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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: enter image description here

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?

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    $\begingroup$ God I hope so. What a beautiful little puzzle. $\endgroup$
    – Sneftel
    Sep 6, 2023 at 13:44
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    $\begingroup$ BOUNTY NOTIFICATION: I plan to offer a bounty of 200 on this wonderful puzzle. $\endgroup$ Sep 7, 2023 at 3:57
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    $\begingroup$ @WillOctagonGibson username checks out :) $\endgroup$ Sep 7, 2023 at 9:59
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    $\begingroup$ @WillOctagonGibson I was about to do the same :) $\endgroup$ Sep 7, 2023 at 12:20
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    $\begingroup$ This is definitely one of the most aesthetically pleasing geometry puzzles ever. $\endgroup$ Sep 7, 2023 at 17:07

4 Answers 4

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Partial Answer:

It is possible, though I'm not sure how many tilings exist.

Solution:

If we label the pentagons like so: Original image of tiles, labeled a-l from top-left to bottom-right

This net will satisfy the requirements:
a net of a dodecahedron with tiles labeled

Here's a version with lines showing which faces will be connected:
the same net, without labels but with extra lines showing connections As a bonus, here's an animation of the resulting dodecahedron I threw together in Blender: an animation of the net being folded into a dodecahedron

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:
net of solutions that are mirrors of each other The same can be said of the second and fifth solutions: nets of solutions that are mirrors of each other 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: enter image description here 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.

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    $\begingroup$ Nice partial solution, you found it fast! And the animation is really good. It is the only solution with a mirror symmetry. Btw, in the net you made a little fault, you actually have to exchange the lettters j and d. I also should have put the i-tile in the first row and the c-tile in the second row, this would have been more logical... $\endgroup$ Sep 7, 2023 at 0:29
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    $\begingroup$ I want this dodecahedron ! $\endgroup$
    – Evargalo
    Sep 7, 2023 at 9:13
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    $\begingroup$ @Evargalo If DqwertyC prints their version on a template like the one from Wikipedia, they can make a nicer version of the cruddy one I threw together in MSPaint and Excel. $\endgroup$ Sep 7, 2023 at 12:15
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    $\begingroup$ @HerbertKociemba Thank you for pointing that out, I've fixed the labels on the net. This was found by trial and error assuming there would be mirror symmetry, so back to square one on the rest of the puzzle. $\endgroup$
    – DqwertyC
    Sep 7, 2023 at 16:17
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    $\begingroup$ I regard my puzzle as solved now. In some sense it is a cooperate answer together with Weather Vane who published the 5 nets in his answer which you finally sucessfully reduced to 3 nets. Sadly I can give the green checkmark to one answer only. $\endgroup$ Sep 8, 2023 at 18:05
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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

enter image description here

and the other half is just the same with the antisymmetry applied

enter image description here

The first half of both the two other asymmetrical solutions can be made identical and look like this

enter image description here

The second half of the first asymmetrical solution again is just the antisymmetry applied to the first half:

enter image description here

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.

enter image description here

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.

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    $\begingroup$ Could you please talk a bit about how you designed this puzzle. $\endgroup$ Sep 10, 2023 at 19:58
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    $\begingroup$ The process was not straightforward but more like poking in the fog. My goal was to find some tiling of the dodecahedron where the set of tiles are are defined as a complete set with a certain property and the tiles are split up in different colored parts such that the result looks appealing. I tried several possibilities and this here was the tiling I liked most because all tiles are different, the tiling is not trivial and the solutions look nice. I found an alternative tiling with 12 identical tiles which also is quite nice and I will make another puzzle using this tile. $\endgroup$ Sep 11, 2023 at 9:11
  • $\begingroup$ I look forward to the identical tiles puzzle. $\endgroup$ Sep 11, 2023 at 9:15
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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.

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

Found by exhaustive permutation.

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    $\begingroup$ Are all these 5 solutions essentially different or are there rotations/reflections that map one arrangement to another? $\endgroup$ Sep 8, 2023 at 16:01
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    $\begingroup$ According to @DqwertyC's updated answer, 2 pairs are symmetrical, giving 3 solutions. $\endgroup$ Sep 8, 2023 at 16:19
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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

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  • $\begingroup$ I am not completely satisfied with this answer. Hint: I also consider two tilings essentially the same when they are mirrored versions of each other. $\endgroup$ Sep 8, 2023 at 12:03
  • $\begingroup$ Then the answer must be one less than what I have above, since you mention elsewhere that there's only one solution with mirror symmetry :) $\endgroup$ Sep 8, 2023 at 12:21
  • $\begingroup$ @AaronWindsor There may also be solutions that are mirrors of each other, but don't have mirror symmetry within themselves $\endgroup$
    – DqwertyC
    Sep 8, 2023 at 14:49
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    $\begingroup$ Can you please share your program? $\endgroup$ Sep 8, 2023 at 17:53
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    $\begingroup$ @BenjaminWang the code is here. It uses a set cover solver in the same repo. Hopefully the instructions in the code are enough to get you going. $\endgroup$ Sep 9, 2023 at 12:04

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