This is a recent meme that’s getting shared on facebook and elsewhere. Repeating the image from the question source:

This is a question that has gone viral recently. Most people answer “G”.

But look closely, as the question says. Many of the pipes are blocked - the line that blocks off D from C is not a mistake.

To find the real answer:

From A to B to C is straightforward. None of them can fill before the next one.

J is a bit more complex.But as you fill J, as soon as it gets to the outlet to L then it will overflow to L. So it can never get any higher so though its level also rises in its outlet tube leading to I, it can never get high enough to overflow to I, not at a low flow rate.

So it flows to L, which in turn fills F.

So which fills first, L or F?

By the time F is full, L will only be partly full (with same water level for both).

So your F is the answer.

This video from THE FLOW... by CorneliaXaos shows the idea, click to restart the animation if necessary.

Which will fill first? Read More CorneliaXaos · 475,745 views · 481 comments

That answers the question, since it shows a dripping tap at a slow flow rate. But let’s go off on a tangent.

WHAT HAPPENS IF THE WATER IS POURED INTO A AT A FASTER FLOW RATE

If the flow is very fast then obviously it will be A, but could any of the others fill up first before F and after A?

It’s’ governed by the Hagen–Poiseuille equation so long as

• the flow is due to a pressure difference
• the fluid is incompressible and Newtonian (water is, approximately);
• the flow is laminar (not turbulent) as it is with water if it flows slowly through a narrow pipe.
• through a pipe of constant circular cross-section
• that is substantially longer than its diameter;
• and there is no acceleration of fluid in the pipe.

All those conditions seem to apply. The pipes are substantially longer than their diameter which is one of the most important requirements. And they are narrow, the fluid is water.

Under those conditions

• If the outlet is above water, the flow rate is proportional to the height of the head of water above the inlet to the pipe.
  
  If the outlet is below water, it’s proportional to the difference in height between the water above the inlet and the water above the outlet.
  
  The difference in height of the water here is often called the “head” of water.
• It is inversely proportional to the length of the pipe.

Or in short, the flow rate for laminar flow, in a pipe significantly longer than its diameter, is proportional to the pressure difference, and so to the head of water, but it is also inversely proportional to the length of the pipe.

(it also depends on the radius of the pipe and the viscosity of the water but those are the same for all our pipes).

Techy details. The equation simplifies to Q = c ΔP / L where c is a constant that is the same for all the pipes, depending on the radius (which is the same for them all) and the viscosity of the water. There ΔP is the difference in pressure between inlet and outlet, which is proportional to the head, Q is the flow rate, L is the length. I’ve put the rest of the techy details into a comment

Double the length of the pipe and you halve the flow rate. Double the head and you double the flow rate.

So now for instance, can L fill at any flow rate?

It’s outlet is a very long pipe. Even if L is nearly full of water ,the head of water in F will mean the difference in heads between L and F is quite small even if F is nearly full and L is likewise.

It’s inlet is a much shorter pipe. Whether L can fill will depend on whether we can get J to have a high head to increase the flow rate of its inlet pipe to more than that of its outlet pipe. But such a high flow rate could mean that J, or A fill first.

So - it’s quite a finely balanced question, and hard to answer.

J could fill first also. A obviously can fill first with a very fast flow rate, just fill it faster than it can empty.

Well we can actually try this out with a real world experiment :).

Prozix has made a 3D printed version of the puzzle. If you have a 3D printer you can download it here and print it out and test it yourself: Answer to the question Which one fill First / water equilibrium system by prozix

I don’t have a 3D printer but he has uploaded some videos.

First this is what happens with a slow flow rate

Note that at 22 seconds in, J nearly fills briefly.

A bubble forms in the outlet from J to L, which makes sense, it’s a downward pipe and air is buoyant. So - if the pipes are very thin - or the flow is just right - that might lead to J filling right there, if you can arrange it to fill before the bubble disperses.

But what happens at faster flow rates? I asked in a comment to the video, and Prozix was interested and answered with a new video

At 28 seconds in, at one of the flow rates, then L and F fill at the same time.

Here, it all makes sense up to J. Also J can’t fill (apart from that possibility due to the bubble) because at this point with J only half full, the head (difference in levels) between the water in J and the water in L is around double the head for L to F, but the pipe from L to F is much more than double the length of the pipe from J to L. So the flow rate from J to L should be more than the flow rate from L to F.

In summary, the pipe from L to F is three times the length of the pipe from J to L. The head is about double when L and F are both full. So you expect to have L fill faster than F because if they were the same level, the outflow would be two thirds of the inflow for L.

So this is what we expect in the ideal situation (no bubbles, laminar flow, so no turbulence).

But how can L and F fill simultaneously? That’s more mysterious.

At this stage with J half full and L and F nearly full, the head between J and L is double that between L and F but the pipe from L to F is about three times the length of the pipe from J to L.

So the flow rate from L to F should be about two thirds of the flow rate from J to L when J is half full and L and F are almost full.

So, I don’t think they can, with a steady flow into L. There must be something going on that doesn’t fit our assumptions of laminar flow, or something else such as a bubble forming.

Let’s look at what lead up to this. If you look at the video, L fills faster than F to start with, keeping nearly the same head from L to F as from J to L.

L is clearly filling faster than F and is on track to beat it. There is no sign of any bubbles in the inlet to L.

But then a little while later you get this (25 seconds in)

Now F is filling faster than J. Something has happened to reduce the flow rate into L, which then permits the two levels between L and F to equalize.

But the head going into L hasn’t changed. Also the input pipe to L is full and there are no bubbles. I think the only possible answer is turbulence.

You can see waves forming in J so maybe that means there’s a bit of turbulence impeding the flow from J to L, especially since the water level for J is exactly at the level for the outlet to L. What are your thoughts?

This is what happened with a moderately fast flow rate:

All of A, B, C, J, L and F are just about full. B, L and F started to overflow first and I think L just about beat the other two though it was almost simultaneous. In this frame you can see L just about to overflow and the other two though they have the water raised above the level of the top, haven’t yet actually started to flow down the side.

So how do we understand that as a possible state in terms of the flow rates? Back to our diagram again

With A, B, C, J, L and F all filled, then A to B to C to J all have the same length of pipe and same head (height difference of the water in the tanks above inlet and outlet) so have the same flow rate. J to L has around 2.5 times the head of C to J, and the pipe is around 2.5 times the length, so the flow out of J is about the same as the flow into it, and the difference in head between the top of J and the outlet to I is small. From L to F, the difference in head is about the same as for C to J but the pipe is far longer, so L shouldn’t be able to empty as fast as it fills, so long before J fills,

So if the flow rate is high enough for J to fill like this, L should fill before J and F doesn’t get a look in.

So how could it happen? Well it must be the bubble from J to L, slows down the flow out of J so that J fills first before L. As for F filling?

Here is the video starting at that point.

Even K can fill, though it is pretty hard to do. This is with a very strong flow into A, and several of the others have been overflowing for some time. They have turned off the inlet pipe at this point.

Amusingly, in the real world, E ends up half full too after some time of running it at a high flow rate with the water overflowing from A.

Here is the complete video

So far the only confirmed alternatives to F are A (obviously) and L (pretty sure it wins at the moderate flow rate).

That’s just a start. There are many other things to try

• Varying flow rate. Can you get, J, say, to fill first or even K by turning the flow rate up and down at critical points during the filling process? This could cause bubbles to form, as well as adjust the heads of the various tanks.
• What happens if you scale the whole model up, or scale it down to a very small size? Scaling it down could make the flow rates out of some of the pipes very slow. It could also mean that bubbles like the one from J to L take a long time to disperse too. Scaling up could lead to more possibility of turbulent flow through the pipes.
• Try adding sugar for viscosity
• What if it is really hot, and you use a slow flow rate so that the water evaporates quickly?
• What if it is really cold so that the water freezes? That would seem to be a way to fill even B first, if the water freezes by the time it gets to B to C but remains unfrozen as far as the flow from A to B.

NOTE

If you see anything in this to correct, however small or important it is, please either suggest an edit for my answer or say in a comment. Thanks!