AQUIMAN/THEBIGZ: Over the last several weeks, I attempted to perform a comprehensive analysis of AK's (et al) cipher theories. It turned out to be a very difficult task and I now hate math. What follows will, no doubt, be quite a boring read for most; however, I believe it may shed some light on how useful probability analysis can (or cannot) be in regards to proving or disproving theories. For those without a math background, I hope I've explained things well enough to understand probability analysis in general. For the math geeks (like myself), I hope I don't bore you... but at least you will be able to correct any mistakes I may have made along the way. Comments, criticisms, and suggestions are welcome. Be gentle.
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First, I want to discuss some basic probability and then delve into why this was not the typical probability exercise. Let’s take the name “Theodore J. Kazcynski.” For each of the 18 letters in his name, we have a bag filled with all 26 letters of the alphabet. If you reach into the first bag, the odds of pulling out the “T” are one in 26. Reaching into the second bag, the probability is again 1/26 that the “H” will be chosen. The overall probability that a “T” and an “H” will be chosen from the first and second bags respectively is 1/26 x 1/26 = 1/676. Someone choosing one letter from each bag, hoping to spell out the entire 18 letter name should expect that the odds are very thin; (1/26)^18 to be precise.
Let’s make the odds a little better. You can now choose one letter from each bag, but you don’t have tp choose the letters in “name order.” You can then rearrange them any way you like to try and spell the name. The number of possible arrangements of 18 different letters would be 18! (i.e., 18x17x16…x2x1). All the letter in THIS name, however, are not unique; there are two E’s, two O’s, and two K’s. Therefore, the number of unique arrangements is given by 18!/(2!x2!x2!). The overall probability then of choosing 18 letters (each from a different bag) and obtaining any one of those arrangements is (1/26)^18 x 18!/(2!x2!x2!).
This is where things begin to get tricky. AK, Kite, et al have proposed that you can use a Caesar shift (of -9, -6, -3, +3, +6, or +9) on each letter, essentially giving you six additional letters from each bag; not just any letters, but those within the constraints of the allowed shifts. The difficulty now lies in determining the number of permutations of 18 sets of seven letters, several of which may be duplicated between sets.
The total number of arrangements when choosing one of seven letters from each of the 18 sets is given by 7^18; however, they are not all unique. Many arrangements are duplicates, not just because of the three pairs of duplicate letters in the name, but also because of the Caesar shifts. Take the letters “H” and “E” for example. With Caesar shifts, they form the following sets: {Q, N, K, H, E, B, Y} and {N, K, H, E, B, Y, V}. There are six duplicates just within those two sets. Eighteen sets will share many more.
A “closed-form” mathematical solution to this problem is not readily apparent (and likely does not exist). Therefore, a brute-force computer program was developed to determine the number of unique arrangements and corresponding probability of obtaining at least one of those arrangements. For 18 letters with Caesar shifts, the number of possible arrangements borders on infinity. To obtain a “true” probability for the proposed solution of 18 letters, it would literally take years to run all the permutations, even if I had the computing power; my computer could only handle nine letters with Caesar shifts. The results of the analysis of two nine letter sets are shown below.
I could only give an estimate as to what the outcome might be for the full 18 letter set, based on the trends of two subsets of nine letters. I can theorize that the general trend would continue downward (as I extend a trend line past nine letters), falling to maybe 1% or so. I'm sure it would bounce around a bit on the way as seen in the two corresponding graphs, but likely be small in the end. Now, if the final 18 were something like ODEHTJROENKZAIKSCY, where THEODORE and KAZCYNSKI were in separate nine letter groupings, then I'm allowed to multiply the two probabilities together to get 0.176%. Otherwise, I can only hazard a guess at this time. I would not theorize that the probability is going to rise substantially (if it rose at all) from nine to 18 letters anyway. So <5% is a pretty good bet.
For the sake of argument, let’s say the probability is 1%. What does this really mean? Well, it means that out of a near infinite number of unique arrangements, 1% of them can form the name “Theodore J. Kazcynski.” This will be in the neighborhood of 3x10^23 possibilities (a rough guess), infinity in and of itself. So while I can pat myself on the back for a valiant effort to “prove” something, I’m not sure I was able to prove anything one way or another; there are just too many variables, too many possibilities. The one true result of this analysis is that you cannot obtain any name out of any set of code when using Caesar shifts and anagramming.
I do applaud AK’s efforts, unsurpassed dedication, and pursuit of the truth. There are many other reasons why TJK should not be dismissed as a suspect; however, I’m not sure the results here do much to support the cipher theory. AK has additional suggestions, which I will look at as time permits. They are good suggestions that may be able to substantially narrow down the possibilities as to whether or not TJK authored these ciphers. I encourage AK and others to uncover all possibilities so we can get that much closer to the truth. I’m keeping an open mind and will help if I’m able. If I have not explained these results well enough, please ask and I’ll do my best to clear things up.
AK WILKS: Great and many thanks again to Aquiman/ThebigZ for doing the excellent and time consuming probability study.
Since all the critics said many names can be produced and the name TJK just appears by chance, I thought the point of the probability studies was to determine if the chances were what the critics were saying (100%, 90%, 80%, 51%) or what I was saying (under 10%, perhaps under 5%, maybe under 1%). But now the argument seems to be "Yes it was 5% or 1% or 0.003%, but those numbers are meaningless because I can get LKANE out of 13 Caesar shifted letters and many other names."
Yes MANY MANY NAMES CAN BE PRODUCED FROM THESE LETTERS. That was shown two years ago. But NONE (I think) can be SEEN and READ by the average person. All the other names are hodge podge mixed up, extreme anagrams. Nobody can seriously argue any were placed.
In other words, to a computer Z CAK NY THEO IKS is just one of thousands of possible combinations and thus insignificant. But to a human eye and mind, the name appears in readable English, and out of the thousands and thousands of names that can be produced, it is the only one that has that quality.
What I argue is significant is that we have THEO as is no anagrams and the last name backwards by syllable. If you don't find that significant then it is one of just many names and has no meaning.
Same thing with the unsolved 18 - either the geographic patterns and internal math have meaning to you or they don't.
The results for closely bunched THEODOREJ and KACZYNSKI in Moby Dick were 4 finds out of 1,176, which is 0.003%, or 3/10 of 1%, and Doranchak admits the results would be similar if not lower for the finding of Z CAK NY THEO IKS in Moby Dick as it is found in the 3rd line of the 1986 Graysmith Raw Solution to the Zodiac 340. Aquiman/thebigZ may be able to soon give us the actual number, along with his reason for why he finds that number significant or not.
BENTLEY: I'm not sure if that's good or bad AK. If TJK made a deliberate attempt to put his last name in the first part and first name in the last, why aren't all 9 correct in the beginning, as well as the end. Not like he needed another level to mask it. Mistakes? Pretty simple to swap the two and get it right.
AK WILKS: Good point. I agree. Unlikely to be a mistake. If he did it it was intentional to incease deception. He may have worried that 9 of 9 on each side was too much of a chance of recognition. Also the switch creates "oriet" and "piti" which suggest "orient" and "pity".
NO ANAGRAMS, NO SCRAMBLES, CORRECT SPELLING, CORRECT ORDER, LEFT TO RIGHT READ.Of
two aliases used by an informant even if he had nothing to do with the murders definitely used ZODIUS one year before ZODIAC, and who we now know DID HAVE inside knowledge of the case making him a major suspect in the murders. Murders in which the mysterious "ROEBERT" from San Francisco put dots under the EBE in his name. And it is EBE that starts the Unsolved 18 in the first Zodiac code.
Wouldn't the probability be similar to the closely bunched results? And here they are WAY BETTER than just closely bunched, they are in order, correct read, NO ANAGRAMS.
So probability wise we are AT LEAST talking 0.003% for ONE NAME to appear, and maybe at least 0.000009% for TWO NAMES to appear?
Aqui, Doranchak, anyone - thoughts, criticisms, ideas?
O...........E..........G..........(D)............C..........R..........W.........O...........M....(+30)
E..........(U).........W..........T..............S..........H..........M.........E............C....(+20)
U...........K..........M...........J.............(I).........X...........C.........(U)........(S)...(+10)
K...........A..........C...........(Z).............Y........(N)..........S.........K............I.....(0)
A...........Q..........S............P.............(O).......(D)........(II).......A.......... Y.. (-10)
Q...........G.........(I)...........F..............E..........T...........Y.........Q...........O...(-20)
(G).........W........Y...........V...............U..........J...........O.........G...........E...(-30)+20..N.......B.......Y.......I
+10.(D)....(R).....O......Y
**0..T.......H.......E......O
-10..J.......X........U......E
-20..Z.......N.......K.......UA January 1969 Detroit News article mentioning the August 1968 tip from an informant, who DID HAVE inside knowledge of the June 25, 1968 Robison Family Murders, and the informants message that he asked to be place in a classified ad.
The ad mentions ZODIUS, almost one year before the ZODIAC announced himself to the world. Both come from the Latin word ZODIACUS.
Body of Mrs. Robison covered by a blanket. The bodies of Domigos and Edwards were also covered by a towel, as were some victims of the EAR/ONS. Like the bodies of Domingos and Edwards, the bodies of Mr. Robison and one of his sons were placed on top of each other. Former FBI Agent John Douglas says posing occurs in less than 1% of murders. Like Valerie Percy, Mrs. Robison had her dress lifted up to expose her.
Police ballistics reports in the Robison Family Murders indicate a .22 with six lands and grooves, with a right twist. This matches both Domigos and Edwards Santa Barbara 1963 and Zodiac 12/20/68.
Roebert's voice on the phone was described as "monontone", "frequent pauses" and "robotic". This is similar to the reports by police dispatcher Nancy Slover on the voice of Zodiac.
Mr. Roebert from San Francisco claimed to represent an organization called SUPERIOR TABLE. This memo was found on Mr. Robison's office desk.
If Zodiac did kill the Robison Family, than this "Mr. Roebert" from San Francisco may have been Zodiac, and this memo may be from Zodiac.
If you look at the name ROEBERT, there are *** under the letters "EBE". The letters "EBE" start the Unsolved 18 from the first Zodiac code. The Robison murders took place in EMMET County. "EMET" is also in the "EBE" string.
AK Wilks: I consider the time of the police as very valuable, and I consider my time valuable as well. I have never officially contacted police in the Robison case specifically. I am considering doing so now, in part because of the announced police interest in another midwest family murder (Bricca, Ohio) based on evidence I sent them in 2010 and which they may have found additional material supporting a possible Zodiac connection, new evidence casting doubt on the favored local suspect Scolaro and a 10 based Caesar shift from Kaczynski discovered by Kite which produces both of the two aliases used by a tipster who was considered a POI based on inside knowledge, GUIDINI and ZODIUS.
Are any of you interested in taking a look at this?
Now I could predict that your opinions would be negative to mixed at best based on your feelings on prior work with Caesar shifts but here is why this may be different. Both alias names are produced in left to right read, with NO ANAGRAMS.
Does that make a difference?
Can we calculate the odds of nine letters producing both aliases in order no anagrams? Would such a calculation be meaningful? Is there another way to look at this?
Should I submit this code work to the police in the Robison case, along with the MO matches, ammo and weapon matches and other items?
Doranchak: I can't speak to the other details, but here is my analysis of the code work:
My recommendation is that you don't mention these grid findings to authorities, because there is no reason to believe that they are intentionally placed or selected messages. I will explain why.
First, some definitions:
- A "shift group" is a set of numbers used to construct the Caesar shift grid. For example, {-30,-20,-10,0,10,20,30} is a shift group.
- A "shift multiple" is the common factor shared by all the numbers of the shift group. For example, "10" is the shift multiple for the shift group {-30,-20,-10,0,10,20,30}. "3" is the shift multiple for {-9,-6,-3,0,3,6,9}.
You claim that "APIAS TUBEROSA" appears in THEOKACZYNSKI's shift group with shift factor 10. But:
1) APIAS TUBEROSA also appears in 9 other shift groups for THEOKACZYNSKI, for these shift factors: 2, 4, 8, 12, 14, 16, 18, 22, and 24. (See
http://zodiackillerciphers.com/ak/robison-1.txt)
2) APIAS TUBEROSA appears in 38% of randomly generated words, if we scan all shift multiples from 1 to 25. If we look only at shift multiple 10, then APIAS TUBEROSA appears in 6% of randomly generated words. (Go to
http://zodiackillerciphers.com/ak and look at robison-2.txt through robison-51.txt)
Therefore, APIAS TUBEROSA appears purely by coincidence.
Your next claim is that the words GUIDINI, ZODIAC, and ZODIUS all appear in order (with no anagrams) when KACZYNSKI is shifted using shift factor 10. But:
1) More than 50 other six-letter words also appear in order, with no anagrams. Here are the top 50 in descending order of popularity in English: forces, studio, duties, tories, pursue, actors, versus, easter, denies, jersey, gasped, austin, fences, verses, guided, ponies, austen, dorcas, easton, euston, spence, venous, juries, wipers, zodiac, porous, tedium, ousted, podium, gaston, widens, dorigo, quiver, esters, purses, dorsay, gdynia, titius, pixies, torsos, auspex, furies, tenses, givers, guider, formes, ouster, jetsam, jencks, pudsey
2) You have cherry picked 3 specific words, but there are 50 other words to pick from above. This forms a "deck" of 53 "cards". There are 23,426 ways to make a hand of 3 cards from this deck. You have only considered one way.
3) If I generate 200,000 random words, only 4 of them produce all three words (GUIDINI, ZODIAC, and ZODIUS) in order with no anagrams. By your logic, this is rare enough to be of interest to you. (see
http://zodiackillerciphers.com/ak/all-t ... -words.txt)
4) But let's look at the list of 50 other words above, and choose "gasped", "quiver", and "torsos" as sufficiently Zodiac-like. If I generate 200,000 random words, NONE of them contain all three words with no anagramming. And yet all three appear in the grid for KACZYNSKI, with no anagramming:
== kaczynski ==
gwyvujoge
qgifetyqo
aqspodiay
kaczynski
ukmjixcus
euwtshmec
oegdcrwom
Match: gasped (gasped), Shifts: -30 0 -10 -10 -20 -10 null null null , Group: -30,-20,-10,0,10,20,30
Match: quiver (quiver), Shifts: -20 20 -20 -30 -20 30 null null null , Group: -30,-20,-10,0,10,20,30
Match: torsos (torsos), Shifts: null null null 20 -10 30 0 30 10 , Group: -30,-20,-10,0,10,20,30
5) That means my three words were intentionally placed or considered, by your logic. However, all of the 23,426 ways to pick 3 words can be shown to be similarly rare. Therefore, you'd have to conclude that all 50 words were intentionally placed or considered. But this conclusion is ridiculous, because every 9 letter sequence of letters in addition to KACZYNSKI will produce many words in order with no anagrams. Here are 3 examples, produced from the names Alexander, Nathaniel, and Sebastian:
http://zodiackillerciphers.com/ak/words ... -names.txt Each of those names produces many possible words.
6) I've only considered the appearances of six-letter words and names. More words and names are possible if we consider other lengths.
Therefore, there is no evidence to support your claim that 3 specific words were specially considered by your suspect.
AK Wilks: Doranchak - Thanks. Very interesting.
I probably will not stress the Apias Tuberosa, as your work shows it could be a random chance happening and Ted may have selected that name by a process other than 10 based Caesar shifts. I have learned from experience that dropping bad information or even information or evidence that may be good but is very speculative increases the chances the police will evaluate the good evidence you are submitting. I have had some success when I follow this policy, as seen in the Tylenol and Bricca cases, also with some investigators on Zodiac as well.
Doranchak: "3)
If I generate 200,000 random words,
only 4 of them produce all three words (GUIDINI, ZODIAC, and ZODIUS) in order with no anagrams.
By your logic, this is rare enough to be of interest to you. (see
http://zodiackillerciphers.com/ak/all-t ... -words.txt) ."
AK Wilks: OK, you say by "my logic", but isn't this rare enough by anyone's logic?
Out of the 200,000 words you looked at, the four that worked, I do not recognize any of them. The fifth word is KACZYNSKI. Someone wrote to the newspaper using GUIDINI and ZODIUS, and later as ZODIAC.
"Kaczynski" is a word that 10 base Caesar shifts to these three words, something that only 1 out of 50,000 or so words can do.
This strikes me as compelling! I am leaning towards focusing on the ammo matches (both Zodiac LHR, SB 63 and Robison had .22 LR Super X six lands and grooves right twist), MO matches, body stacking, other case matches, the use of ZODIUS with its similarity to ZODIAC and evidence of possible Zodiac and/or TJK links to other midwest murders like Percy and Bricca. I am also leaning towards mentioning the GUIDINI/ZODIUS caesar shift info, with a summary of my thoughts and your thoughts and a link to your data.
If anyone else has any thoughts, please let me know.
Thanks again for taking a look at this, and I have an idea now of how I will proceed.
But in all seriousness, one question.
If you do not consider a 1 in 50,000 event as low enough random odds to be worth further investigation, what number would you consider low enough? 1 in 100,000? 1 in 500,000? Or is there no number?
Doranchak: A low probability isn't enough when many alternatives are possible, as I've demonstrated. You could pick any three six-letter words and show that the probability of them being produced in a given shift group is very low.
Once again, there is an important difference between "the probability of these specific 3 words appearing" and "the probability of any 3 words appearing", just as there was before when considering names. I.e., the probability of a specific name being produced is low, but the probability of any name being produced is high.
Another example: The probability of the known solution of the 408 being random is extremely low. What is the probability of randomly producing ANY OTHER solution of the 408 that is as equally readable as the known solution? So far, it's been zero. That's low enough.
AK: "Once again, there is an important difference between "the probability of these specific 3 words appearing" and "the probability of any 3 words appearing", just as there was before when considering names. I.e., the probability of a specific name being produced is low, but the probability of any name being produced is high."
I understand your points and I am considering them, in some cases agreeing, in more cases not.
But when you say "these 3 specific words", again, look at the context.
I did not pick these words. There is evidence to suggest one person with a .22 six lands and grooves right twist, who tied couples with pre-cut rope, and stacked bodies atop each other, used 3 aliases he sent to newspapers - DR GUIDINI, ZODIUS, ZODIAC. All 3 of those names appear in a 10 based shift from the name of a known serial killer with connections to both areas where these crimes happened - Michigan and California. Your data shows that is a rare event, happening by chance in only 1 out of 50,000 names.
Percentage wise that works out to 0.0002% of the time by chance. Is that right?
The names GUIDINI and ZODIUS are not common names. Yet it appears the killer used them.
Doranchak: "Percentage wise that works out to
0.0002% of the time by chance. Is that right?"
That's correct. It's also a higher percentage than other selections of 3 words, because ZODIAC and ZODIUS are very similar (only the last two letters need to differ to find a match in another tested word). For instance, I found "gasped", "quiver", and "torsos" occurring for KACZYNSKI, but none of the 200,000 randomly generated words produced the same three words.
You say that you did not pick the words, but you and/or Kite sure did look for them. Eventually you can find anything you want using these techniques. Not enough words appearing? Create more lines in the grid. Still can't find something? Change the shift group values. Still not satisfied? Try strings that are longer than "KACZYNSKI". Relax a few constraints and all sorts of things will pop out.
AK: "You say that you did not pick the words, but you and/or Kite sure did look for them."
No. William T Rasmussen wrote a book called Corroborating Evidence III I think in 2012 in it which he showed pictures of an August 1968 Detroit News article about a tip in the June 1968 Robison Family Murders in which the tipster used the aliases DR GUIDINI and ZODIUS. The tipster had unpublished inside knowledge of the crime and police thought he could be the killer. So I did not look for words. This happened, it is real. Now on aspects of the code work, yes Kite and I have been complete and explored many angles, also asking you, thebigZ and others to examine and critique the work. I don't think there is anything wrong with being complete.
The ad mentions ZODIUS, almost one year before the ZODIAC announced himself to the world. Both come from the Latin word ZODIACUS.
Posted: Wed Nov 06, 2013 9:14 pm
