Divide a "base" edge of a regular pentagon into three equal parts. Then draw two lines from the base to the center of the other edges such that the lines do not intersect. This splits the pentagon into three parts. When you color these parts with two colors you get exactly 12 different tiles:
Oh, exactly 12 tiles! Can you tile the surface of a dodecahedron with these 12 tiles such that all lines and colors match up? How many essentially different tilings exist?
Partial Answer:
It is possible, though I'm not sure how many tilings exist.
Solution:
If we label the pentagons like so:
This net will satisfy the requirements:
Here's a version with lines showing which faces will be connected:
As a bonus, here's an animation of the resulting dodecahedron I threw together in Blender:
Other Solutions:
Other people have done the bulk of the brute force work, but it looks like there are 3 "Essentially Different" solutions instead of the 5 others have come up with. What hasn't been taken into account is that some solutions that don't have rotational symmetry still have mirror symmetry. If we rearrange the solutions from @WeatherVane to have the "I" tile in a fixed position and orientation, it becomes clear that some of these solutions are mirror images. For example, here's his first and fourth solution rearranged:
The same can be said of the second and fifth solutions:
The third solution (The one I had already found) is a mirror of itself, so it doesn't have a mirror image.
Fun Stuff:
Here's an animation of the first mirrored pair:
I've also put together a folder of higher-quality renders of the different solutions, the vector graphics I used to make them, and a PDF with papercraft versions of the different nets. You can find them all here.
Designing the puzzle took much more effort than solving it. While the mirror symmetrical solution can be found manually, for a complete survey a computer aided search is necessary. This can be done quite easily using a backtracking algorithm and the 5 "raw" solutions are found within seconds. I think the details of my program are not very interesting and there are surely other approaches which also solve the problem.
Up to rotation/reflection there are 3 essentially different solutions. This was pointed out in detail in the solution of DqwertyC. The exchange of the colors blue <-> yellow is an operation which can be described in mathematical terms as an antisymmetry. All three solutions have beautiful properties concerning antisymmetry which can be seen best if you build the models manually. The main goal of my answer is to invite you to build these models. If you print each net half fitting on a DIN A4 page the edge length is about 5 cm which is quite handy. The resolution of the pictures is quite high, you can right click and save them.
There are essentially only two(!) different types of nets of a dodecaedron half you need to build all three solutions. Lets start with the symmetrical solution. One half net looks like this
and the other half is just the same with the antisymmetry applied
The first half of both the two other asymmetrical solutions can be made identical and look like this
The second half of the first asymmetrical solution again is just the antisymmetry applied to the first half:
When you build the model you can find a unique twofold symmetry axis such that that rotation about this axis just exchanges all colors. In other words rotating and applying the antisymmetry leaves the dodecaehdron unchanged.
The second half of the second asymmetrical solution is just the mirror image of the second half of the first asymmetrical solution directly above.
When you build the model you can see that a point reflection at the center of the dodecahedron has the same effect as exchanging the colors. In other words applying the point reflection and the antisymmetry together leaves the dodecahedron unchanged.
I have found more than 1 solution.
Here are my 5 arrangements.
They are labelled the same way as DqwertyC's answer.
Face a is in the same location and orientation in them all.
Found by exhaustive permutation.
I wrote a program to enumerate the ways the pentagons can be rotated and placed into a Dodecahedron net like the one illustrated in @DqwertyC's solution. I get
300 distinct nets
But this overcounts the number of essentially different Dodecahedrons, of course, since each rotation in 3-D space shows up as a distinct 2-D net. The internet tells me there are 60 different distinct rotations of a Dodecahedron, so I guess my answer is:
300 / 60 = 5 distinct colorings
