Encoding Morse Code

Question

The letters of a message are converted to Morse code, with a space between each pair of letters and two spaces between each pair of words. Each pair of elements (for example dot-space, dash-dash, and so on) is then assigned a number from 1 to 9. A message written in this system is as follows:

512537293228231463737325742346428147

What pair of elements corresponds to each number, and what does the message say?

Hint 1:

Start by analysing 2282 .

It is meant to be a space between pairs of letters and not between letter right? — Wen1now, Mar 07 '17 at 10:05
I think it's meant to be .-../../-.-/.//.../--- without spaces (/) at the beginning or at the end of the phrase. — M Oehm, Mar 07 '17 at 10:17
Is this even possible to solve without brute-force or guesswork? — Techidiot, Mar 07 '17 at 11:22
@Techidiot: As with substitution ciphers you can make some conjectures about symbol pair frequency, but the message is rather short. And I hope it's not some proverb in Latin or Esperanto. — M Oehm, Mar 07 '17 at 11:30
@noneuclideanisms, is 2282 a full word? There are already 56 different possibilities just for it, and if it doesn't at least stand alone, it doesn't help to start there in particular. — anonymous2, Mar 07 '17 at 16:31
@anonymous2 Start by eliminating the possibilities that are impossible. — noneuclideanisms, Mar 07 '17 at 23:25
@noneuclideanisms, I've already done that, and it leaves me with about 10 just among the stand alone letters. If 2282 flows into one or both of the 3's on either side, there are an unbelievable number of possibilities. And btw, none of the 10 options made a complete word. — anonymous2, Mar 08 '17 at 00:29
@MOehm, are you suggesting that the code stated in the puzzle contains errors? Did you solve this but not post an answer? — humn, Mar 23 '17 at 15:49
@humn: Yes, that's what I'm suggesting and I had hoped that the OP would address it. After trying for quite a while, I threw a program at it and it came up with rot13(JRYV ORTHA VFUNYS QBAR), which has two obvious typos. The pairwise Morse code for the corrected sentence is quite different, because the typo in the fourth characer introduces a parity mismatch. And I don't see how 2282 helps in deciphering. The alternating chain 37373 seems more promising. I didn't post an answer, because I used a program. — M Oehm, Mar 23 '17 at 16:50
Thanks, @MOehm! Now I'll try to solve your version. The premise of this puzzle is truly delightful, and Morse code has some intriguingly skewed statistics but they don't seem to help with the stated example. — humn, Mar 23 '17 at 17:07
The premise is promising, but I found the actual solving - what little progress I made - rather tough. Good luck! — M Oehm, Mar 23 '17 at 17:12

score 13 · Accepted Answer · edited Apr 13 '17 at 12:50

The secret message and digit correspondences are . . .

    W   E   L   L       B  E  G   U  N     I S       H   A   L   F       D  O  N  E
    .-- . .-.. .-.. / -... . --. ..- -. / .. ... / .... .- .-.. ..-. / -.. --- -. .
 .--/./.-../.-..//-..././--./..-/-.//../...//..../.-/.-../..-.//-../---/-././
 5 1 2 5 3 7 4 2 9 3 2 2 8 2 3 1 4 6 3 7 3 6 3 3 7 1 5 3 7 5 2 9 3 9 8 9 2 2


                 -/   ./   ..   -.   .-   //   /.   --   /-

                 1    2    3    4    5    6    7    8    9    </pre></p>

This puzzle is built on a superb premise. &hairsp; Morse code’s natural 9 dot&hairsp;/dash&hairsp;/space pairs may be recoded as digits for a perfect combination of encryption and clue-idity. &hairsp; Morse code’s statistical skew provides a nicely shaped search space, like slopes and valleys that concentrate hunches.

The puzzle needed minor rebuilding. &hairsp; Unfortunately the original encryption contained imperfections. &hairsp; Fortunately M Oehm figured out (!) the intended message computationally. &hairsp; Unfortunately that message’s encryption has an odd number of dots&hairsp;/dashes&hairsp;/spaces, so the rules need to say that the message ends with a space.

Revised encryption and rules. &hairsp; The 38 digits of 51253742932282314637363371537529398922 represent a combination of 76 dots, dashes and spaces. &hairsp; Every Morse code letter is followed by a space while an additional space separates each pair of words. &hairsp; For example, Morse code for LIKE THIS would be .-../../-.-/.//-/..../../.../, where each / represents a space.

Solution begins by doubling the encoded digits and profiling a statistically likely solution.

       5511225533774422993322228822331144663377336633337711553377552299339988992222

       -/-/-/././/../../../-./.-/-.//.../.-./-../---/---//..--/..-./.-../.-../-..-/

The -/-/-/ . . . /-..-/ profile has approximately 5 letters per word, approximately equal counts of letter lengths, and favors . over - as is statistically expected. &hairsp; Here is the distribution of code digits and, without elaboration, likelihoods of their counts if they were specific dot&hairsp;/dash&hairsp;/space pairs.

               Pair:           ..    -.    .-    --    ./    /.    /-    -/    //

             Expected:        5.5    4    3.5   2.5   6.5    6     4     4    1.5

         Actual code digits                      Probabilities
         3 3 3 3 3 3 3 3 3    .05   .01   .005  .0005 .09   .07   .01   .01   .00001
(last)    2 2 2 2 2 2 2 2     .09   .03   .01   .002  .13   .11   .03   .03   .00008
(first)       5 5 5 5         .16   .21   .14   .07   .11   .13   .21   .21   .05
              7 7 7 7         .16   .21   .14   .07   .11   .13   .21   .21   .05
              9 9 9 9         .16   .21   .14   .07   .11   .13   .21   .21   .05
               1 1 1          .11   .20   .22   .22   .06   .08   .20   .20   .13
                4 4           .05   .14   .18   .26   .02   .04   .14   .14   .26
                6 6           .05   .14   .18   .26   .02   .04   .14   .14   .26
                8 8           .05   .14   .18   .26   .02   .04   .14   .14   .26

Morse code within-word statistics. &hairsp; Strongly skewed in favor of dots. &hairsp; (Derived from Wikipedia)


Frequencies of pairs:      ..  15.9%       -.  10.5%       /.  17.1%       ./  18.0%
                           .-   9.7%       --   7.4%       /-  11.1%       -/  10.3%

  Letters grouped by beginnings           Letters grouped by endings        Total by length
----------------------------------    ----------------------------------    ---------------
 E  /.    12.7%    T  /-     9.1%      E     ./ 12.7%    T     -/  9.1%        1   21.8%

 A  /.-    8.2%    M  /--    2.4%      I    ../  7.0%    A    .-/  8.2%        2   24.3%
 I  /..    7.0%    N  /-.    6.7%      N    -./  6.7%    M    --/  2.4%

 R  /.-.   6.0%    D  /-..   4.3%      D   -../  4.3%    K   -.-/  0.8%        3   32.0%
 S  /...   6.3%    G  /--.   2.0%      G   --./  2.0%    O   ---/  7.5%
 U  /..-   2.8%    K  /-.-   0.8%      R   .-./  6.0%    U   ..-/  2.8%
 W  /.--   2.4%    O  /---   7.5%      S   .../  6.3%    W   .--/  2.4%

 F  /..-.  2.2%    B  /-...  1.5%      B  -.../  1.5%    J  .---/  0.2%        4   22.0%
 H  /....  6.1%    C  /-.-.  2.8%      C  -.-./  2.8%    Q  --.-/  0.1%
 J  /.---  0.2%    Q  /--.-  0.1%      F  ..-./  2.2%    V  ...-/  1.0%
 L  /.-..  4.0%    X  /-..-  0.1%      H  ..../  6.1%    Y  -.--/  2.0%
 P  /.--.  1.9%    Y  /-.--  2.0%      L  .-../  4.0%    X  -..-/  0.1%
 V  /...-  1.0%    Z  /--..  0.1%      P  .--./  1.9%
                                       Z  --../  0.1%
         -------           -------              -------          -------
     /.   60.7%        /-   39.3%          ./    63.6%       -/   36.4%

Solution proceeds by testing most-likely matches. &hairsp; The clear place to begin is with the 22 at the end. Not only must it end with a / space, but it is also so numerous that the most-likely-of-all ./ pair is the best candidate.

code
|
|  substitution
|  |
|  |  5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./  5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
      |                     e                                                   e
      result

The next code to assign is the most-numerous 3, and the statistically favored candidate is /.

      5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./  5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
3 /.  5511./55/.7744./99/.././88.//.114466/.77/.66/./.771155/.7755./99/.998899././
                      ^^ i  e ^^ //                                     ^^^^^^

  998899     means that 88 or 99 must contain a / space.
/i/e/88.//   prevents anything meaningful if 88 contains a / space.
 /99/i/e     prevents 99 from being // or -/
             So the next assignment to try is  9  =  /-    ( ./ and /. are already taken)

      5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./  5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
3 /.  5511./55/.7744./99/.././88.//.114466/.77/.66/./.771155/.7755./99/.998899././
9 /-  5511./55/.7744.//-/.././88.//.114466/.77/.66/./.771155/.7755.//-/./-88/-././
                                                                   //t e     n  e
//t/-88/n/e  cannot be a word.                                     ^^^^^^^^^^^^^^

So much for 3 being /.. &hairsp; The next most likely candidate for 3 is ...

5511225533774422993322228822331144663377336633337711553377552299339988992222 2 ./ 5511./55337744./9933././88./33114466337733663333771155337755./9933998899././ 3 .. 5511./55..7744./99..././88./..114466..77..66....771155..7755./99..998899././ ^^^^^^^ ^^^ e /99..99 means that 99 must contain a / space. 99./e/ makes /- the most likely candidate for 99. ("ne" vs. "ee" or "ie")

   5511225533774422993322228822331144663377336633337711553377552299339988992222

2 ./ 5511./55337744./9933././88./33114466337733663333771155337755./9933998899././ 3 .. 5511./55..7744./99..././88./..114466..77..66....771155..7755./99..998899././ 9 /- 5511./55..7744.//-..././88./..114466..77..66....771155..7755.//-../-88/-././ // b e h // d ^^^ n e

From here the solution unravels neatly enough without further narrative.

8 -- 5511./55..7744.//-..././--./..114466..77..66....771155..7755.//-../---/-././ // b e g ^^ h ^^ ^^ // d o n e

7 /. 5511./55../.44.//-..././--./..114466../...66..../.1155../.55.//-../---/-././ // b e g ^^ h // d o n e

6 // 5511./55../.44.//-..././--./..1144//../...//..../.1155../.55.//-../---/-././ // b e g // i s // h ^^^^ ^^ // d o n e

1 -/ 55-/./55../.44.//-..././--./..-/44//../...//..../.-/55../.55.//-../---/-././ e // b e g u ^^// i s // h a // d o n e

4 -. 55-/./55../.-..//-..././--./..-/-.//../...//..../.-/55../.55.//-../---/-././ 5 .- .--/./.-../.-..//-..././--./..-/-.//../...//..../.-/.-../..-.//-../---/-././ w e l l // b e g u n // i s // h a l f // d o n e

So how did the actual statistics compare to predictions? &hairsp; Very well! &hairsp; Only two digits, representing .. and -., varied much from expectation and created just one false lead.

                   Pair     Predicted              Actual
                                           Digits          Morse
                    ./         17%        8    21%       15    20%
                    /.         16%        4    11%       12    16%
                    ..         15%        9    24%       14    19%
                    /-         11%        4    11%        6     8%
                    -.         11%        2     5%        9    12%
                    -/         11%        3     8%        4     5%
                    .-          9%        4    11%        7     9%
                    --          7%        2     5%        4     5%
                    //          4%        2     5%        4     5%

The mechanics of the puzzle were too good to let go, puzzler @Techidiot, for weeks. I want to plagiarize it. — humn, Mar 31 '17 at 11:08

Encoding Morse Code

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