Given 6 points that are connected with each other with a resistor of resistance $R$, find the resistance between any two points. (Answer: $R/3$)

(All the conducting wires have the same resistance $R$.)

I know that such a wording immediately implies that these 6 points are absolutely identical, which makes it possible to apply symmetry arguments that will help in reducing the network to a simpler one.

That is, after choosing any two points in the network, the remaining four points will still be identical, so we can swap any of them and the network will remain the same. Thus, we can remove the resistors which are connected between these other four points, since the points are identical.

However, we can also swap the two chosen points, and the system will still remain the same. So why can't we also remove the resistor between the two chosen points?

I'm told of the following analogy: The system of these 6 points is like a system of 6 absolutely similar balls painted, say, in white. By choosing two points, we paint them in black, thus the system loses some symmetry level, but its certain elements are still symmetrical under certain rearrangements.

Specifically, any two of the white balls can be swapped without changing the system in any way, so all the white balls are identical and we can ignore any resistors between them. But swapping the two black balls still won't change the system, so why can't we follow the same logic and ignore the resistor between them as well?

I'll generalize the question a little bit: why don't we care about other symmetries in the system?

(I'm looking forward a simple explanation, without involving advanced math, as I'm just a self-taught beginner and I'm only familiar with calculus. So I try to avoid matrices and anything advanced which students learn in advanced courses of electronics. I just want to get the idea and the concept itself.)