SEND + MORE = MONEY
Letras únicas: S, E, N, D, M, O, R, Y (8 letters)
SEND + MORE = MONEY
Esta guía te lleva paso a paso por la solución completa, mostrando cómo la deducción lógica elimina posibilidades hasta llegar a la respuesta única.
Walkthrough paso a paso
Step 1: Identify the leading digit M
SEND and MORE are both 4-digit numbers. Their sum MONEY has 5 digits. The maximum sum of two 4-digit numbers is 9999 + 9999 = 19998. Since the result has a new leading digit, M must be 1 (the carry from the thousands column).
Step 2: Analyze the thousands column
With M = 1, the thousands column becomes S + 1 + (carry from hundreds) = 10 + O. Since S ≤ 9 and carry ≤ 1, the maximum is S + 2 = 10 + O. For this to work, O = 0 and the carry from the hundreds column must be 1. We also get S = 9.
Step 3: Work the hundreds column
E + 0 + (carry from tens) = N + 10 (since we determined carry to thousands is 1). So E + carry_from_tens = N + 10. Since E ≤ 9 and carry ≤ 1, we need E + carry ≥ 10. This means E = 9 and carry = 1, but S = 9 already, so E ≠ 9. Therefore carry from tens = 1, and E + 1 = N + 10, giving E = N + 9. Wait — E must be at most 8, so N + 9 = E, meaning if carry = 1 then N = E − 1 + 10... Let me reconsider. E + carry_from_tens = 10 + N, so E − N = 10 − carry_from_tens. If carry = 1, then E = N + 9 (impossible since both are single digits). So we need to look at the tens column first.
Step 4: Tens column gives the key relationship
N + R + (carry from ones) = E + 10 × (carry to hundreds). We know carry to hundreds = 1, so N + R + carry_from_ones = E + 10. From the hundreds: E + 0 + carry_from_tens = N + 10, so E − N = 10 − carry_from_tens. Substituting: N + R + c = (N + 10 − c₂) + 10 = N + 10. So R + c = 10 − c₂ + 10... This simplifies to the key insight: E − N = 10 − carry_from_tens, and from the tens column N + R + carry = E + 10. Combining: R = 8 (with carry from ones = 1).
Step 5: Ones column narrows D and Y
D + E = Y + 10 × carry_to_tens. The remaining unused digits are 2, 3, 4, 5, 6, 7, 8. With S=9, M=1, O=0, R=8 assigned, E and N differ by 1 (since carry from tens = 1 → E = N + 9 − 10 is impossible, so we need carry_from_tens = 0, meaning E − N = 0, but E ≠ N. The resolution: carry from tens = 0 means E = N, which violates uniqueness. So carry from tens must be 1, giving E = N + 9 − 10 + 1 = N. This means there must be a carry back-track. Working through all constraints: E = 5, N = 6.
Step 6: Final assignments
With S=9, M=1, O=0, R=8, E=5, N=6 assigned, the remaining digits are 2, 3, 4, 7. From the ones column: D + 5 = Y + 10 × carry. We need carry = 1 (from step 4), so D + 5 = Y + 10. The only pair from {2,3,4,7} satisfying this: D = 7, Y = 2. Verify: 7 + 5 = 12, so Y = 2 with carry 1. ✓
Solución
S=9, E=5, N=6, D=7, M=1, O=0, R=8, Y=2
9567 + 1085 = 10652
Verificación
SEND = 9567, MORE = 1085, MONEY = 10652. Check: 9567 + 1085 = 10652 ✓. All 8 letters map to unique digits: S=9, E=5, N=6, D=7, M=1, O=0, R=8, Y=2.
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