Work energy homework question 
Equal force $(F>mg)$ is applied to the string in all three cases. Starting from rest point of application of force moves a distance of 2 metres down in all cases. In which case the block has maximum kinetic energy? The diagram is shown 

My approach:
I did this as an experiment with some thread and support at home and concluded that in case two, the distance by which the mass m moves up is half of the distance by which the string is pulled. This certainly reduces the kinetic energy of case 2 as compared to 1 because of work-energy theorem; the work force F is doing is going into raising the kinetic energy of mass and since work is force times displacement, case 1 has higher displacement than 2 and thus more kinetic energy.
But I am unable to imagine to extend this for case 3. The solution my teacher gave is a one liner; he let's tension in string be T in each case (which I still don't understand why it should be equal for all 3 cases) and then proceeds to draw free-body diagram of all the three. He draws a 3T force upwards on 3, a 2T upwards on 2 and a T force upwards on 1 and simply concludes 3 has more tension pulling it up and thus most kinetic energy.
I need to understand how to do this in a more intuitive way!
 A: The energy input to the system is from the rope being pulled.  Work equals force multiplied by distance.  As both of these quantities are fixed, the total energy input to the system must also be identical for each.
The energy is going into only two places, the kinetic and gravitational potential energy of the masses.  So the system with the smallest increase in GPE will have the largest increase in KE.  Which will move upward the least for a given distance the rope is pulled?

He draws a 3T force upwards on 3, a 2T upwards on 2 and a T force upwards on 1 and simply concludes 3 has more tension pulling it up and thus most kinetic energy.

This is a poor argument to me.  It holds if the tension operates over the same distance, but that is not true.  In the 2T case, the force on the block is double, but the distance it acts over is only half.  If we were to ignore the potential energy in the system, all 3 cases would give identical KE.
To see which one rises the most and which rises the least, take a look at the wikipedia article on the mechanical advantage of pulleys.
A: When dealing with these problems, go from the following fact:
If you have a force $F$ pulling an idealized pulley, then the force in each string supporting the pulley is $\dfrac{F}{2}$. (the same is true if you flip the pulley around)

If you want a reason as to why, it's because, amongst some other assumptions, the pulley is massless. By Newton's third law, the forces on either side are simply action-reaction pairs for the objects they're attached to. If this confuses you, don't worry, when you learn about torque and moment of inertia it will all make sense.
The reason that in case (3) the 3 tensions pulling up the block are the same:
Let's say the rightmost string pulls up with tension $T$. Then, by the rule I explained above, the middle pulley exerts a force of $2T$ on the block, which implies the total upwards force is $3T$.
