This is the net map of the outer surfaces of the 20 edge cubes that represents
a 3 x 3 x 3 'parent' cube (like in a Rubik cube).

A B C D E F G H A D F I J M N O F G H J K O P Q H E C K L Q R S C B A L I S T M O P Q N R M T S

Assign unique values from 1 to 20 to the letters A-T such that the sum of each pair of diametrically opposite cubes is to be the same as all others while the sum of each set of edge cubes may not differ from that of any other set by more than one.

Eg, diagonals:
                    A + Q = N + E = C + O ...(etc)
and edge cubes:     A + B + C = C + E + H = F + G + H ...etc,
                or (A + B + C) ±1 = C + E + H
                                  = F + G + H ....(etc) 

Note: The problem's development and my solution used a spreadsheet; as such a well-constructed sheet could enable a solution. Although this problem may lend itself to a programmed solution I would appreciate seeing attempts of a more manual basis within the first 24-48 hrs.