COFFEE × 3 = THEOREM
Step-by-step solution
Three identical six-digit numbers add up to a seven-digit number. The repeated E and F digits give us a place to start; we can solve this as 3 × COFFEE = THEOREM.
Each letter represents a digit from 0 to 9. Matching letters keep the same value; different letters cannot share a digit. The leading letters C and T cannot be zero.
Jump to the answer1. Start with the two E digits
In the units column, E + E + E ends in M and sends a carry to the tens column. A carry in this three-number addition can be 0, 1, or 2.
In the tens column, E + E + E plus that carry must end in E again. Checking the ten possible digits against both columns leaves E = 0 or E = 9. Zero would also make M = 0, which is forbidden because E and M are different letters.
So E = 9. The units total is 27, giving M = 7 and a carry of 2. The tens total is 29, writing E = 9 and carrying 2 again.
3 × 9 = 27; 3 × 9 + 2 = 29
2. Narrow down the repeated F digits
The hundreds column is 3 × F + 2. Its last digit is R, and its carry enters the thousands column. The thousands column is then 3 × F plus that new carry, ending in O.
F cannot be 7 or 9, which already belong to M and E. F = 0 makes O = F; F = 4 makes R = F; F = 5 makes R = M. F = 6 or 8 makes R and O equal. Each of these breaks the different-letter rule.
That leaves three candidates. Keep their carries: they matter in the next column.
| F | R | O | Carry out |
|---|---|---|---|
| 1 | 5 | 3 | 0 |
| 2 | 8 | 6 | 0 |
| 3 | 1 | 0 | 1 |
3. Use the next E to choose F = 1
The ten-thousands column adds three O digits and the carry from the table. Its result digit must be E = 9.
For F = 1, the column gives 3 × 3 + 0 = 9. For F = 2, it gives 18, ending in 8. For F = 3, it gives 1. Only the first candidate fits.
We now have F = 1, R = 5, and O = 3. There is no carry into the C column.
3 × O + carry = 3 × 3 + 0 = 9
4. Finish with C, H, and T
The remaining left-hand calculation is 3 × C = 10 × T + H. C cannot be zero, and the unused nonzero digits for C are 2, 4, 6, and 8.
C = 2 gives 6, so there would be no seventh digit. C = 4 gives 12 and C = 6 gives 18; both would make T = 1, already assigned to F.
C = 8 gives 24. Thus T = 2 and H = 4, both unused. Every letter is now determined.
3 × 8 = 24 → C = 8, T = 2, H = 4
Answer and verification
- C
- = 8
- O
- = 3
- F
- = 1
- E
- = 9
- T
- = 2
- H
- = 4
- R
- = 5
- M
- = 7
831199 + 831199 + 831199 = 2493597
All letters have distinct digit values, no leading digit is zero, and the equation checks out.
Takeaway
Repeated letters link the columns. Keep track of each carry, then use a later column to rule out candidates from an earlier one. The cases above exhaust the possibilities, leaving one solution under the stated rules.