I have this weird idea for a puzzle/toy (or torture device, depending on how you look at it) I've been trying to make for years now. I happen to be worse at this kind of math as I thought; and I'd be delighted to get some help.
This puzzle is made of 64 cubes, with a peg or hole on each side (such as, you can connect them like Lego bricks). These pegs and holes have different shapes, call them, if you will, symbols (just bear in mind there's an "up" and "down" symbol pair on each connection). The goal of the puzzle is simple, build a solid $4\times 4\times 4$ cube, with no gaps in it.
The problem is, how can I generate a set of these cubes with a low number of possible solutions (Ideally just 1), while using the fewest amount of symbols?
P.S. Sorry if there was some weird grammar or tone here, I'm not a native speaker.