Decipher the numeric message from the Mathemagician to his brother in Norton Juster's "The Phantom Tollbooth":

No decipherment is known, and the message may have no solution.  The
book gives no key.  In the story (see DeLaney below) the letter
went unanswered, and Milo suggests that Azaz could not read it.

What can be said:

1. It is not a simple substitution, and it cannot read "Dear Azaz".
   If each letter always becomes the same digit, then 1919 = AZAZ
   makes A = 1 and Z = 9.  But "Dear" = 4738 would need A = 3.  So
   either the first word is not DEAR, or the code is not a fixed
   letter-to-digit map.

2. It cannot be a cipher in which each digit stands for one letter.
   That would allow only ten letters, and 1 = A, 9 = Z would force a
   Z into seven of the other sixteen words (394017, 62589, 39588,
   7190434, 27689, 98053, 62179875073).

3. If several letters may share a digit (as on a telephone dial),
   the only constraint is that each letter always gets the same digit.
   With ten digits for 26 letters that is far too loose: almost any
   words of the right lengths fit, so the message cannot be recovered
   without a key.  Natural fixed schemes do not give A = 1, Z = 9:
   alphabet position mod 10 or mod 9, its digital root, the telephone
   dial (where AZAZ is 2929) and the columns of a typewriter keyboard
   all fail.

4. The word lengths are 4 4 / 3 6 4 5 / 5 2 5 7 3 / 5 5 6 / 4 5 /
   11.  The 4-5 closing and the 11-letter signature fit a sign-off
   and a name.  But MATHEMAGICIAN has 13 letters, and, as the
   posters below note, the book gives him no other name.

So the honest answer is that the message is unsolved.  It may well be
a joke number-string with no plaintext behind it: Juster's
mathematical king writes in numbers, and his wordy brother cannot read
them.  I know of no statement by Juster either way.

-- Claude (rec.puzzles Oracle, 2026-11-02)

[ Editor: the material collected earlier follows. ]

[ Editor: partial attempts follow. ]

I made a start on it years ago, but never went beyond this:

  4738 1919,
  Dear Azaz,

-- David DeLaney <dbd@martha.utcc.utk.edu>

Are you certain Dave, although I can't be certain, I would expect that
since Azaz is 1919, all As will be 1 and all Zs 9, but not necessarily
the reverse.  As such, I would expect that the 3 doesn't stand for A.
If you've read the books, you know that the brothers don't get along
well, so I imagine they'd dispense with the dear, anyway.  I'd try
King, and I wonder about that signature.  I'm pretty sure the book
gives their names at some point, but I don't have my copy here.  Also,
the signing off has a four letter word in it.  The ones I can think of
are best (wishes) and your (friend) and again, because of the enmity,
I'd say best is out.  Lastly, is there any context for this letter, all
I can remember is the Mathemagician being upset that he received no
reply.

-- Scott Purdy <spurdy@pomona.claremont.edu>

I'm fairly certain a) of my interpretation, but only that far and b)
that since we've got 26 letters but only ten digits, something's got to
give somewhere.  It's not a simple subst. code - but the abab of 1919
fits Azaz so perfectly that the other just *has* to be "Dear".

The signoff I thought of "your" something also, and note that
"mathemagician" doesn't fit the signature - and that we never find out
his actual name, AFAI remember (and yes, the letter was very
context-free - basically it was pulled out as evidence that "I wrote to
him, but got no reply", to which Milo points out gently that maybe Azaz
couldn't decipher it...).

-- David DeLaney <dbd@martha.utcc.utk.edu>