I have a 10 sided "cube", which is made by "Wonderful Puzzler" (they also
make crappy cheapo regular cubes). The guts are essentially the same as
the regular cube, but the corners are cut off.

If you look down from the top, you see an octogon. The edges are all the
same as a regular cube, but since the corners are cut, they are one color
(thus the 10 colors). For example, there are red, blue, and orange
faces. On an ordinary cube you would have a red-blue-orange corner cubie,
but on this in its place is a pink face. To make this clearer, here is
the coloring of the thing:

red
 *
  light-blue**gold**yellow**blue**pink**orange**violet**green
 *
      white

(red on top, white on bottom, looking at the blue face, back face is green,
right face is orange, left face is gold).

The interesting thing about this is that unlike the ordinary cube, every cube
does not have a place, you dont know that the pink corner goes between the
blue and orange faces (in an ordinary cube it is the red-blue-orange corner
so you know where it goes). To solve it, you just put the corners in some
order, solve it using the usual transformations, and then if you get a
"parity error" you must go back to the top layer but switch two of the corners
and solve it again. Thus in general you have to solve it almost 2 times!
(almost because you dont have to redo the top layer or half of the second
layer (this assumes you solve top down, which I do.)).

What I mean by parity error is that if the corners are switched you can get
a configuration that in an ordinary cube would tell you the cube is put
together wrong. For example, you can be solving it and get to a point where
an odd number of edges must be fliped.

There may be a transformation to flip an odd number of edges with this cube,
but I have not found it. Anyway its more interesting to solve and it
changes it shape in general with each transformation. (unlike the cube
which stays a cube; this is a octagonal prism).

Alan

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