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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
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T |
W |
2 |
+ |
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T |
W |
2 |
- |
- |
- |
- |
- |
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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
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T |
W |
3 |
+ |
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T |
W |
3 |
- |
- |
- |
- |
- |
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1 |
3 |
U |
R |
There is no solution, because both R and T must correspond to 6.
- If O=4, then
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T |
W |
4 |
+ |
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T |
W |
4 |
- |
- |
- |
- |
- |
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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
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T |
W |
5 |
+ |
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T |
W |
5 |
- |
- |
- |
- |
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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
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T |
W |
6 |
+ |
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T |
W |
6 |
- |
- |
- |
- |
- |
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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
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T |
W |
7 |
+ |
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T |
W |
7 |
- |
- |
- |
- |
- |
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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
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T |
W |
8 |
+ |
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T |
W |
8 |
- |
- |
- |
- |
- |
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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
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T |
W |
9 |
+ |
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T |
W |
9 |
- |
- |
- |
- |
- |
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1 |
9 |
U |
R |
This means R=8 and T=9, but this gives no solutions because O=9.
So, the only solutions are:
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7 |
3 |
4 |
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7 |
6 |
5 |
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8 |
3 |
6 |
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8 |
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6 |
+ |
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7 |
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4 |
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+ |
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7 |
6 |
5 |
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+ |
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8 |
3 |
6 |
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+ |
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8 |
4 |
6 |
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1 |
4 |
6 |
8 |
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1 |
5 |
3 |
0 |
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1 |
6 |
7 |
2 |
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1 |
6 |
9 |
2 |
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8 |
6 |
7 |
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9 |
2 |
8 |
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9 |
3 |
8 |
+ |
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8 |
6 |
7 |
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9 |
2 |
8 |
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+ |
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9 |
3 |
8 |
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- |
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1 |
7 |
3 |
4 |
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1 |
8 |
5 |
6 |
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1 |
8 |
7 |
6 |