Multiplication Puzzle

Introduction

The multiplication puzzle is an excellent adventure in basic math over the set of nonnegative integers less than 10 under standard addition and multiplication modulo 10. A multiplication problem is shown with all digits replaced by letters. Your task is to guess which letter represents which number.

Strategy

In all multiplication puzzles, the following are true:

1. There is a one-to-one correspondence between the elements of the set (ABCDEFGHIJ) and the ten integers 0 through 9.
2. The puzzle, when interpreted in standard grade-school arithmetic notation, is mathematically correct.

So there are many hints to guide the gussing. For example, given a puzzle like this:

    I H H
X     H G
-----------
  H C I G
C B J J
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E E C G G

For this particular example, the following are true:

1. HxG = G (mod 10) That is because the units digits in the 2 multiplicands (H in IHH and G in HG), when multiplied, gives the result G (the units digit in HCIG).
2. HxH = J (mod 10)
3. I+J = G (mod 10)
4. C+J = C (mod 10) (if there is no carry), or C+J+1=C (mod 10) (if there is a carry)

From the fact 3 above, we can see that J cannot be 0. Combines with fact 4, that gives J=9. Place J=9 into fact 2, we have HxH = 9 (mod 10). That indicates H is either 3 or 7. Now look at fact 1, knowing that H is either 3 or 7, and G can not be 0, that will gives us G=5. Now we have solved J and G. Then from fact 3, we immediately get I = 6. Put all the digits we have solved so far into the puzzle, we get the following:

 
    6 H H
X     H 5
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  H C 6 5
C B 9 9
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E E C 5 5

Now we see that 6HHx5 = HC65. From previous analysis, H is either 3 or 7. So H has to be 3. The rest of letters can be solved straight forward. So we solved the puzzle from mathmatical analysis without any gussing. The final solution is:

     6 3 3
X     3 5
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  3 1 6 5
1 8 9 9
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2 2 1 5 5