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 .
Answer
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
This puzzle is built on a superb premise. Morse code’s natural 9 dot /dash /space pairs may be recoded as digits for a perfect combination of encryption and clue-idity. Morse code’s statistical skew provides a nicely shaped search space, like slopes and valleys that concentrate hunches.
The puzzle needed minor rebuilding. Unfortunately the original encryption contained imperfections. Fortunately M Oehm figured out (!) the intended message computationally. Unfortunately that message’s encryption has an odd number of dots /dashes /spaces, so the rules need to say that the message ends with a space.
Revised encryption and rules. The 38 digits of 51253742932282314637363371537529398922 represent a combination of 76 dots, dashes and spaces. Every Morse code letter is followed by a space while an additional space separates each pair of words. 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. Here is the distribution of code digits and, without elaboration, likelihoods of their counts if they were specific dot /dash /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. Strongly skewed in favor of dots. (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. 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 /.. 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? Very well! 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%
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