SEND + MORE = MONEY. Then what is the value of M + O + N + E + Y ?

Concept

In a cryptarithmetic (alphametic) puzzle, every distinct letter stands for a unique digit from 0 to 9, the same letter always represents the same digit throughout, and no word's leading letter can be 0. Such puzzles are solved by adding column by column (units, tens, hundreds, ...), tracking the carry generated into the next column, and using these constraints (together with trial and error within the narrowed possibilities) to pin down each letter's digit uniquely.

Application

1. SEND and MORE are both 4-digit numbers, but their sum MONEY has 5 digits. Adding two 4-digit numbers can only produce a 5th leading digit if the leftmost column carries into a brand-new place value, and that new leading digit can only ever be 1. So M = 1.

2. Look at the leftmost column: S + M (plus any carry from the column to its right) must produce that carry into M's new place, i.e. S + 1 (+carry) is at least 10. For this to happen without reusing a digit already fixed as 1, the column must settle with O = 0 and a carry of 1 passed into the M column.

3. Working the remaining columns (E + O, N + R, D + E) under the rule that all eight letters (S, E, N, D, M, O, R, Y) must be distinct digits 0-9, and tracking which columns must carry into the next, narrows the possibilities enough that trial and error yields exactly one consistent assignment: S = 9, E = 5, N = 6, D = 7, M = 1, O = 0, R = 8, Y = 2.

4. Substituting these digits: SEND = 9567 and MORE = 1085.

Cross-check

9567 + 1085 = 10652, and reading M = 1, O = 0, N = 6, E = 5, Y = 2 from this sum gives exactly MONEY = 10652 - confirming the column-wise assignment is consistent.

Result

M + O + N + E + Y = 1 + 0 + 6 + 5 + 2 = 14.