The Pentagonal Icositetrahedron as a 24-sided Die

David and I were talking about polyhedra, as one does, and the dual of the snub cube, the Pentagonal Icositetrahedron came up somehow. I noticed it had 24 faces and was isohedral so I was like... oh, is this the standard shape for a 24-sided die? Turns out not. Henry Segerman here shows off what is, the deltoidal icositetrahedron.

I wondered "why not also the Pentagonal Icositetrahedron?" and it became quite clear after I wrote some quick and dirty typescript to generate an .stl/.obj and 3d-printed it:

Just like the tetrahedral d4, when a face lies flat on the table, you don't get a canonical face pointing up, but rather a vertex! So it's annoying to write numbers around the vertices.

But then I wondered whether you could assign numbers to faces such that the sum of the three upmost faces would give you a unique number 1-24. This was easy thing to ask z3. Assuming this diagram is correct:

and assuming I correctly transcribed it to this smt2 file, z3 finds the solution

sat
((min 1)
 (max 24))
((a 5)
 (b 0)
 (c 0)
 (d 8)
 (e 4)
 (f 2)
 (g 2)
 (h (- 1))
 (i 8)
 (j 2)
 (k (- 3))
 (l 1)
 (m (- 1))
 (n (- 4))
 (o 14)
 (p 1)
 (q 2)
 (r 0)
 (s 17)
 (t 3)
 (u 18)
 (v 0)
 (w 19)
 (x 3))
((sum1 17)
 (sum2 11)
 (sum3 7)
 (sum4 1)
 (sum5 8)
 (sum6 10)
 (sum7 5)
 (sum8 6)
 (sum9 12)
 (sum10 18)
 (sum11 2)
 (sum12 9)
 (sum13 19)
 (sum14 16)
 (sum15 3)
 (sum16 4)
 (sum17 13)
 (sum18 14)
 (sum19 15)
 (sum20 20)
 (sum21 21)
 (sum22 22)
 (sum23 23)
 (sum24 24))

Update

It works!

Sharpie unfortunately leaks into the layer lines.