. It then overflows, preventing any of the taller ones from filling. 7 is shallower but won't start filling until 3 gets fuller than 5, which it never will. I assume the blockage between 2 and 3 is a mistake.
5, but it also depends on the circumstances. What liquid is used, temperature, viscosity, etc. There's some material science stuff that's far beyond the intended scope of this question.
5 will be the only one that will ever fill up unless you really crank up the pressure in which case 7 will also fill but very slowly. 5 and 7 are open containers and there's a hole in the bottom of 4. But if it's water coming out of a tap then only 5 will fill .
Wouldn't scale and viscosity play a role? Seriously, imagine a river vs a capillary tube. Also how many dimensions? And forces involved? Is that a blockage between 2 and 3? Are the walls breakable? How will the fluid hold air? Are the lines into structure 5 lower than the walls? Is this in a vacuum?
I got it! First, the free floating faucet will drop into bucket one. The impact will certainly break its connecting tube and broken 1 + faucet collapse into 4. Therefore 4 will be broken but full of shards.
Ha. Trick question! All of them are already full of air, and niether the flow rate nor the direction (or lack) of gravity was specified anyway. You lose. :)
The number of people in these comments who already understand the self-siphoning nature of water with zero explanation required makes me so proud to be here among them.
Vessel 5 is the lowest point, so it might seem like it would fill first.
However, if the faucet pours water faster than it can flow through the tube from Vessel 1 to Vessel 2, and as the faucet is depicted as having a larger opening, Vessel 1 will fill up first because the water can't flow out quickly enough to the other vessels.