Accidents of small $n$ In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that there is no general algebraic solution for polynomials of degree $n$ except when $n \leq 4$.
It seems that there are many interesting examples of these special accidents of structure that occur when the objects in question are "small enough", and I'd be interested in seeing more of them.

I would include in this list a discussion of $G(n)$ and $g(n)$ in the Waring Problem. 
The point is that when it comes to representing integers as sums of powers of non-negative integers, it seems to happen that some (smallish) integers require more powers to represent them due to some some (presumably unknown) peculiarity of small integers.
For example, in the case of representing integers as sums of cubes, it has been proved that 9 cubes are sufficient, and some numbers require 9, so that $g(3)=9$.
On the other hand, calculations suggest that almost all numbers from some point onwards are sums of at most 4 cubes (so that $G(3)$ might be 4, but this is not proved), and it appears that it is an "accident of small $n$" that there are some (smallish) numbers which require more than 4 cubes.
