> Given 6 points that are connected with each other with a resistor of
> resistance $R$, find the resistance between any two points.

So 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 reducing the network to a more simple one. That is, after choosing any two points in the network, we can remove the resistors which are connected between the other four points, as they remained identical, and they keep the symmetry, thus we can change between them and the network will remain the same. However, we can also swap the two chosen points, and the system will still remain the same. 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 looses some symmetry level, but its certain elements are still symmetrical under certain dispositions. But swapping the two black balls still won't change the system, so why can't we follow the same logic and remove the resistor between the two chosen points? 

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