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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.]

  • 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