Studying Mathematics Thread

[Fireball]

Yes, I get a great deal of satisfaction when I solve something for myself. Here is the case for four blocks. The 7th and 8th are mirror images; they are not rotationally symmetric.

OK, there is a fair amount that can be said here.

First, this question makes sense in every dimension. For dimension 1, use line segments (this is a trivial case), for dimension 2, put together squares, dimension 3 uses cubes (this is your case), etc.

It is clear that the number of arrangements for n 'cubes' in dimension d+1 is at least as large as that for n 'lower dimensional cubes' in dimension d: just fatten up any arrangement in dimension d to a similar arrangement in dimension d+1.

So, while the dimension 1 case is trivial, it turns out that even the dimension 2 case is highly non-trivial. The figures in dimension 2 are known as polyominoes.

https://en.wikipedia.org/wiki/Polyomino

Even in the dimension 2 case, the growth rate is exponential. A similar proof works in dimension 3, where, if A_n is the number of arrangements of n cubes, we automatically get

A_n A_m <= A_{n+m}

We can see this by just attaching the 'upper right front' cube of an arrangement of n cubes to the 'lower left bottom' cube of an arrangement of m cubes.

This inequality is enough to prove exponential growth and gives a lower bound. So, based on the 2 dimensional results, the number in 3 dimensions grows at least as fast as (4^n)/n.

I also suspect that the use of 'twigs' can give a lower bound for any dimension, but have not yet worked through that.

I hope this helps!

Interesting. Wiki has an entry on polycubes that shows me that I missed two of the pentacube configurations. Oh, well. I hadn't looked at this in years, but the solution existed before I started working on it. I just didn't know where to look.