
I have a device like this one:
http://www.youtube.com/watch?v=kA_hw_lY-OY (All of them work the same)

Here's an image:


I want to make it spin without touching it with hands.

I tried using magnets of several kinds, and tried to put them in many places around it but it didn't make it spin. It just made it to vibrate a little.

I guess there is a point in which the "pen" will spin and spin. Basically, I prefer to make it rotate not with magnets, but with an electric current. Most likely, I will use an AC generator, and it also makes sense, because I thought that in some point there will be same poles, and then the pen will spin a little bit, and then there will be different poles, but the alternate current changes the poles each time, so the poles will always be the same.

My question is: in what point should it happen?

This is not paradoxical and it is not necessary for any physical phenomenon to a priori have to obey any particular law. Some phenomena do have to obey inverse-square laws (such as, particularly, the light intensity from a point source) but they are relatively limited (more on them below). 

Even worse, gravity and electricity don't even follow this in general! For the latter, it is only point charges in the electrostatic regime that obey an inverse-square law. For more complicated systems you will have magnetic interactions as well as corrections that depend on the shape of the charge distributions. If the systems are (globally) neutral, there will still  be electrostatic interactions which will fall off as the inverse cube or faster! The van der Waals forces between molecules, for instance, are electrostatic in origin but go down as $1/r^6$.

It is for systems with a conserved flux that the inverse-square law must hold, at least at large distances. If a point light source emits a fixed amount of energy per unit time, then this energy must go through every imaginary spherical surface we think up. Since their area goes up as $r^2$, the power per unit area (a.k.a. the irradiance) must go down as $1/r^2$. In a simplified picture, this is also true for the electrostatic force, where it is the flow of virtual photons that must be conserved.
