In the cryptarithmetic addition DOW + RED = ERRE, where each letter stands for…

In a cryptarithmetic (alphametic) addition, every distinct letter represents one fixed digit from 0-9, and the column-wise addition must hold exactly in every place, including any carry passed to the next column. When adding two 3-digit numbers produces a 4-digit result, the new leading digit of the result can only be 1, because a single column (two digits plus an incoming carry of at most 1) can never generate a carry greater than 1 -- and the column equations, not any assumption about which letters are allowed to be zero, are what pin down every remaining letter.

1. The sum DOW + RED = ERRE has four digits while both addends are written with three letters each, so the thousands-place digit of the result is exactly the carry out of the hundreds column. A column carry cannot exceed 1, so E = 1.

2. In the hundreds column: D + R + (carry from tens) = R + 10*E = R + 10. Cancelling R from both sides gives D + (carry from tens) = 10. Since D is a single digit (at most 9), the carry from the tens column must be 1, which forces D = 9.

3. In the tens column: O + E + (carry from units) = R + 10*(carry from tens) = R + 10. Substituting E = 1 gives O + 1 + (carry from units) = R + 10, i.e. O + (carry from units) = R + 9. Since O is at most 9, R + 9 cannot exceed 10, so R can only be 0 or 1; R = 1 is already taken by E, so R = 0, giving O + (carry from units) = 9. (R coming out to 0 is the only value consistent with every column equation, even though it leaves the leading letter of RED equal to zero.)

4. In the units column: W + D = E + 10*(carry from units) = 1 + 10*(carry from units). With D = 9, trying carry from units = 0 gives W = -8, which is impossible, so the carry from units must be 1, giving W = 2. Feeding this back into the tens-column result, O + 1 = 9, so O = 8.

Substituting every digit back into the original letter equation confirms this is the only consistent solution: DOW reads as 982, RED reads as 019 (its two significant digits being 1 and 9, since its forced leading digit is 0), and 982 + 19 = 1001, which reads exactly as ERRE with E = 1, R = 0, R = 0, E = 1.

The unique digit assignment forced by the column equations is D = 9, O = 8, W = 2, R = 0, E = 1. The word error is spelled with the letters e, r, r, o, r, so e + r + r + o + r = 1 + 0 + 0 + 8 + 0 = 9.