In my engineering class our teacher gave us a challenge to draw a sketch of every unique combination of 3,4,5, and 6 cubes. I spent about 30 minutes on that before I realized that I was making lots of duplicates. I wanted to find a way to tell if one would be unique without having to actually make it. I'm not good at rotating it in my head so I wanted something like a formula for uniqueness. I was thinking maybe something like the polynomials, sets, and numbers used to represent knots would work, but I haven't been able to figure out how to do that.

The approaches I have tried are as follows:

Have a set of numbers that just contains how many cubes it touches

Have a set of ordered pairs/trios/etc that are at junctions that say how far it goes in each direction

Have a sets of ordered pairs like above just group them with a number that says what the shortest straight line distance between the junctions are.

I saw that all of these failed. How else could I do this and would a representation allow me to generate all possible cubes without brute forcing it.