High School Are the second and third circuits same as the first one?

[donaldparida]

It seems at first sight of the first circuit that all the resistors of resistance r are connected in parallel and i have thus simplified it to the second circuit. But then on numbering the nodes and redrawing the circuit i got the third circuit.
I got the same equivalent resistance for all the three circuits which i think is a coincidence because i do not not know why the second circuit is equivalent to the first one (i drew that way because the in the first circuit it seems that all the resistors are in parallel).
I basically have two questions:Is the first circuit equivalent to the second circuit or the third circuit or both? Why is the second circuit equivalent to the first one if it is so?
https://drive.google.com/open?id=0B3IenBvKWb4PWWZxdHdPVEZIQ3M

 

[berkeman]

It seems at first sight of the first circuit that all the resistors of resistance r are connected in parallel and i have thus simplified it to the second circuit. But then on numbering the nodes and redrawing the circuit i got the third circuit.
I got the same equivalent resistance for all the three circuits which i think is a coincidence because i do not not know why the second circuit is equivalent to the first one (i drew that way because the in the first circuit it seems that all the resistors are in parallel).
I basically have two questions:Is the first circuit equivalent to the second circuit or the third circuit or both? Why is the second circuit equivalent to the first one if it is so?
https://drive.google.com/open?id=0B3IenBvKWb4PWWZxdHdPVEZIQ3M

Yes, they are the same. All of the resistors are in parallel. Each resistor connects to the same two nodes.

 

[Baluncore]

Colour all wires connected to the (+)ve terminal red.
Colour all wires connected to the (–)ve terminal black.
Note that every resistor now has different coloured wire at each terminal.
If a component had a wire that was not coloured, then it would be connected to a third “node” in the circuit and you would need another colour. You do not need colours if you number the nodes wherever they connect to a component.

Two circuits are topologically the same if the components are the same and are connected to the same nodes.If you can work out the resistance of a network try this one. Imagine a cube with nodes at the corner points. Place a 12 ohm resistor on each edge to connect with the ends of two other edge resistors. Draw the cube of resistors. There will be 8 corner nodes and 12 edge resistors.
Pick two of the diagonally furthest apart corners of the eight resistor cube. Find the resistance between them.