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The key to this sort of problem is to be systematic, and Andrei explained very clearly how he worked through every possibility. Here is his solution:
I started from the observation that I must add two 3-digit numbers, and the result is 4-digit one, so that I must assign the value 1 to the letter F.
My method is to give values to the letters from right to left, the way additions are performed. [O is also a good place to start as it occurs three times. - Ed.]
T | W | 2 | ||
+ | T | W | 2 | |
- | - | - | - | - |
1 | 2 | U | R |
T | W | 3 | ||
+ | T | W | 3 | |
- | - | - | - | - |
1 | 3 | U | R |
T | W | 4 | ||
+ | T | W | 4 | |
- | - | - | - | - |
1 | 4 | U | R |
T | W | 5 | ||
+ | T | W | 5 | |
- | - | - | - | - |
1 | 5 | U | R |
T | W | 6 | ||
+ | T | W | 6 | |
- | - | - | - | - |
1 | 6 | U | R |
T | W | 7 | ||
+ | T | W | 7 | |
- | - | - | - | - |
1 | 7 | U | R |
T | W | 8 | ||
+ | T | W | 8 | |
- | - | - | - | - |
1 | 8 | U | R |
T | W | 9 | ||
+ | T | W | 9 | |
- | - | - | - | - |
1 | 9 | U | R |
So, the only solutions are:
7 | 3 | 4 | 7 | 6 | 5 | 8 | 3 | 6 | 8 | 4 | 6 | |||||||||||
+ | 7 | 3 | 4 | + | 7 | 6 | 5 | + | 8 | 3 | 6 | + | 8 | 4 | 6 | |||||||
- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | |||
1 | 4 | 6 | 8 | 1 | 5 | 3 | 0 | 1 | 6 | 7 | 2 | 1 | 6 | 9 | 2 | |||||||
8 | 6 | 7 | 9 | 2 | 8 | 9 | 3 | 8 | ||||||||||||||
+ | 8 | 6 | 7 | + | 9 | 2 | 8 | + | 9 | 3 | 8 | |||||||||||
- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | ||||||||
1 | 7 | 3 | 4 | 1 | 8 | 5 | 6 | 1 | 8 | 7 | 6 |