Say I have n cubes. I am allowed to glue the faces of these cubes together, but the faces must line up perfectly. How many unique shapes could I make? All orientations of one shape are considered to be the same shape.
I'm doing an engineering assignment where I must draw the different possible shapes, but I'm curious as to what the mathematical formula is to determine the maximum number of shapes.
This is an open problem; no general formula is known. The shapes are called "polycubes".
However, the values are known for small $n$; you can find them listed up to $n=16$ the in OEIS.
Note that there are two sets of answers, depending on whether you consider two polycubes to be the same, or different, when they are mirror images of one another.