The answer is:

																6 ? ?          6 4 5
          x     ? ? ?    x     7 2 1
        _____________  _____________
                ? ? ?          6 4 5
            ? ? ? ?        1 2 9 0
          ? 5 ? 5        4 5 1 5
        _____________  _____________
          ? ? 5 ? 4 ?    4 6 5 0 4 5

I obtained this solution as follows:

First, I assigned variable names to each unknown digit in the original problem => 6ab x cde. I then went about solving each digit.

Since there are only 3 digits in the first number under the top line, it followed that e must be equal to 1, since anything larger would result in 4 digits. This yielded:

                6 a b
          x     c d 1
        _____________
                6 a b
            ? ? ? ?
          ? 5 ? 5 
        _____________
          ? ? 5 ? 4 b

Since the third number under the top line ends in 5, then either c or b must be 5, and the other number (b or c) must be odd. Trying 5 for c quickly shows no way to obtain the ?5?5 for the third number under the top line, therefore b must be 5 and c must be odd. This yields:

                6 a 5
          x     c d 1
        _____________
                6 a 5
            ? ? ? ?
          ? 5 ? 5 
        _____________
          ? ? 5 ? 4 5

Now to get ?5?5 for the third number under the top line, c must be either 7 or 9. c=9 would work only if a=1, but we can disprove this possibility by realizing that in order to get the 4 in the last line, a+5d must end in a 4, and no integer d will allow this. Therefore, c must be 7 by default. This yields:

                6 a 5
          x     7 d 1
        _____________
                6 a 5
            ? ? ? ?
          4 5 ? 5 
        _____________
          4 ? 5 ? 4 5

In order to get ?5?5 for the third number under the top line, a must be either 4 or 5. Only 4 will satisfy the previously mentioned condition that a+5d must end in a 4, so a must be 4. This yields 645 x 7d1.

                6 4 5
          x     7 d 1
        _____________
                6 4 5
            ? ? ? ?
          4 5 1 5 
        _____________
          4 ? 5 ? 4 5

For the final number (d), we realize that in order for the final number to have a 4 for its second to last digit, d must be even. Plugging in numbers identifies 2 as the correct solution, yielding:

                6 4 5
          x     7 2 1
        _____________
                6 4 5
            1 2 9 0
          4 5 1 5 
        _____________
          4 6 5 0 4 5

It is, of course, debatable whether it was quicker to reason out an answer in this fashion or to write a program to search out all possibilities for the original 5 ?'s above the first line, since there would be only 100,000 variations to try.

Dave Arnett

Copyright © 1996-2009 Alexander Bogomolny

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