Is there is a way of putting the numbers 1 to 20 on 20-sided die such that there exists a constant K such that for every face f, the value on face f and the values of all three faces adjacent to f add up to K?
Here's (a perl script that outputs some octave code which computes) the answer
The thing that surprised me is that the face adjacency matrix of the icosahedron is actually nonsingular. I guess I expected it to be singular because of... symmetry or something? Dunno.