The OP clearly is an ESL (as I gather you are also ) and is not managing to adequately explain the concept. I think we should understand "puzzle" as "simulation" and "unsolvable" as "irreversible" for a given run of the sim.
I think I get it. That is neat. They are cups that come out of a tray. The LEDs are installed in the bottom of the cups and the leads make contact with strips in tray. To play you lift the cups?
I am guessing the cups are secretly a multiplex LED matrix? You have 5 columns and 5 rows for a total of 10 pins coming back to the Arduino? You have not built the interface circuit to the Arduino yet? You can still install the resistors and/or transistors near the Arduino?