.

he 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:
• 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
olvers 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                        246         178/43978             356              837              712              1258              1088               170

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