HERE = COMES – SHE, (Assume S = 8). Find the value of R + H + O.

Concept:

In a cryptarithmetic (alphametic) puzzle, every distinct letter stands for a unique digit from 0-9, and the arithmetic must hold true digit by digit, including every carry between columns, exactly as in ordinary column addition. The leading digit of any multi-digit number in the puzzle can never be 0. Solving means working column by column, usually starting from the units place, and using any given clue to pin down each letter.

Application:

HERE = COMES – SHE can be rewritten as an addition: HERE + SHE = COMES. Lining the letters up by place value:

H E R E

+ S H E

------------

C O M E S

Solving column by column from the units place, with S = 8:

1. Units column: E + E ≡ S (mod 10). With S = 8, either 2E = 8 (E = 4, no carry) or 2E = 18 (E = 9, carry 1). Taking E = 9 forces a contradiction later in the thousands column (it would require H to equal E), so E = 4 with no carry out of this column.

2. Tens column: R + H + 0 ≡ E = 4 (mod 10), so R + H is either 4 or 14.

3. Hundreds column: E + S + carry = 4 + 8 + carry = 12 + carry, so this column always carries 1 into the thousands column, whichever case from the tens column applies.

4. Thousands column: H + 1 (the carry from the hundreds column) must give O in this column and carry a digit into the ten-thousands column. Since COMES has five digits, that ten-thousands carry must be exactly 1 (giving C = 1), which is only possible if H + 1 is at least 10 — so H = 9 and O = 0.

5. Back in the tens column, H = 9 rules out R + H = 4 (R would be negative), so R + H = 14 must hold, giving R = 5 and confirming the carry of 1 into the hundreds column used above.

6. With that carry, the hundreds column gives M = 12 + 1 = 13, so M = 3.

Cross-check:

The digits are C = 1, O = 0, M = 3, E = 4, S = 8, H = 9, R = 5, all distinct as required. HERE = 9454, SHE = 894, and 9454 + 894 = 10348 = COMES, so COMES – SHE = 10348 – 894 = 9454 = HERE, confirming every digit.

R + H + O = 5 + 9 + 0 = 14.