“Magic” Dodecahedron

[Toast0042]

Are “magic” dodecahedrons possible? I.e. similar to the idea of a magic 3x3 square where each row/column adds up to 15. But in this case, if you numbered each of dodecahedron sides with 1-12, any 6 sides facing up would add up to 39.

 

[Dr.Peterson]

Interesting idea! We might call this a "face-magic dodecahedron", as oppposed to "vertex-magic", because you're putting numbers on the faces. You could do the same thing labeling the vertices of an icosahedron. And I found different kinds of "magic" polyhedra, with numbers on vertices and edges, here:

Magic 3-D Polygons & Graphs

Perimeter Magic Platonic Solids

Your idea is definitely worth thinking about. Rather than do all the thinking, I'll let you tell us what ideas you have about making one, or proving it's impossible. (I have no idea yet.)

You have already done some good thinking, in realizing what the sum would have to be. The next thought I have is to consider what happens if you roll it onto the next face. Since the sum of the faces now upward is equal to the original sum, I observe that the sum of opposite pairs of sides (the pair that move to the bottom, and the opposite pair that moves to the top) must be the same. I don't know where that will lead, but it's worth pondering.

Let us know what further thoughts you have!

 

[Toast0042]

I had two theories, but I haven’t been able to get either to work yet. My first was to have opposite faces add up to 13, similar to how regular 6 sided die add up to 7. But there are simply too many combinations to test efficiently.

My second idea, which worked better, was to place the 6 pairs of integers adding up to 13 next to one another. I was able to get 10 of the 12 faces to work using this method.

I’ve seen the references you’ve posted as well, and though interesting, I’m more interested in the faces themselves.