# Solution

##### Age 11 to 14

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