Patented by Zoltan and Robert Vecsei 1993, purchased from Mefferts 2007.
(plastic, 3.5 inches)

This puzzle has twenty faces that meet at 12 vertices. Mefferts has made this puzzle in a number of color schemes, including the 12 color "Dogic 1" (on the left above) where each vertex is colored, the 10 color "Dogic 2" (on the right above), where each face is colored (and each color appears on two of the 20 faces), the 5 color "Dogic 3", the 2 color "Dogic 4" and "Dogic 5", and the 20 color "Dogic 6". A set of 5 faces can be rotated about a vertex, or just the five triangles that meet at a vertex can be rotated.

Here are what a few moves of Dogic 1 look like:

Here are what a few moves of Dogic 2 look like:

Jaap's Page gives a solution that solves one tip at a time, and then uses the solution for the Impossiball to solve the faces.

Further Reading
Meffert's Page, from:
Jaap's Page, from:
Vecsei HU Patent, from: - patent no. HU214709