I didn't understand the comparison. In pushing it, you didn't transfer a particle from one side to the other; you only caused particles to move (the particle on the other side only moved a few meters, not faster than the speed of light). So how exactly did you cause anything to exceed the speed of light? And how exactly are you transmitting information from one place to another this way?

"Pushing it in a certain direction would seemingly also move its other end immediately."

The movement of the other end would not be immediate.

Ishay,
I don't have a good answer to the question that was raised, but on the face of it your suggestion also isn't a solution. The effect on the position of the far end of the rod seems equivalent to transmitting a signal. You can imagine, for example, two friends standing at the two ends of the rod who agreed in advance that a slight movement of the rod to the right on their side means "yes" and a movement to the left means "no." And there you have it: information has been transmitted faster than the speed of light. So at least seemingly, the paradox stands.

Doron,
When you manage to move a rod 300,000 km long in less than a second, then we'll have something to talk about. In practice the rod won't move, and neither will you.

Rabbi Michi: Thanks, I wasn't familiar with this. Where can I find a place that discusses it?
Ishay: See Doron's reply.
Lemi: Why not?
Doron: Thanks, that's what I meant.
Y.D.: That's not an acceptable answer to questions of this kind. The question is on the level of theoretical understanding, and technical difficulties are not an answer.

Search, for example, long rod light speed.

Okay, I dug around online and found a solution to the paradox, seemingly…. The claim is that material bodies (rods) are never perfectly rigid, and therefore even short objects do not move simultaneously. When I push a body, there is necessarily (because of the laws of physics) a disturbance in the medium that travels from one end of the rod to the other. This is in fact the motion of a signal or information. In the case of such a long rod, that disturbance will in any case travel no faster than the speed of light (and possibly much more slowly).
But… this raises the question: does this limit of the speed of light have logical status, or only physical status? That is, perhaps in another possible world governed by slightly different laws there are ideally rigid rods, and there one could in fact surpass the speed of light. Also: how does special relativity view this last question? As I understand it, relativity blurs the boundary between a priori (logical) considerations and empirical physical considerations, and as such it maneuvers itself into a position that is uncomfortable for itself. Because according to its own internal logic there isn't even logical room for such a possible world. If that is indeed what it claims (even if only implicitly), then maybe there is a problem here too.

By the way, the thought experiment has a perhaps more successful version: a very strong flashlight casts a shadow on an object at an astronomical distance. The question is whether moving the flashlight would immediately change the location of the shadow on that object.

Thanks to everyone who replied; I found the answer, including a detailed explanation:
https://physics.stackexchange.com/questions/2175/is-it-possible-for-information-to-be-transmitted-faster-than-light-by-using-a-ri

Moving a flashlight will never change a shadow at more than the speed of light.
Besides, what's the problem with assuming that by means of a rod you could transmit messages faster than the speed of light?

As for the rod, I gave an explanation that seems very plausible to me as to why there is a problem with bypassing the speed of light. At least a physical problem. As for the flashlight and the shadow, it seems that this limitation is explained in the same way.

It may be that there is a physical limitation; the point is that if there were no limitation, it wouldn't pose a difficulty for anything.
As for the limitation itself, does it also exist when pushing forward? According to that, if we push a rod forward at a speed close to the speed of light, it would contract to zero size. According to what was brought in the link, which is speaking about a much lower speed, fast airplanes and missiles should have contracted significantly because of the slowness with which the push is transmitted.

A shadow is the absence of light, and moving a flashlight shouldn't move the light that has already left it, so there is no basis for the question.

In my opinion you're mixing together two different issues (which admittedly are both connected to special relativity). The first is a change in the measured dimensions of a moving object as predicted by relativity, and the second is the elasticity of bodies, whose rigidity is never absolutely ideal.
As I understand it, even before Einstein we could say that there was seemingly a paradox here. But it was not between the speed of light as an upper limit and the simultaneous motion of the two ends of the rod; rather, between the intuitive (but mistaken) assumption that the two ends must move simultaneously and the laws of elasticity, which forbid that. The solution to the old paradox is good for the new one as well, and there's no need to bring relativistic considerations into it here. That's how I understand it.

I meant somewhat different things from what you understood.

And I didn't quite manage to understand why you call a mistaken intuition a paradox.

Do you agree with what I wrote about the flashlight?

Regarding the flashlight, maybe you're right. I brought the example from memory from a YouTube clip I saw a long time ago. Maybe I remembered it incorrectly.

As for the basic issue:
I said that even in the pre-Einstein world they might perhaps have thought there was a paradox here. After all, there seem to be two conflicting views here: on the one hand, our intuition tells us that both ends of a rod—even a hypothetical rod as long as you like—move at the same time (if you pushed one end, the far end must necessarily move immediately); on the other hand, they already knew how to say back then that matter is not absolutely rigid, and therefore there is a "delay" in the motion passing through it from one side to the other (the far end will begin moving some time after the near end has already begun its movement).
In any case, I assume that even in the pre-Einstein period they understood that this was only an apparent paradox. The intuition as though the ends of the rod move simultaneously is created by mistake because in reality rods are so short that one cannot actually discern that supposed delay. By contrast, in our thought experiment it is easy to see that the elasticity of the material ceases to be negligible, and then a real "delay" arises.

Take a long rod, tie it at one end so it can't move, and put on the other end a machine that keeps pulling it. The moment you release it from the tie, it will already start being pulled.

I already wrote earlier that I agree with Ishay that there is no problem here at all, but whoever sees a problem in it is welcome to look for an answer.
Yeshiva Student

For the original example of a shadow moving faster than the speed of light (in English):
http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html#3