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From: lief@hpfcrs.UUCP (lief)
Newsgroups: net.puzzle
Subject: Re: Orphaned Response
Message-ID: <15400001@hpfcrs.UUCP>
Date: Thu, 18-Oct-84 22:01:00 EDT
Article-I.D.: hpfcrs.15400001
Posted: Thu Oct 18 22:01:00 1984
Date-Received: Thu, 18-Oct-84 19:23:18 EDT
References: <-83700@ihuxe.UUCP>
Lines: 47
Nf-ID: #R:ihuxe:-83700:hpfcrs:15400001:37777777600:1666
Nf-From: hpfcrs!lief    Oct  8 18:01:00 1984

In response to the SEND + MORE = MONEY puzzle:

   SEND
 + MORE
 ------
  MONEY

(1)      D + E = 10*C1 + Y
(2) C1 + N + R = 10*C2 + E
(3) C2 + E + O = 10*C3 + N
(4) C3 + S + M = 10*M  + O

    where C1, C2, and C3 are the corresponding carries.  These can be rewritten:

(4) C3 = 9*M + O - S
(5) C2 = 90*M + 9*O - 10*S + N - E
(6) C1 = 900*M + 90*O - 100*S + 9*N - 9*E - R
(7) D  = 9000*M + 900*O - 1000*S + 90*N - 91*E - 10*R + Y  (subs for C1 in (1))

    assume C3=1, then from (4) M = (S - O - 1)/9 where M,S, and O are integers
    less than or equal to 9.  This equation is impossible for this stipulation
    so therefore C3=0, and S=9, O=0, and M=1.

    plug this result into (5) and C2 = N - E.  We know that N is not equal to
    E so therefore C2 must be 1 and E = N -1.

    plug the above results into (6) and C1 = 9 - R.  Since S is already 9, R
    must be 8 and C1 must be 1.  (Of course you agree that C1 can only be
    either 0 or 1).

    finally, play with (7) and D - Y = 10 - E.  Let's assume that N = 7.  Then
    E = 6 since E = N - 1.  Thus we have D - Y = 4.  The only integer pairs
    that can solve this is D,Y = 4,0   5,1   6,2   7,3   8,4   9,5.  However,
    one of the numbers in each pair is already defined so none of these will
    work.  Let's assume that N = 6 and of course E = 5.  Then we have to fit
    D - Y = 5.  The possible solutions are D,Y = 5,0   6,1   7,2   8,3   9,4.
    Note that both 7 and 2 have not been defined yet, so this pair is a
    solution.

O=0  M=1  Y=2  E=5  N=6  D=7  R=8  S=9

    9567
   +1085
   -----
   10652      which is a correct solution (I didn't prove it was unique!)

Lief Sorensen