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

  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: