Secret Message
Copyright 1997 by Paul Niquette.  All rights reserved
.
 
The solver of this puzzle will see the first part of the student's message...
    SEND + MORE = MONEY
...as hiding the key for decoding the numbers. He or she might then proceed as follows:

Assumptions:

  • Each letter corresponds to a distinct decimal digit.
  • The left-most digit of each number is not zero.
     
    SEND
    +MORE

    MONEY
If two four digit numbers are added together to produce a five digit number, the left most digit of the sum must be 1. Thus, by inspection, M = 1.
     
    SEND
    +1ORE

    1ONEY
The sum S + 1 must be at least 10. There are two ways for that condition to be met...
  • S = 9 or... 
  • S = 8 AND the additions to the right produce a carry into the left-most position.
To be systematic about the matter, let us name that left-most carry digit A and, while we're at it, we should give names to other possible carries as indicated.
     
    ABC-
    SEND
    +1ORE

    1ONEY

    If A = 0, then S = 9, since M = 1, then O = 0.
    If A = 1 and S = 9, then O = 1 = M. BLAGH!
    If A = 1 and S = 8, then O = 0, which is OK.

So we know that whatever the value of A, O = 0.
     
    ABC-
    SEND
    +10RE

    10NEY
We still don't know if S = 8 or 9.
If B = 0, then N = E. BLAGH!
If B = 1, then N = E + 1, which is OK, so we know that B = 1.
 
A1C-
SEND
+10RE

10NEY
If A = 1, then E + 1 = 10 and N = 0 = O. BLAGH!
So we know that A = 0, which means S = 9.
     
    01C-
    9END
    +10RE

    10NEY

    If C = 0, E = N + R - 10 (that -10 is the effect of B = 1).
    Since N = E + 1, then E = E + 1 + R - 10, and R = 9 = S. BLAGH!
    If C = 1, E = N + R + 1 - 10, then R = 8, which is OK.

       
      011-
      9END
      +108E

      10NEY
    The available digits for D, E, N, Y are 2, 3, 4, 5, 6, 7.
    Since N = E + 1, then 2 < N < 8 and 1 < E < 7.
    Now, Y = D + E - 10 (that -10 is the effect of C = 1).
     
  • If E = 2, then D = 9 or D = 8. Either way, BLAGH!
  • If E = 3, then D = 7 and Y = 0. BLAGH!
  • If E = 4, then D = 6 and Y = 1. BLAGH!
  • If E = 5 and D = 5 BLAGH! or D = 6, then Y = 1 BLAGH!
  • If E = 5 and D = 7, then Y = 2, which is OK. Hoo-hah.
     
    SEND
    +MORE

    MONEY
    9567
    +1085

    10652

  ...and the keys are...

    0


    O
    1


    M
    2


    Y
    5


    E
    6


    N
    7


    D
    8


    R
    9


    S
Solvers are all set to decode the two messages and discover the implications.
The Student's Message

20, 1006 10751. 
857 751069 0875857 107586 780659!
7001 95519 9006. 
60952 7081 7007.
The Mother's Reply

065 9085 70608 9559 15992 8001, 206758.
60 1085 10652 087589 85675857. 
90882, 101.
 

Solution to bonus puzzle, from old file found 5 April 2015.
Message: 81773 210625

        

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