APPLE+GRAPE = CHERRY. Find the value of C+H+E+R+R+Y.

In a cryptarithmetic puzzle, each letter stands for one fixed digit from 0-9, no two letters share a digit, and a word's leading letter can never be 0. Solve it exactly like ordinary column addition: work from the units column leftward, and let the carry each column produces pin down the next letter, rather than guessing values ahead of time.

    A P P L E
  + G R A P E
  -----------
  C H E R R Y

Applying this column by column, using c1 to c5 for the carry out of each column:

1. Column 6 (a brand-new leading digit): the only thing that can land there is the carry out of column 5, and a carry out of adding two digits (plus at most one carry-in) is never more than 1, so C = 1.

2. Column 1 (units, E + E = Y with carry c1): the doubled letter E fixes a small set of possible (E, Y, c1) triples once checked against the other columns.

3. Column 5 (A + G + c4 = 10 + H, since the carry out of this column must be 1 to make C = 1): this pins down A, G and H together, since their sum has to land just past 10.

4. Columns 2 and 3 both produce the SAME letter R (L + P + c1 = R + 10c2, then P + A + c2 = R + 10c3). Matching these two equations against each other rules out all but one consistent choice for L, P and R.

5. Column 4 (P + R + c3 = E + 10c4) ties the column 2 and 3 choice of P and R back to the E already fixed in column 1, closing the system.

6. Only one assignment satisfies every column at once: A = 6, P = 3, L = 7, E = 4, G = 9, R = 0, C = 1, H = 5, Y = 8, nine different digits, exactly as the puzzle's rules require.

Substituting back, APPLE = 63374 and GRAPE = 90634, and C + H + E + R + R + Y = 1 + 5 + 4 + 0 + 0 + 8 = 18.

Independent check: 63374 + 90634 = 154008, which reads exactly as CHERRY (1-5-4-0-0-8) with every column carry matching what was used above, confirming the digit assignment and, with it, the total of 18.