Circuit Problem: How Many Ways to Light All the Bulbs?

[mjsd]

see my diagram when it is ready...hopefully it will convey what I mean before.

case I and II are effectively the same after redrawing (unless you have some weird rules governing your system)

 

[pixel01]

No, it is not like that. You can not stretch the wire that way. Imagine, the bulbs are far from each other, so in the case I, you connect from B to A1 and then from A1 to A2. It is completely different from your circuit.
You have created one more node in your drawing. The requirement here is that the wire can only be connected to the battery or a certain bulb, not to common node.

 

[mjsd]

tell me how you can connect two wires to one terminal without "making one more node". My point is coming from elementary circuit theory point of view.. if you know what I mean. you want to connect two wires to the same terminal of the device, the "new node" that you have created is the point where the two wires and the terminal meet, before connection there is no (essential) node. You have a (essential, 3-way) node after connection because conservation of "stuff" holds at that junction (eg. KCL)...
you said that wires can only be connected to battery/device... true... but nothing forbids current to go from A1 to A2 without entering B in between, so there will be a direct link beween A1 and A2 even when u have B in between. This is the effect of connecting two wires together a same point.

I brought this issue up because there may be problems in defining what are equivalents and what are not when you have tree structure that may have some rotational symmetry in them.