Formally, a "buckyball" is not a single shape, but actually a class of polyhedra known as Goldberg polyhedrons. However, the most popular shape in the class (which describes the puzzles on this page) is technically named a truncated icosahedron (or perhaps a regular duotrigahedron). That is quite a mouthful, so many people just call it a buckyball. This polyhedron has the following key properties:
 32 faces = 12 regular pentagons (each with 5 identical sides) + 20 regular hexagons (each with 6 identical sides)
 90 edges (each a line where two faces meet) = 60 pentagon/hexagon edges + 30 hexagon/hexagon edges
 60 vertices (each a point where three edges meet)
Like all twisty puzzles discussed on my site, each face is divided into
multiple facets which are (for this class) a variety of colored polygons (mostly kites and isosceles trapezoids, but also pentagons and hexagons). Technically the facets are not polygons because their sides are curved. The puzzles discussed on this page are often constructed with 32 unique colors (one for each face), but some have been manufactured which duplicate colors and thus have fewer unique colors (those usually share colors between the pentagonal and hexagonal faces because their facets can not intermix).
I know of 2 variants of this puzzle, each described in some detail further below. But a quick summary is:
 Void Tuttminx [n=3] has 10 polygons per pentagonal face plus 12
polygons per hexagonal face for a total of 360 facets [10*12+12*20], and 60 "corner" blocks plus 90 "edge" blocks for a total of 150 surface
blocks
 Tuttminx [n=3] has 11 polygons per pentagonal face plus 13 polygons per hexagonal face for a total of 392
facets [11*12+13*20], and 32 "center" blocks plus 60 "corner" blocks plus 90 "edge"
blocks which totals 182 surface blocks
An important thing to note about the overall shape of this puzzle class is that it is not a Platonic solid.
This goes against my original intent of limiting the puzzles I talk about on my site. Although it is not a "perfectly regular" polyhedron (there are only 5 in the universe, each discussed on a separate page), the shape is "very regular" and is one of only 13 [or 14] Archimedean Solids (that count excludes the 5 Platonic Solids). The shape is a specific Goldberg polyhedron: GP_{V}(1,1).
Beyond math, the shape is interesting in other respects. Artistically, the shape is customarily seen on International Footballs (American soccer balls). In chemistry, the predicted (in late 1960's) Carbon60 allotrope was discovered/confirmed (in 1985) and named a buckminsterfullerene. Since then, many other fullerenes have been discovered. Some of them have the same "cage" or "psudospherical" form (each a Goldberg polyhedron) as the original, and a few form a cylindrical shape ("nanotubes"). Anyway, the Void Tuttminx puzzle is a great model of the first (and most common) fullerene (C_{60}).

"Buckyball"
(truncated icosahedron) (duotrigahedron)

Most importantly, unlike all other twistypuzzles discussed on my site, this class of puzzles features two (instead of just one) shapes for faces (pentagons and hexagons). This makes describing and playing such a puzzle more challenging. In particular, there are two types of "edge" blocks. The edgetype depends on which types of faces it borders:
 Pentagon/Hexagon edge, or simply "pentagon edge": there are 5*12 = 60 of these
 Hexagon/Hexagon edge, or simply "hexagon edge" (or verbosely "nonpentagon edge"): there are 9060 = 30 of these
Also important for playing is that in most "buckyball" puzzles (all discussed here), the rotation of each face depends on the shape of that face. In particular:
 Pentagon faces may be turned by 72° (as expected)
 Hexagon faces may be turned by 120° (unexpected)
Mathematically, a hexagon should be able to be twisted by only 60°. But when you do that on this class of puzzles, the edgetypes get mixed up. In most puzzles (the ones detailed on this page), the puzzle will become "locked" if you rotate a hexagon face by only 60°. In other words, you need to make two 60° rotations of a hexagon face (120° total) to keep the edgetypes synchronized (and prevent "lockup"). Note: I have read about a buckyball puzzle ("Futtminx") that actually allows the player to perform only 60° twists of the hexagon faces! I don't know enough about it to give a detailed discussion on this page, but I presume this puzzle must have 32 unique colors (because the "nomixing rule" has been abolished) and it would have many more combinations than the puzzles described below.
This faceturning (n=3, singlelayer) puzzle is very complex to play, but it makes a nice model of a Carbon60 molecule if you just want to look. Each pentagonal face has 10 facets (colored "polygons") and each hexagonal face has 12 facets. Both face types have the "same" two polygons:
 Kites on the corners
 Isosceles trapezoids on the edges
The edge facets are not truly the "same" because there is a variation in size between the "pentagon edges" and the "hexagon edges". Because there are 12 pentagonal faces and 20 hexagonal faces, this puzzle has a total of 360 visible facets (12*10 + 20*12).
The Void Tuttminx is composed of 3 block types:
 60 "corner" blocks (each with 3 visible facets [colored kites] and only one orientation)  they represent carbon atoms in a C_{60} molecule
 60 "pentagon edge" blocks (each with 2 visible facets [colored isosceles trapezoids] and only one orientation)  they represent singlebonds in a a C_{60} molecule
 30 "hexagon edge" blocks (each with 2 visible facets [colored isosceles trapezoids] and two orientations)  they represent doublebonds in a C_{60} molecule
Thus, this puzzle has 150 surface blocks (60+60+30) and 360 colored facets (60*3+60*2+30*2). It has an average of (exactly) 2.4 visible facets per surface block (360/150=12/5). An obvious feature is zero "core" and "center" blocks.
I haven't been able to confirm, but I believe the Void Tuttminx has roughly 205.417959 googol googol (10^{200}) combinations. That value is calculated as: 60!/2 ("corners") * 60!/2 ("pentagon edges") * 30!/2*2^{29} ("hexagon edges") / 60 ("orientation").
Because this puzzle has no "center" blocks, there is no fixed orientation of the full puzzle. This can create "parity errors", but the problem isn't as bad as similar puzzles. Parity errors are limited by the "nomixing rule" and the fact of only 1 "edge block" per geometric edge. If you find a 16color version, it should have more parity errors.

Void Tuttminx 
This faceturning (n=3, singlelayer) puzzle is very complex to play,
but the fixed center pieces make it slightly easier to play (good for mental orientation of the full puzzle). Each pentagonal face has 11 facets (colored "polygons") and each
hexagonal face has 13 facets. The two face types each have a uniquelyshaped "center" polygon: a pentagon at
the center of pentagonal faces and a hexagon at the center of hexagonal
faces (quite logical). Both face types share the "same" two
polygons:
 Kites on the corners
 Isosceles trapezoids on the edges
The edge facets are not truly the "same" because there is a variation in size
between the "pentagon edges" and the "hexagon edges". [Also, the facets of a "hexagon edge" look more like two squares than the facets of a "pentagon edge" which resemble a trapezoid and rectangle.] Because
there are 12 pentagonal faces and 20 hexagonal faces, this puzzle has a
total of 392 visible facets (12*11 + 20*13).
The Tuttminx is composed of 6+ block types:
 60 "corner" blocks (each with 3 visible facets [colored kites] and 1 orientation)
 60 "pentagon edge" blocks (each with 2 visible facets [colored isosceles trapezoids] and 1 orientation)
 30 "hexagon edge" blocks (each with 2 visible facets [colored isosceles trapezoids] and 2 orientations)
 20 "hexagon center" blocks (each with 1 visible facet [colored regular hexagon] and no orientation)
 12 "pentagon center" blocks (each with 1 visible facet [colored regular pentagon] and no orientation)
 ?? (unknown) "core" blocks (each with no visible facet nor orientation)
Thus, this puzzle has 182 surface blocks (60+60+30+20+12) and 392 visible facets (60*3+60*2+30*2+20+12). It has an average
of (approximately) 2.153846 visible facets per surface block (392/182=28/13). I don't own one, so I don't know how many "core" blocks exist (and thus the total number of blocks is unknown too).
The Tuttminx has roughly 12.3250775616 googol googol thousand (10^{203}) combinations. That value is calculated as: 60!/2 ("corners") * 60!/2 ("pentagon edges") * 30!/2*2^{29} ("hexagon edges"). That value seems reasonable, but I only have one dubious source (this wikipedia article which lacks references).

Tuttminx

© 2017, H2Obsession
Diagram of truncated icosahedron ("buckyball") © 2005 "Cyp" from Wikipedia ( http://en.wikipedia.org/wiki/File:Truncatedicosahedron.jpg)
Photograph of "Void Tuttminx" © 2002 TwistyPuzzles.com ( http://www.twistypuzzles.com/cgibin/puzzle.cgi?pkey=3689)
