In the cryptarithmetic puzzle NO + NO + TOO = LATE, each letter represents a…

In a cryptarithmetic (verbal-arithmetic) puzzle, every distinct letter stands for exactly one digit from 0–9, no two letters share a digit, and the leading letter of any number can never be 0. The puzzle is solved by working through the addition column by column, exactly as in ordinary long addition, keeping track of the carry generated by each column.

Setting up the sum by place value:

N O

N O

T O O

———————

L A T E

1. Units column: O + O + O = E + 10 × carry1. With E = 2: carry1 = 0 gives 3 × O = 2 (not a whole digit); carry1 = 1 gives 3 × O = 12, so O = 4; carry1 = 2 gives 3 × O = 22 (not divisible by 3). So O = 4, with carry1 = 1 passed into the tens column.

2. Thousands column: LATE is a 4-digit number, so its leading digit L can never be 0. The only digit carried into the thousands place is the carry out of the hundreds column, carry3 — so carry3 = L, and since L ≥ 1, carry3 = 1.

3. Hundreds column: T + carry2 = A + 10 × carry3 = A + 10, so T = A + 10 − carry2.

4. Tens column: N + N + O + carry1 = T + 10 × carry2, i.e. 2N + 5 = T + 10 × carry2. Substituting T = A + 10 − carry2 from the hundreds column gives 2N = A + 5 + 9 × carry2. Testing carry2 = 0 needs T = A + 10 > 9, which is impossible; carry2 = 2 needs A ≤ 1, but A = 1 repeats L's digit and A = 0 makes 2N = 23 (not a whole number) — both fail. Only carry2 = 1 works: A = 0, T = 9, and 2N = 0 + 5 + 9 = 14, so N = 7.

5. Check: all six digits — O = 4, E = 2, L = 1, A = 0, N = 7, T = 9 — are distinct, and every leading letter (N, T, L) is non-zero, so this is a valid, unique solution.

Cross-check: NO = 74, NO = 74, TOO = 944, and 74 + 74 + 944 = 1092, which matches LATE with L = 1, A = 0, T = 9, E = 2 — the sum checks out exactly.

Therefore O + L + E = 4 + 1 + 2 = 7.