In the addition LETS + WAVE = LATER, each letter stands for a unique digit…

A cryptarithm (or alphametic) turns a valid arithmetic sum into a puzzle by replacing every digit with a letter: the same letter always stands for the same digit, different letters stand for different digits, and the leading letter of any number cannot be 0. Solving one means applying ordinary column-by-column addition — units, tens, hundreds, and so on, together with their carries — alongside these uniqueness and non-zero rules, usually starting from the column that is most constrained.

  L E T S
+ W A V E
---------
 L A T E R

1. LETS and WAVE are both 4-digit numbers, so their sum is at most 9999 + 9999 = 19998 — a 5-digit number whose leading digit can only be 1. Since LATER has 5 digits, this forces L = 1.

2. Working column by column from the right, call the carries into the tens, hundreds, and thousands columns k1, k2, and k3, and the carry into the brand-new leading digit k4. Since that leading digit is L = 1, k4 = 1.

3. In the thousands column, L + W + k3 = A + 10·k4, i.e. 1 + W + k3 = A + 10. A carry of k3 = 1 into this column would force the hundreds-column digit T down to 0, clashing with A; so k3 = 0, which fixes A = 0 and W = 9.

4. In the hundreds column, E + A + k2 = T + 10·k3. With A = 0 and k3 = 0, this simplifies to T = E + k2.

5. In the tens column, T + V + k1 = E + 10·k2. Substituting T = E + k2 shows k2 must be 1 (k2 = 0 would force V = 0, already used by A), giving V + k1 = 9. Since W = 9 is taken, V cannot be 9, so k1 = 1 and V = 8.

6. In the units column, S + E = R + 10·k1 = R + 10. With the given S = 7, this gives R = E − 3.

7. The digits L=1, A=0, W=9, V=8, and S=7 are already used, leaving {2, 3, 4, 5, 6} for T, E, and R, with T = E + 1 and R = E − 3. Checking each remaining value of E shows only E = 5 keeps both T = 6 and R = 2 inside that set without repeating a digit already used.

8. This gives the full assignment L=1, A=0, T=6, E=5, R=2, S=7, V=8, W=9.

Substituting these digits back into the original addition confirms the assignment:

  1 5 6 7
+ 9 0 8 5
---------
 1 0 6 5 2

which reads L A T E R = 1 0 6 5 2, matching exactly.

So V × S × E = 8 × 7 × 5 = 280.