Confused about wiring
I'm a little confused by the dimmer switch based on your post, so hopefully you'll correct that and I'll finish this answer.

However, to start you off on a new front that you might not have considered: go ahead and draw in 3-wire cabling between every item. You already know the layout that they have to be in and the maximum wires. Although it's for bonus credit, it can be used to your advantage.

There should be 2-wire going into the first light as your homerun, that's easy. Then leaving that first light to the switch will be 3 wires, hot wire nutted with the line, one tied to the light common, and then the neutral. Then you know that one wire from each of the other 3-wire cables has to be used to carry the neutral as well.

Without knowing the switching that is desired, I can't come up with a working solution based on my guess just yet. But this might help you find the answer yourself for now.