If we attach a fan at the end of a boat and switch it on, will the boat move? I've been across a lot of questions like these, but I'm never quite able to get them. In this question, I reasoned that if I consider the boat and the fan to be a system, then there is actually no external force acting on it, so there is no reason for it to move. But, I have this gut feeling that the boat should actually move, but I'm not able to convince myself of this mathematically. Help will be appreciated. Do try and really break it down for me. I'm an eleventh grader, and I've only just started delving into the realms of physics. 
 A: If your fan-boat is in vacuum, they won't move. In the air, they will. Your assumption conflicts with your intuition is because you isolated the system from the air, which should not.
A: There is no doubt that such a system would move, as other people here say, moreover,  such a system can be quite practical and is actually used in marshy/shallow water (https://en.wikipedia.org/wiki/Airboat )
A: The boat may obviously move forward. That's how airplanes move, too. The turbines etc. make the fluid move backwards so the boat or aircraft has to move forward. A more interesting fact is that it may also move forward by creating "wind in the forward direction" by its own fan if this wind is reflected from the sail.
Mythbusters have checked this question in this real-world video

https://www.youtube.com/watch?v=uKXMTzMQWjo

and the result is that they could make it move although the speed was unremarkable. There exist many approximate pictures in which the forces are guaranteed to cancel but none of those calculations is quite exact in the real world.
Ultimately, there have to be terms that don't cancel in the real world. And these terms have a similar effect on the boat as the motions of our hands when we swim. 
Swimming clearly doesn't contradict the laws of physics. When we push our hands through water behind our bodies, we are creating a higher pressure of the water on the rear side and lower pressure on the front size of the hands. That's why the net force goes forward.
The boat with the fan ultimately does nothing else than that.
One may imagine that the air is composed of "particles" (it's the atoms in the air but imagine it's marbles) and the fan is picking them from the air, shooting to the sail in front, and they get reflected from the sail. The reason why the boat is moving to the front is that the marbles it shoots are ultimately moving "mostly backwards", after they're reflected from the sail, but when they're picked from the fan, they're more or less caught as "marbles coming from all directions". So the boat is changing the marbles' velocity to be "more to the back" after the maneuver than before that which is why the boat itself is moving forward.
This picture of "marbles" would completely disagree with ideal fluids etc. If all fluids (air and water) were perfectly ideal etc., the effects would have to cancel, I think. But the nonzero viscosity and similar things makes the fluids behave "a little bit as if they were marbles", and a little bit of this effect is enough to get the boat moving.
A: The fan moves air (pushing it from the front of the boat to the back). Conservation of momentum say that "if something moves one way, something else must move in the opposite direction". In this case - air moves back, boat moves forward. If you were holding the fan in your hands you would feel a force (as it is pushing against the air). That force is transmitted to the boat, and causes it to move.
A variation on this is "if a fan mounted on a boat blows on the sail of that boat, does the boat move?". That is a little trickier - the answer can be "yes or no", and it depends on the configuration of the sail and fan. If the fan blows at the sail that is perpendicular to the boat, then the air bouncing off the sail goes sideways, and the boat may remain stationary (the forces on sail and fan balance). But if you angle the sail such that the net result is that air is pushed towards the back of the boat, the boat will again move forward.
A: Of course the the boat will move, you are converting the electrical energy from the battery into the rotational energy of fan, which produces almost a rectilinear motion of air particles, thus the momentum of air particles is changed by the the motion of blades of fan, which means that the blades of fan would be applying a force on the air particles, and the air particles would apply an equal amount of force to the fan in opposite direction and thus the boat would move. 
In short you can say that the energy stored in battery is eventually converted into into the kinetic energy of air particles and boat.
Think it in this way: it is very much identical to all the activities that we humans do in our everyday life. Let's say that you are sitting on a chair and you want to stand up, now you may argue by saying that there is no external force and you cannot stand up, of course you can stand up, what you actually do is that, you convert the potential energy stored in you muscles as fat and carbohydrates into kinetic energy of bones and other moving part of your body and eventually into gravitational potential energy as the center of mass of you body lifts up.
In your case if we consider the boat, fan and the air particles as a system, then we would observe that the the momentum this is conserved.
There are many examples in our everyday life, like movement of small pellets from a firecracker when it bursts, movement of two blocks connected to a compressed spring, when the spring is released, etc; in all these examples the potential stored in the system is converted eventually into kinetic energy, but if we include everything in the system: all the pellets and both the blocks in respective above examples, we would see that the momentum remains conserved although the kinetic energy has changed - because momentum is a vector quantity whose vector sum may be zero, as everything gets cancelled out eventually, but kinetic energy is added algebraically.
In your case the change in kinetic energy of the boat and the air particles was due to the Sun, you are actually using the Sun's energy to move your boat. For the Earth the only external source of energy is Sun, other than nuclear energy. The electrical energy that the boat used was actually from the sun. The electrical energy would have been fed into the battery either by burning coal, water falling on turbines, harnessing wind energy, etc. All these sources of energy are deriving their energy from the Sun. Like the burning the coal is releasing the energy that was once stored into the plant through photosynthesis, the falling of water on turbines derives its energy by harnessing the change in gravitational potential of water as it falls though a height, now this water was lifted up by sun as it evaporated the water.
So, yeah the boat would move!
A: You are in a boat. You throw a ball outside boat with speed $v$. It will push the boat foward by conservation of momentum. Thus, assuming initially boat and ball are at rest, the boat will have opposite momentum of the ball, and then boat will move foward.
$$
\mathbf p_{boat} = -\mathbf p_{ball}
$$
This is mathematically OK. There is no doubt that boat indeed moves forward. Now, with your fan, you are doing the same, but with smaller little balls: Air particles. Thus, boat also moves forward.
A: Ridiculous idea!  How could that possibly work?

