Sandy Thompson sent the following solution for the tetrahedron:
My answers for this question use the following flattened version of a
tetrahedron. I know it might look a little strange, but if you fold that
thing up, you'll find that it does work out.
a
/ \
h i
/ \
b - e - c
/ \ / \
/ j f \
/ \ / \
/-- g - d ------\
Sides: ahb, aic, agd, bec, bjd, cfd
Faces: abcehi, bcdefj, abdghj, acdfgi
Phew!
First, faces with the same sum. All 14 possibilities are listed here, from
the smallest to largest sum. Only one of the 24 possible transformations
for each is listed. All lists of integers are in order from a to j.
Face sums of 31:
[1, 2, 3, 8, 6, 5, 4, 9, 10, 7]
[1, 2, 4, 7, 9, 6, 8, 10, 5, 3]
Face sums of 32:
[1, 2, 5, 10, 8, 3, 6, 9, 7, 4]
[1, 2, 6, 9, 8, 4, 7, 10, 5, 3]
[1, 3, 4, 10, 8, 2, 6, 7, 9, 5]
[1, 4, 6, 7, 2, 5, 3, 9, 10, 8]
[2, 3, 4, 9, 8, 1, 6, 5, 10, 7]
Face sums of 34:
[1, 6, 9, 10, 3, 2, 5, 8, 7, 4]
[1, 7, 8, 10, 3, 4, 5, 9, 6, 2]
[2, 5, 9, 10, 3, 1, 4, 7, 8, 6]
[2, 7, 8, 9, 3, 6, 5, 10, 4, 1]
[4, 5, 7, 10, 9, 2, 8, 6, 3, 1]
Face sums of 35:
[3, 8, 9, 10, 5, 2, 7, 6, 4, 1]
[4, 7, 9, 10, 2, 1, 3, 5, 8, 6]
Edges with the same sum. I didn't find any combination of the numbers that
resulted in all 6 edges having the same sum... and my program tried all the
possible combinations. There aren't any!