Use Quake 4!
Here's the map: cqroom.pk4 (not intended for playing). Please excuse the crappy mapping, lighting, scale and use of textures.
I experimented a bit with scripting for Doom3 the other day. The potential of the engine is just incredible. Anyway, this is what came out: Rubik's Cube for Doom 3
It's a small test map, so don't expect real singleplayer gameplay.