Cryptarithm - Addition problem

  MEMO
+ FROM
------
 HOMER



Number the columns as follows:
 12345
 -----
  memo
+ from
------
 homer


We note that O + M = R (column 5)
and also O + M = E, (column 4) so column e must generate a carry of 1,
and E = R + 1.
Then from column 3,
E + R + 1 = M (column 4 generates a carry also)
and M = R + 1 + R + 1 = 2 R + 2.

So we have candidates for R E M as follows
R E M
0 1 2
1 2 4
2 3 6
3 4 8
4 5 0
5 6 2
6 7 4
7 8 6
8 9 8 - eliminated by duplication
9 0 0 - eliminated by duplication

R - M = O, so we add a column for O

R E M O
0 1 2 8
1 2 4 7
2 3 6 6 - eliminated by duplication
3 4 8 5
4 5 0 4 - eliminated by duplication
5 6 2 3
6 7 4 2
7 8 6 1

We add (*) which would be the carry from E + R + 1 = M, either 1 or 0,
and F from M + F + carry ? = O
R E M O (*) F
0 1 2 8 (0) 6
1 2 4 7 (0) 3
3 4 8 5 (0) 7
5 6 2 3 (1) 0
6 7 4 2 (1) 7 - eliminated by duplication
7 8 6 1 (1) 4

In order to create the H in HOMER, M + F + carry ? must itself generate
a carry
R E M O (*) F
0 1 2 8 (0) 6 - elim, no carry out
1 2 4 7 (0) 3 - elim, no carry out
3 4 8 5 (0) 7
5 6 2 3 (1) 0 - elim, no carry out
7 8 6 1 (1) 4

That leaves
R E M O (*) F
3 4 8 5 (0) 7
7 8 6 1 (1) 4
Since H must then be 1,
the second row is eliminated and the solution is
R E M O F H
3 4 8 5 7 1

 MEMO   8485
+FROM   7358
__________
HOMER  15843