Re: can you figure this out? tried+drive=rivet

  t r i e d
+ d r i v e
  r i v e t

I rewrote the problem in this way to make it easier to understand the relationships between the digits. So, what do we know? Well, looking at the columns, we know that either the sum of the first digits in each column or the sum of the first digits plus one (for all but the rightmost column) equals either the bottom digit or the bottom digit plus ten (for all but the leftmost column), for example if d is 5 and e is 1, then t must be 6 (5 + 1 = 6), but if d is 5 and e is 7 then t must be 2 (5 + 7 = 12 = 2 + 10). So here are the columnar equations separately:

d + e = t OR t + 10
e + v OR e + v + 1 = e (+10)
i + i (+1) = v (+10)
r + r (+1) = i (+10)
t + d (+1) = r

There are three very interesting features of the problem that immediately stand out:

If e + v (+1) = e (+10), then v must be 0 or 9: e + 0 = e; e + 9 + 1 = e + 10.

If i + i (+1) = v (+10) and v must be 0 or 9, then for v = 0, i = 5 (5 + 5 = 0 + 10) and for v = 9, i = 4 OR 9 (4 + 4 + 1 = 9 OR 9 + 9 + 1 = 9 + 10): assuming v and i cannot both be nine, this implies that if v = 9 then i = 4.

If r + r (+1) = i (+10), then for i = 4, r + r = 4 (+10), so r = 2 OR 7; and for i = 5, r + r + 1 = 5 (+10), so r = 2 OR 7.

If t + d (+1) = r, then r cannot be two, since t and d cannot both be 1 and neither can be 0 (assuming your brother is not writing the numbers with extraneous zeros as their leftmost digits).

For r = 7, i = 4 OR 5
For i = 4, v = 9; and for i = 5, v = 0
For t + d + 1 = 7, t + d = 6, so:

t and d must be 4 and 2 OR 2 and 4, if i = 5
t and d must be 5 and 1 OR 1 and 5, if i = 4

For v = 0, d + e = t
For v = 9, d + e = t + 10.

Taken together, we see that for d + e = t: v = 0; i = 5; and t and d are 2 and 4 OR 4 and 2. If d + e = t, then d < t, so d = 2 and t = 4. HOWEVER, for 2 + e = 4, e = 2, and 2 is already assigned to d, so this scenario cannot work. Therefore:

v = 9
i = 4

Thus, d and t are 1 and 5 OR 5 and 1. Now, d + e = t + 10, so for e < 10 (e is a single digit), d > t, so:

d = 5
t = 1
And, 5 + e = 1 + 10, so e = 6

If we do the substitutions, we can rewrite the answer like this:

  1 7 4 6 5
+ 5 7 4 9 6
  7 4 9 6 1

Looks right to me. This same procedure would work for any similar substitution problem - just work out the simple equations for each column until one or more variables can be assigned, and then work back.