FORTY + TEN + TEN = SIXTY, find the value of T+E+N. F O R T Y T E N + T E N…

In a cryptarithmetic (alphametic) puzzle, every distinct letter stands for a unique digit from 0-9, and the leading letter of any number cannot be 0. Such puzzles are solved by analysing the addition column by column, starting from the units place and carrying forward into the next column, until every letter is pinned to exactly one digit.

Applying this to FORTY + TEN + TEN = SIXTY:

1. Units column: Y + N + N leaves the units digit unchanged as Y on both sides of the sum, so 2N must itself be a multiple of 10 -- meaning N is 0 or 5.

2. Tens column: in the same way, T + E + E (plus any carry from the units column) leaves the tens digit unchanged as T, so 2E plus that carry must also be a multiple of 10.

3. Testing N = 5 first: it sends a carry of 1 into the tens column, forcing 2E + 1 to be a multiple of 10 -- impossible, since 2E + 1 is always odd. So N = 0, and the units column carries nothing forward.

4. With no carry coming in, 2E itself must be a multiple of 10. Since the digit 0 is already taken by N, the only possibility is E = 5, which carries 1 into the hundreds column.

5. Ten-thousands column: F plus any carry from the thousands column equals S. If that carry were 0, F would equal S, which is not allowed since every letter is a different digit -- so the carry must be 1, giving S = F + 1.

6. Thousands column: O plus the carry from the hundreds column equals I plus 10 (since the carry into the ten-thousands column is 1, from the previous step). Since O is at most 9 and the hundreds-column carry is at most 2, their sum can reach 11 at most -- and it must reach at least 11 for I to be a positive digit. So the sum equals exactly 11, forcing O = 9, the carry out of the hundreds column to be 2, and I = 1.

7. Hundreds column: R + 2T + 1 (the carry into this column) equals X + 20 (since the carry out is 2), so X = R + 2T - 19. Checking the six digits left over, {2, 3, 4, 6, 7, 8}, for F, R, T, Y, S, X -- with S = F + 1 and all six distinct -- only one combination keeps X in range and every letter unique: F = 2, S = 3, R = 7, T = 8, X = 4, leaving Y = 6 for the one digit not yet used.

The full digit assignment that satisfies every column is:

Cross-check: substituting back, FORTY = 29786, TEN = 850, and 29786 + 850 + 850 = 31486, which is exactly SIXTY = 31486 -- every column checks out.

Therefore, T + E + N = 8 + 5 + 0 = 13.