Just realized that in a 2d puzzle you have only 4 corner pieces, but in a higher-dimensional (say, 20d) puzzle, almost all pieces would be corner pieces.
Puzzles at higher difficulties are not rectangular, don't have straight sides, or have straight & corner-looking pieces in the middle.
My favorite difficult puzzles are Stumpcraft -- this small business makes wooden, beautiful smelling puzzles with a laser-cutting technique. They're very difficult with intricate and intentionally misleading pieces and geometries. https://www.stumpcraft.com/
Suppose a perfect grid in a hypercube. In d dimensions we have 2^d corner pieces. But if there are k pieces per side, then k^d total pieces (for example 10^d).
If we consider pieces adjacent to a corner, including a diagonal, then these are little cubes of size 2^d sitting at the corners, so a total of (2^d)(2^d) = 4^d pieces close to a corner.
In general taking little cubes out of the corners isn't enough to get most of the pieces. If we think about the diagonals of length k sqrt(d), we've only cut those a little shorter.
However, what does "corner" mean in an N dimensional puzzle? I was taking it to mean that it has N sides without pegs or holes. But if you took it to mean at least 2 sides without pegs or wholes, then indeed the ratio would increase (I think there are 3^N - 2^N - 1 of those) with N. But I think this is the less-satisfactory way to generalize "corner" pieces and these really should be thought of as edge pieces.
More generally, with k pieces on a side, there are k^N pieces total, and (k-2)^N internal pieces, so in the limit, the ratio of internal to total is (1-2/k)^N -> e^{-2N/k}.
So if the number of dimensions gets to be bigger the grid size, almost all pieces are "edge" (i.e. outer) pieces. Even if N=k/2, something like 70% of the pieces are edge pieces.