SEND + MORE = MONEY
Step-by-step solution
A fifth digit appears when two four-digit numbers are added. Start with that carry, then connect the columns to solve SEND + MORE = MONEY without guessing all eight letters.
Each letter represents a digit from 0 to 9. Matching letters keep the same value; different letters cannot share a digit. The leading letters S and M cannot be zero.
Jump to the answer1. The extra digit makes M = 1
Adding two four-digit numbers can create only a carry of 1 in the ten-thousands column. MONEY has five digits and cannot start with zero, so M = 1.
Let the carry entering the thousands column be either 0 or 1. The thousands calculation is S + 1 + carry = 10 + O. With S at most 9, O can only be 0 or 1. It cannot be 1 because M already uses that digit.
Therefore O = 0. S is 9 if the incoming carry is 0, or 8 if it is 1.
S + 1 + carry = 10
2. The hundreds column makes S = 9
The hundreds column adds E + O plus a carry from the tens. Since O = 0, this is just E plus 0 or 1.
Could it carry 1 into the thousands? That would require E = 9 and an incoming carry of 1, making N = 0. But O already uses zero, so that branch is impossible.
There is no carry into the thousands, giving S = 9. In the hundreds, E and N must differ, so there must be a carry of 1 from the tens: N = E + 1.
E + 1 = N
3. The tens column makes R = 8
The tens column adds N + R plus the units carry, writes E, and carries 1. Substitute N = E + 1 and cancel E from both sides.
This leaves R + units carry = 9. A zero carry would make R = 9, already assigned to S. So the units carry is 1 and R = 8.
N + R + carry = E + 10 → R + carry = 9
4. Find E, N, D, and Y
The unused digits are 2, 3, 4, 5, 6, and 7. Because N = E + 1, E can only be 2, 3, 4, 5, or 6.
The units column must satisfy D + E = 10 + Y. The table checks every possible E, keeping all letters different and remembering that zero and one are already used.
| E | N | Units-column check |
|---|---|---|
| 2 | 3 | Even D = 7 cannot produce a carry. |
| 3 | 4 | Only D = 7 produces a carry, but it makes Y = 0. |
| 4 | 5 | D = 6 or 7 makes Y = 0 or 1; both are used. |
| 5 | 6 | D = 7 gives 12, so Y = 2. All letters differ. |
| 6 | 7 | D = 4 or 5 makes Y = 0 or 1; both are used. |
D + E = 7 + 5 = 12 → Y = 2
Answer and verification
- S
- = 9
- E
- = 5
- N
- = 6
- D
- = 7
- M
- = 1
- O
- = 0
- R
- = 8
- Y
- = 2
9567 + 1085 = 10652
All letters have distinct digit values, no leading digit is zero, and the equation checks out.
Takeaway
Do not assume you must start on the right. The extra leftmost digit fixes M immediately; combining that fact with the carries forces the rest. The final table eliminates every alternative, so the solution is unique under these rules.