In the multiplication given below, all the digits, except one, were consistently replaced by another digit:

    617 
*   702 
_______
   1639 
  3733 
 2807  
__________  
 265699

Find the substitution code and restore the original multiplication.

I needed to change to letters.  Numbers was too confusing:

    ABC
*   CDE   
_______     
   BAFG     
  FCFF    
 EHDC    
_______     
 EAIAGG

C must be 1, 5, or 6 from the 3rd partial product C*C ends in C but C cannot be 1 or 5 (all partial products have different last digits) so C=6.

C+F= 9 or 10 from fourth column sum, but since F=3 would not carry F=4.  Hence G=8.

E*6 ends in 8, since 8 is taken, E=3.

Now consider the first partial product we now have
AB6 * 3 = BA48,  to make B6*3 end in 48 it must be B=1,
since there's no carry, A*3 = 1A so A=5

Dividing from the second partial product 4644/516=9=D

For the 3rd column sum I=7
And for the 2nd column sum H=0

I don't care to work out the puzzles replacements except to say 1 is not replaced.  The corrected multiplication is

     516
 *   693
 _______     
    1548     
   4644    
  3096
__________
  357588

I was hoping the fact that only one digit was not replaced would be a vital piece of information.  That would have required some interesting logic. But it is not.

Edited on March 11, 2014, 12:40 pm

2nd edit for formatting

Edited on March 11, 2014, 2:42 pm

Posted by Jer on 2014-03-11 12:21:11