/ \   / \   / \
    C   D E   F G   H
  / \   / \   / \   / \
 L   M N   O P   Q R   S
 W   X Y   Z a   b c   d
  \ /   \ /   \ /   \ /
    h   i j   k l   m
     \ /   \ /   \ /
Trigons are similar to Triominos, except instead of having numbers at each corner, the numbers are on the sides of each triangular piece. Can you take a set of 24 trigons, consisting of all possible configurations of the values 0, 1, 2, and 3, and place them into a hexagon, two units on a side, such that each adjacent side matches correctly? Or, show why it can't be done.

(Note that 1-2-3 is a different trigon than 1-3-2, since neither can be rotated to create the other; while 1-1-2 would be the same as 1-2-1, since the latter can be rotated to obtain the former.)
Since I think the above problem may be easily solved, consider the following extension puzzles:

Source: Original, based on a puzzle in Dell's Math Puzzles and Logic Problems magazine.

Mail to Ken