If “EAT + THAT = APPLE”, what is the sum of A+P+P+L+E?

This is a cryptarithmetic (alphametic) puzzle: each distinct letter stands for a unique digit (0-9), and the letter-addition must hold true digit-by-digit, exactly like ordinary column addition with carries.

Solve it by working one column at a time from the units place leftward, using the fact that a carry out of any single column can only be 0 or 1, since at most two digits plus an incoming carry are being added in each column.

Applying this column by column, aligning EAT (3 digits) under the right-hand end of THAT (4 digits), with the sum APPLE (5 digits):

1. The ten-thousands place of APPLE can only come from a carry out of the thousands column, and a carry out of a single column is at most 1, so A = 1.

2. In the thousands column, T (the leading digit of THAT) plus any carry-in must produce P plus 10 times the carry-out (which is A = 1). Since T is a single digit, this forces P = 0 and T = 9, with a carry-in of 1 from the hundreds column.

3. In the units column, T + T = 9 + 9 = 18, so the units digit E = 8, with a carry of 1 into the tens column.

4. In the tens column, A + A plus the incoming carry of 1 gives 1 + 1 + 1 = 3, so the tens digit L = 3, with no further carry into the hundreds column.

5. In the hundreds column, E + H plus the incoming carry (0) must equal P plus 10 times the carry-out into the thousands column (which is 1, from step 2), i.e. 8 + H = 0 + 10, which pins down H = 2.

6. Checking that every letter (E=8, A=1, T=9, H=2, P=0, L=3) has been assigned a distinct digit confirms the assignment is consistent with the puzzle's constraints.

Substituting the digits back: EAT = 819 and THAT = 9219, and 819 + 9219 = 10038, which matches APPLE = 10038 exactly, confirming the assignment independently of the column-by-column derivation.

With the digits fixed this way, A + P + P + L + E works out to exactly 12.