In the cryptarithmetic puzzle EAT + THAT = APPLE, where each letter stands for…

Concept: This is a cryptarithmetic (alphametic) puzzle -- every distinct letter stands for one fixed digit from 0 to 9, the same letter always represents the same digit everywhere it appears, different letters must represent different digits, and the first letter of a multi-digit number can never stand for 0. Such puzzles are solved by adding column by column starting from the units place, tracking the carry into the next column (which, when adding a small number of single digits, can only be 0 or 1).

1. Ten-thousands column (leading digit): EAT is a 3-digit number and THAT is a 4-digit number, yet their sum APPLE has 5 digits. Adding a 3-digit and a 4-digit number can only produce a 5-digit result via a carry out of the leading column, and that carry can only be 1. So A = 1.

2. Thousands column: with A = 1, the thousands-place letter T, plus whatever carries in from the hundreds column, must be large enough to send a carry of 1 out to the ten-thousands column while still leaving a single digit, P (the thousands letter of APPLE), behind. Since T is at most 9, this only works if T = 9 and the carry coming in from the hundreds column is 1, which forces P = 0.

3. Units column: the units digit of THAT is T and the units digit of EAT is also T (both words end in T), so this column computes T + T. Since T = 9, T + T = 18, so E (the units digit of APPLE) equals 8, and this column carries 1 into the tens column.

4. Tens column: the tens digit of THAT is A and the tens digit of EAT is also A, so this column computes A + A plus the carry of 1 coming in from the units column. Since A = 1, that is 1 + 1 + 1 = 3, so L (the tens digit of APPLE) equals 3, and this column carries 0 into the hundreds column.

5. Hundreds column: the hundreds digit of THAT is H and the hundreds digit of EAT is E, which is already fixed at 8 (it is the same letter as the units digit of APPLE, from the units-column step). Adding the carry of 0 just found, this column's sum must equal P (0, from the thousands-column step) plus a carry of 1 out to the thousands column -- i.e. it must total 10. Since E = 8, H must be 2 to make the column total 10.

6. Collecting every letter now: A = 1, T = 9, E = 8, P = 0, L = 3, and H = 2 -- six different letters, six different digits, no repeats, and no leading letter equals 0.

Cross-check: substituting every digit back in, EAT = 819 and THAT = 9219, and 819 + 9219 = 10038, which matches APPLE = 10038 exactly -- confirming every digit assignment is consistent.

Therefore A + T + L = 1 + 9 + 3 = 13.