I would say that you need a simple data stucture to represent the state of the solution so far, say a hash in which $digit{B} = 1 if you chose B to represent 1, and unchosen assignments are undefs.

Suppose you worked from right to left across the addition problem, choosing all of 0 to 9 as the 1's digit of the first number, then each possible unassigned number as the 1's digit of the second number. Then a third digit choice will be automatic (the 1's digit of the sum of the first two choices). Some of these choices will be eliminated by testing that you haven't assigned the same digit to two letters. Keep working right to left, and it should work out.