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.]*

- If O=0, then R must be 0 too. As each letter stands for a different number, it isn't possible.
- O can't be 1, as F=1.
- If O=2, then
T W 2 + T W 2 - - - - - 1 2 U R - W=0 NO, because 0+0=0, so U must be 0 too.
- W=1 NO, because F=1.
- W=2 NO, because O=2.
- W=3, so U=6. NO, because T=6.
- W=4 NO, because R=4.
- W=5 NO, because this would make U greater than 9.

- If O=3, then
T W 3 + T W 3 - - - - - 1 3 U R - If O=4, then
T W 4 + T W 4 - - - - - 1 4 U R - If O=5, then
T W 5 + T W 5 - - - - - 1 5 U R - W=5 NO, because O=5.
- W=6, U=3 - SOLUTION
- W=7 NO, because T=7.
- W=8, U=7 NO, because T=7.

- If O=6, then
T W 6 + T W 6 - - - - - 1 6 U R - W=0, so U=1. NO, because F=1.
- W=1 NO, because F=1.
- W=2 NO, because R=2.
- W=3, U=7 - SOLUTION
- W=4, U=9 - SOLUTION

- If O=7, then
T W 7 + T W 7 - - - - - 1 7 U R - W=5, so U=1. NO, because F=1.
- W=6, so U=3 - SOLUTION
- W=7 NO, because O=7.
- W=8 NO, because T=8.
- W=9, so U=9 - NO

- If O=8, then
T W 8 + T W 8 - - - - - 1 8 U R - W=0, so U=1. NO, because F=1.
- W=1 NO, because F=1.
- W=2, U=5 - SOLUTION
- W=3, U=7 - SOLUTION
- W=4, so U=9. NO, becuase T=9.

- If O=9, then
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 |