This YouTube video created by Henryk Gasperowicz interested me so I decided to make a replica of the circuit shown in the video.
The circuit appears to be a simple series parallel circuit but behaves differently. The trick is a small oscillator enclosed in the 9V battery snap. This oscillator turns the DC from the battery into a square wave. Hidden diodes on the switches and LEDs steer the LEDs in the circuit. A schematic for this circuit can be found on Henryk Gasperowicz Google+ page. He builds all his circuits dead bug style. Dead bug style circuits are small but also difficult to build. Due to this a PCB was designed for this circuit.
Due to the small size OSH Park charged $1.10 shipped. The 0402 parts were hand soldred without a microscope so it may look a little rough. Here is a shot of the assembled board and the circuit in action