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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 3digit numbers, and the result is 4digit 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 
This means R=4 and T=6. In this case W+W=U, and I analyse the numbers for W one by one:
 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 
There is no solution, because both R and T must correspond to 6.
 If O=4, then


T 
W 
4 
+ 

T 
W 
4 
 
 
 
 
 

1 
4 
U 
R 
This means R=8 and T=7. In this case, W+W=U, and analysing as previously, I find that W cannot be 0, 1, 2, 4, and cannot be greater than or equal to 5. There is a solution with W=3 and U=6.
 If O=5, then


T 
W 
5 
+ 

T 
W 
5 
 
 
 
 
 

1 
5 
U 
R 
This means T=7 and R=0. Now, W must be at least 5, because W+W+1=10+U.
 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 
This means R=2 and T=8. W+W+1=U, so W=4.
 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 
This means R=4 and T=8. W+W+1=10+U. So, W=5.
 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 
This means R=6 and T=9. W+W+1=U. So, W=4.
 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 
This means R=8 and T=9, but this gives no solutions because O=9.
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 