Number Theory Linking Zodiac Murder Dates

Re: Number Theory Linking Zodiac Murder Dates

Postby akwilks on Mon Feb 13, 2012 5:27 pm



Zodiac 11 9 69 Bus Bomb Letter To SF Chronicle.jpg
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KITE: What might be an interesting visual presentation are the dates Zodic acted on. If you take a Zodiac symbol and assign ONE thru TWELVE on the points. Now change that to match the Zodiac dates, or in other words:


So this is not subjectively placing numbers at the point but matches the ONE thru TWELVE and is just MULTIPLYING by FIVE and, after all, MULTIPLES OF FIVE are what this is all about. Zodiac marked the points 6,8,9,10,11 with X's in a November 9, 1969 communication. With the adjusted multiplied by five Zodiac symbol, he is giving us the numbers: (80),(90),(95),(100),(105). And the 4 dates he has already acted on up to this time are:

12/20/68 (12+20+68)=100

So, you have a match for 4 of the 5 numbers and as it will turn out, the next date 3/22/70(3+22+70)=95, is the extra 95 and so it ends up as the 5 numbers(80,90,95,100,105) being the 4 dates already acted on and what will be the next. And, Zodiac wrote next to the Zodiac symbol with the X's........... (be sure to print the part I marked out on page 3 or I shall do my thing), so the extra X=95 that turns out as the next date (he at least alludes to) might express the threat left by the Zodiac symbol.

I think it would take an astronomical coincidence, no pun intended, for all the number matches or otherwise numerical allignment to be a coincidence. And then add to that that the next date is also a 95.....6/19/70(6+19+70)=95.

What are the odds of that meaning that a repeat 95 follows a 95 that Zodiac would probably consider an unsuccessful attempt? In other words, he still needed a 95 and what I mean by that is I believe the result Zodiac wanted was to put SIX LINES thru all 12 points of the Zodiac symbol, being symbolic of all 12 signs of the Zodiac.

So if you were to draw a straight line thru the Zodiac points(80,90,95,100,105) and their opposite counterpart, so to speak. In other words point 80 connects to point 110(or point 6 to point 12). Point 90 to point 60(point 8 to point 2). Point 95 to point 65(point 9 to point 3). Point 100 to point 70(point 10 to point 4). Point 105 to pint 75(point 11 to point 5).

So, here, at this point!, you only have POINT 85 to POINT 55(or point 7 to pint 1) to complete the ZODIAC CIRCLE and that is exactly what you get with the Donna Lass case and the date 9/6/70(9+6+70)=85!


Zodiac Symbol As A Clockface Murder Dates Number Theory 1st Round.jpg
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KITE: DES JULY AUG SEPT OCT=7 and then adding up those months numbers-for-letters=(DES+JULY+AUG+SEPT+OCT=223). So, that results as 223=7. And then that matches in that 2+2+3=7. I didn't notice this the first time, but by it being 2+2+3=7, that would be like Zodiac saying "See, you need to add the dates I left on the car door in that fashion". In other words:


AK WILKS: Kite - Good analysis! AweShucks - Great work! Looks very clean and concise. Both the clock chart and the postcard graphic get across complex information in a straightforward way.

If you can do one other version of the clock/zodiac symbol, I might suggest adding what I consider to be a possible "second round" of murders in 1970.

Judith Hakari = 3/7/70 = 80. 80 Divided by 5 = 16 - 10 = 6

Nancy Bennallack 10/25/70 = 105. 105 Divided by 5 = 21 - 10 = 11

Zodiac has been mentioned by police, reporters, Graysmith and researchers most often as perhaps responsible for these crimes in 1970:

Judith Hakari 3/7/70
Kathleen Johns 3/22/70
Richard Radetich 6/19/70
Donna Lass 9/6/70
Nancy Bennallack 10/25/70

Zodiac claimed credit for Johns, probably hinted at Radetich, probably hinted at Lass with the Pines card, possibily hinted at Bennallack with the "14" on the Halloween card. No hint that I am aware of at Hakari, but I do think he upped the victim count after her death.

All five match into the number theory. That is interesting, as all the four confirmed Z crimes also match into the number theory.

So it goes 9 for 9.

What are the odds of that???


Zodiac Symbol As A Clockface Murder Dates Number Theory 2cd Round 1970 Cases.jpg
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Re: Number Theory Linking Zodiac Murder Dates

Postby KITE on Tue Feb 14, 2012 10:32 am

The number theory possibly also known as THE MULTIPLES OF FIVE BY THE ZODIAC DATES is comprehensively shown in an excellent way in that post AK. Concerning the:
DES JULY AUG SEPT OCT=7 (223=7). That is left at the end of the 11/8/69 letter, while the 5 X's along the Zodiac symbol is left at the end of the 11/9/69 letter. So there may be some allignment there. And, there are 5 dates(Des July Aug Sept Oct) to match 5 X's(on the Zodiac symbol), a clue or possibly misdirection, but the 223=7(2+2+3=7), would be a huge coincidence considering the allignment and how it matches what you would do with the dates(7+4+69=80), for example. And, the inclusion of AUG and possibly upping the count from 5 to 7, might suggest alluding to the 8/3/69 date, another match (8+3+69=80), and then possibly 10 for 10.
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Re: Number Theory Linking Zodiac Murder Dates

Postby akwilks on Wed Feb 15, 2012 9:59 pm

ODDS WISE: If we say Zodiac just did the four confirmed murders, it is a 1 in 625 chance that all four create divisible by five numbers.

If we say Zodiac did the four confirmed and the five 1970 possibles it is 1 in 2,000,000.

Looking at what I call the Bentley Weekend Limitation Factor (BWLF), which I think is largely sound from both a criminal MO and math point of view, it changes the probability more than I thought.

Sticking to just the four confirmed cases, Doranchak stated and BigZ verified that the odds are 1 in 625 that four dates would be divisible by five. But that is based on 1 out of 5 dates being divisible by five, or 20%. Right?

But if we accept the BWLF we have 104 possible Fridays and Saturdays out of which only 18 are divisible by five. So I get that as about 17%. Which is about 1 in 6. Right? If so you would think there is little difference between a 1 in 5 chance or a 1 in 6 chance, at least I did, but you would be wrong.

Using a 1 in 6 chance with the BWLF, sticking to just the four confirmed cases, I come up with a 1 in 1296 chance that all four would be divisible by five.

If we keep the BWLF and apply it to all nine cases it is 1 in 10,000,000.

What do you make of this question from Aquiman?

Aquiman: One thing that throws a wrench into this theory for me is the tick mark at 9 o'clock. Z first claims 7 victims in the Dripping Pen card (11/8/69), where he states, "Des July Aug Sept Oct = 7". He then reiterates his claim in his letter to the Chronicle (11/9/69), where he states, "Up to the end of Oct I have killed 7 people." All of the 5 confirmed murders can be accounted for on the clock face: 6:00 (Ferrin), 8:00 (Stine), 10:00 (Jensen/Faraday), 11:00 (Shepard). The one we can't account for is the 9:00 position. I don't believe Z was predicting that this would be the "time" he would take his next victim; I would assume (if the number theory held true), that at least one of the two unconfirmed murders would have ALREADY accounted for the 9:00 position (and within the timeframe of Dec '68 - Oct '69).

So, if Z truly established a "system" starting with the Faraday/Jensen murders in "Des", then there are very few "murder dates" (that fall on a weekend) which would fill the 9:00 position on his clock:

Saturday, 1/25/69
Sunday, 3/23/69 (assuming Z was a "weekend warrior", so as not to confine his activities to only Fri or Sat)
Friday, 6/20/69
Saturday, 7/19/69

Note that none of those dates occur during the month of August, which has long been a question of who Z could have been referring to. According to the number system, the only possible weekend murder dates during August of '69 would have been:
Sunday, 8/3/69 (6:00 position)
Friday, 8/8/69 (7:00 position)
Saturday, 8/23/69 (10:00 position)

If Z were hinting that his next victim would fill the 9:00 position, then how do we account for August?

Edit: The only date in August that fulfills the number theory requirement to obtain "9" is 8/18/69 - a Monday.

This was my answer, but I am looking for any thoughts you or others have:

AK WILKS: The question is does Z mark the 9:00 position as a boast for what he has already done, or is it a threat of what is to come?

After this November 1969 letter with the marked clockface, Zodiac would take credit for the Kathleen Johns incident on 3/22/70, a "95" or position 9 on the clock, and she lived, and many think Zodiac hinted at credit for Radetich, who was killed on 6/19/70, also a "95" or position 9 on the clock. We are "missing" a mark on the 7, and the Donna Lass missing persons/murder case was on 9/6/70, an "85" or 7 on the clock.

Zodiac Symbol As A Clockface Murder Dates Number Theory 1st Round.jpg
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The whole "Des July Aug Sept Oct = 7" has always been confusing and one wonders if this is a Zodiac trick, misdirection or mind!@#$.

Kite has a theory that the cryptic "Des July Aug Sept Oct = 7" actually could mean a letters for number count, as he proposes on the Pines Card:

And these letters "Des July Aug Sept Oct" add up to 223, and 2 + 2 + 3 = 7. Now that is a little bit of an off beat take on it maybe, but it does add up and is worthy of consideration IMO.

If Zodiac is claiming, real or false, to have killed two people in August, many have looked to the Snoozy and Furlong murder. That happened on 8/3/69, so it is an "80" or 6 on the clock, right? A mentally ill person named Karl Warner killed Kathy Bilek in April of 1971 and then confessed to those 8/3/89 Snoozy and Furlong murders, and the facts seem to indicate he did it, but there is not a smoking gun fullproof conviction in regards to Snoozy and Furlong. I don't think Z killed those 2 girls, but did he want us to speculate he did? I don't know.

I am open to alternate ideas but the Z himself making those marks is IMO a very interesting bit of evidence, seeing as how they do tie in to the 4 confirmed cases and the "next two most mentioned possibles" with Johns and Radetich.
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Re: Number Theory Linking Zodiac Murder Dates

Postby akwilks on Mon Mar 05, 2012 5:23 pm

Still reading the 600 page book by Scott Bartz "The Tylenol Mafia".

There was an April 3, 1982 death of William Pascaul in Philadelphia but that is listed by police as a suicide, based on a letter he wrote, and Bartz accepts it as a suicide. That date would not fit the number theory, though I am not sure if he actually died on 4/3/82 (which appears to be the case) or 4/4/82. But it seems to perhaps be a suicide, though we do not have all the facts. But Bartz accepts it as a likely suicide as do the police, unlike the others listed below which he thinks are murders, which I agree also are murders and so do the police.

These are the 1982 Cyanide in Tylenol Murders:

7/26/82 Jay Mitchell Sheridan, Wyoming

8/15/82 Mary Louise Watkins Chicago, IL

9/14/82 Mark Husted Father was Village Attorney for Carpentersville, IL

9/29/82 Seven Victims Suburban Chicago

12/1/82 Galen Pariott Skokie, IL


Looking at these dates, what do they have in common?







7/26/82 = 115

8/15/82 = 105

9/14/82 = 105

9/29/82 = 120

12/1/82 = 95


All five 1982 Cyanide murder dates are divisible by five.

All four confirmed Zodiac murders fit the divisible by five number theory, as do all five possible 1970 cases.

Dornachak says and thebigZ confirms that the four confirmed cases would randomly happen that way only 1 in 625 times. If we think all five possible 1970 cases were either done by Zodiac or he claimed/hinted at credot for them, we are in the 1 in 2,000,000 range for random.

I think there is something here. I think Zodiac, for whatever reason, selected murder dates that were divisible by 5 and fit into his symbol, AS HE HIMSELF MARKED THE SYMBOL IN THIS WAY GIVING US A CLUE.

Zodiac Symbol As A Clockface Murder Dates Number Theory 2cd Round 1970 Cases.jpg
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Ted Kaczynski is a suspect in both the Zodiac killings (see and the Tylenol Murders.

Now it appears that there MAY be a link between this Zodiac Number Theory and the 1982 Chicago Tylenol Murders.

Interestingly, the Zodiac Killer said in one of his later letter letters that his future killing would be disguised as "routine robberies, killings of anger, + a few fake accidents, etc,", which fits the Tylenol Murders. At first, many of the 1982 and 1986 Tylenol and Cyanide deaths were thought to be of natural causes, accidental overdoses, suicides or killings by family members.

Zodiac was also suspected of killing a woman in Sonoma County, California named Caroline Davis by Strychnine poison, and in perhaps the first ever case of product tampering, the Fall 1969 sickening of a Martinez, California man named Daniel Williams of BUSH Street who drank Arsenic put into a Seven Up soda pop bottle.

In 1984 Thomas Dresser of Manitowoc, Wisconsin would die after drinking from a Coca - Cola bottle with Cyanide placed in it. An item found in Kaczynski's cabin indicated he may have visited Manitowoc, Wisconsin, as discussed above in the section on the 1986 deaths.

It appears the tainted Tylenol bottles were planted in the Chicago area starting late at night on September 27, 1982. Which is the 13th anniversary of the Zodiac attack at Lake Berryessa, California.
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Re: Number Theory Linking Zodiac Murder Dates

Postby akwilks on Tue Sep 25, 2012 12:12 am

Had some very interesting exchanges with doranchak of over at Morf's forum on the Zodiac Murder Dates Number Theory. This number theory was found by KITE and further developed by myself AK Wilks, Bentley and others, with work also done on probabilities by Aquiman/thebigZ. Thanks to Doranchak for doing this great work and having a good debate:

Doranchak: I can't remember if I wrote about this before. But this one still interests me. The low probability of the dates fitting the "divisible by 5" scheme is compelling. But I think we need to answer a broader question:

In a set of dates, what are the odds of finding any interesting pattern?

Adding the date values and seeing the if the sum is divisible by five is only one such pattern. But what about other patterns formed by all the different possibilities? For instance, divisibility by other factors. Or all possible numerological procedures.

Consider the birthday paradox. The birthday paradox question is: In a room with only 23 people in it, what are the odds that two people in the room have the same birthday? Intuitively, it seems like the chances would be very low. But actually there's about a fifty-fifty chance. Why? Because you are looking for any two people that share the same birthday, and the birthday can happen any time of the year. And there are 253 possible pairings of the 23 people in the room. The details of the full calculation can be found here: ... robability

So, back to this "divisible by 5" problem. We've already calculated the odds of all the dates having the divisible by 5 property. This is like calculating the odds that two people have July 4 as their birth date. But we need to examine the more general question: What are the odds of the dates having ANY interesting pattern? This is like calculating the odds that two people have the same birth date, where the date can be selected from any day of the year.

I think it would be interesting to try to come up with this calculation. Anyone want to give it a try?

AK WILKS: Doranchak both you and thebigZ did look at the probabilities on this, and we debated what they may mean, but we are all in agreement on the actual probabilities. Which were 1 out of 625 for four murders.

I think you have the right idea though, in asking the question, what are the odds that ANY pattern - divisible by five or three or four or some other thing, appear in four random dates converted to numbers and in nine random dates converted to numbers.

I did look at the murder dates of other serial killers and saw no pattern of divisible by five, and no readily apparent other pattern, but I did not look intensely for other possible patterns.

Doranchak: Let's consider the simpler matter of adding the numbers in the dates, then determining if the result is divisible by some number. We'll limit this to four dates for now for simplicity.

Now we must decide which numbers are significant. If all four dates result in numbers divisible by two, is that significant? What about three? Four? Etc.

Let's start with two for now. The odds of four dates producing numbers that are divisible by two is 1/(2*2*2*2) = 1/16.

The odds of four dates producing numbers that is divisible by three is 1/(3*3*3*3) = 1/81.

Four dates divisible by four: 1/256.

Four dates divisible by five: 1/625.

And so on. So, what are the odds of four dates all being divisible by SOME number? For example, if you picked four dates completely at random, what are the odds that they will produce numbers that are all divisible by some specific number, not necessarily the number 5?

The answer: 1/16 + 1/81 + 1/256 + 1/625 + ... etc ...

This can be expressed with a simple formula: sum( (1/n)^4, n from 2 to 100). Wolfram alpha gives us the final answer: 8.2%. (I stopped at 100 because it felt like a good stopping point. Going further than 100 doesn't change the odds by much.)

So, if you pick any four dates, there's an 8.2% chance that they produce numbers that are all divisible by the same number.

Now, that's the result when we consider only ONE numerological method: adding the pieces of the dates, then inspecting the sum for divisibility. But what if we include other methods, such as further manipulating the sum by adding ITS digits? For example, today is 9/13/12. 9+13+12 = 34. But we can say 3+4=7. Or what if multiplied the numbers instead? Or added the first two and subtracted the third? Etc. So, each of these possibilities is a variation of the original method, and produces a different set of numbers to inspect for divisibility.

If we assume that a second numerological method exists, that can produce a different sequence of numbers from our original method, then it, too, has a 8.2% chance of finding a pattern in the same four dates. Thus, unless I'm mistaken, the chances of finding a pattern by applying our first OR second method increases to 16.4%.

With enough unique methods of producing numerical sequences, the odds seem to go up quite significantly of finding arbitrary patterns.

Let's increase this from 4 to 6 dates. Here are the re-calculated results:

If you pick any six dates, there's a 1.7% chance of finding a pattern using one numerological method, and a 3.4% chance when using two.

If you pick any nine dates, there's a 0.2% chance of finding a pattern using one numerological method, and a 0.4% chance when using two.

TheBigZ (and everyone else): lemme know if I made any mistakes in my analysis.

AK WILKS: Thanks very interesting! I am going to look through this in detail tomorrow but I have one question or objection first. Since every number must be odd or even I dont think it is fair to include divisible by 2 as a potential number patter it is just too common and too likely to happen by chance. Could you run one analysis without it? Otherwise looks solid in conception and execution thanks again I will have more to say after studying this work.

Doranchak: That's a fair objection. Re-calculating:

The odds of 4 dates forming numbers divisible by another number between 3 and 100: 2.0%
The odds of 6 dates forming numbers divisible by another number between 3 and 100: 0.17%
The odds of 9 dates forming numbers divisible by another number between 3 and 100: 0.0055%

I wrote a quick program to run some experiments. The program generates a list of random dates, then looks for patterns among the dates.

Each experiment generates 10,000 sets of 4, 6, and 9 dates, randomly selected from the years 1968, 1969, and 1970.

In the first experiment, the program checks if the dates form sums that are divisible by some number greater than 2. For example, the date 12/23/69 forms this sum: 12+23+69=104. 104 is divisible by 4. Here are the results: ... tems-1.txt (1.9% form patterns) ... tems-1.txt (0.19% form patterns) ... tems-1.txt (0.02% form patterns)

The predicted results were 2.0%, 0.17%, and 0.0055%. So the experiment wasn't too far off. I think the small difference is accounted for by the fact that sums divisible by 5 within the selected date range, for example, aren't precisely 1 in 5, but close to it. Needs verification though.

Anyway, the first experiment only cared about two-digit years. What if we looked at two-digit AND four-digit years? For example, the date 2/7/68 forms the sum 2+7+1968=1977, which is divisible by 3. The results show, as expected, that more patterns are easily found: ... tems-2.txt (3.65% form patterns) ... tems-2.txt (0.44% form patterns) ... tems-2.txt ((0.03% form patterns)

OK. So this is starting to reinforce the idea that if you go looking for patterns using numerology, you're likely to find them. In the next experiment, I included all the following variations of patterns:

* Adding the date parts together and looking for common factors in the sums
* Multiplying the date parts together and looking for common factors in the sums
* Allowing the day of the month, or the month, to be subtracted instead of added.
* Allowing the year to be two digits or four digits
* When looking at the sum or product, continue to add its digits until a one-digit value remains. For example: 12/21/68 becomes: 1968*12*21=495936 [4+9+5+9+3+6=36] [3+6=9]
* Seeing if all the dates fall on a U.S. holiday
* Seeing if all the dates fall on the same day of the week
* Seeing if all the dates fall on the weekend
* Seeing if all the dates are on the same day of the month

Results: ... ms-all.txt (44.9% form patterns) ... ms-all.txt (20.6% form patterns) ... ms-all.txt (8.5% form patterns)

There are many other numerological systems I haven't tested. For example, what if we simply added the digits of the dates: 7/20/1970 becomes: 7+2+0+1+9+7+0 = 26, 2+6=8. Or you could multiply the individual digits (except for zero). Or you could shift the numbers by some fixed amount before computing the sum. Or you could convert the dates into their names, then convert all the letters of the names into numerical equivalents and perform arithmetic on them. Or you could leave out the day or the month when computing the sum or product. The possibilities just keep going. The discovery of any such pattern for a given set of dates would seem fantastic in isolation, but clearly you have to consider all the possible patterns that can be found.

Still, I cannot conclude that the killer definitely did NOT intentionally pick the dates to fit the divisible-by-5 pattern. I'm just saying that in light of this experiment, there is more room for doubt.

AK WILKS: Thanks!

Very interesting results.

Personally I would tend to reject results with a four digit year, as Zodiac used a two digit year on the car door.

I also tend to reject your last "everything and the kitchen sink" category, reducing dates to a single digit creates too many patterns too easily, and as simple things like work schedules could cause dates to fall on the same day or weekends, thus too easily showing patterns when none were intended.

So far me the most valid study is below, facing the central issue of random occurence of a pattern of divisible by a number larger than two.


In the first experiment, the program checks if the dates form sums that are divisible by some number greater than 2. For example, the date 12/23/69 forms this sum: 12+23+69=104. 104 is divisible by 4. Here are the results: ... tems-1.txt (1.9% form patterns) ... tems-1.txt (0.19% form patterns) ... tems-1.txt (0.02% form patterns)

AK Wilks: Then the question becomes, what is significant?

A 2% chance, something that happens randomly once out of 50 tries. I find that right on the borderline between moderately interesting to potentially significant. But it probably is not enough to catch the eye or major interest of law enforcement or most Zodiac researchers. It could just be chance and most will write it off that way. And they may be right.

Your work does show that patterns with numbers can happen by chance more than one might think.

So this whole theory probably will lose in the court of public opinion, and the results are just not spectacular enough, so I probably will not show them to law enforcement. Also, even if I or someone did, it is not clear what if anything law enforcement could do with the information.

But I happen to think based on both Zodiac's claims/clues and the evidence, the Zodiac did likely kidnap Johns and Lass.

Thus the results for SIX instances of just 0.19% happening by chance I think is significant to very significant. There is only a 1 out of 500 chance of a divisible by number pattern happening in six random dates, but it DOES happen in these six Zodiac crimes - the four confirmed plus the two possible/probables of Johns and Lass.

So IMO it is likely that this number pattern was intentional by Zodiac.

Also, doranchak/thebigZ/everyone, what if anything do you make of what KITE found of Zodiac himself marking numbers on his symbol that CAN MATCH TO THE MURDER DATES NUMBER THEORY?

Look at the 6, 8, 10 and 11 that Zodiac marks are a possible mathematical match to the four confirmed murders.

12/20/68 = 100 @ 10 spot. Because 100/5 = 20 and 20 - 10 = 10.

7/4/69 = 80 @ 6 spot. Because 80/5 = 16 and 16 - 10 = 6.

9/27/69 = 105 @ 11 spot. Because 105/5 = 21 and 21 - 10 = 11.

10/11/69 = 90 @ 8 spot. Because 90/5 = 18 and 18 - 10 = 8.

Is there a better, simpler or more logical way to express what I am trying to say mathematically above?

Also the next cases most often mentioned as possible Zodiac crimes - Hakari, Johns, Radetich. Lass, Bennallack - they ALL fit the number theory. Something your work shows is mighty rare indeed to happen by chance.

Jem = "My thoughts -
Divisibility by 2 - Why you guys think it's okay to toss this?"

AK - Doranchak did do a calculation with divisible by two dates and you can see the results. I argued to him that since every number is odd or even, and all even numbers are divisible by two, could he also do a calculation only with numbers divisible by three or higher. Otherwise IMO you get way too many unintended patterns from numbers divisible by two. Doranchak agreed that perhaps this was a fair point and was nice enough to do a calculation without the divisible by two.

Jem - Kitchen Sink method - with this, you can use ANY number of dates and get the probability as close to 100% as you like. So the thing is, what's an interesting pattern? It should be something simple, something that could or should have been figured out a long time ago, imo. Maybe Z hoped someone would notice and this would cause public alarm? Dunno.

Divide by 5, subtract 10 - I'm not quite comfortable with this. Too complicated. Maybe Dave can figure out another equation that works better, as AK suggests? Or maybe it has nothing to do with the marked Xs.

AK - I agree. I see the pattern, and it seems to me to perhaps be intentional. Yet I am still not comfortable with it! It is a little too elaborate, and Doranchak has shown number patterns can happen by chance more than you may think. To say the pattern is add the numbers in the dates, divide by 5, then minus 10 - well it seems too contrived! It works, yes it does do that. But what does it mean? Why that formula?

The only thing I can suggest is that IF Zodiac viewed his symbol as a clock face, then the formula above would render dates that add up to numbers between 70 and 141 into numbers 1 through 12.

Jem - Five Xs on the diagram - is that a clue meant to point to a divisibility by 5 method of committing murders? As well as whatever else they might mean?

AK - I never caught that! Five marks means divide dates by five! Could be...he had only done four crimes by that point, but makes five marks.

Jem - Weekend dates - probably a good idea to count the number of weekend dates (Fridays, Saturdays and Sundays) in the relevant time period for the 4 confirmed Z crimes. Just to see if the probability is what you'd expect, or higher or lower.

AK - As he indicated, Bentley already did this. If you say Zodiac only did crimes on weekends, then the odds of a divisible by five date go from one in five to about one in six. It changes the odds of four random dates hitting divisible by five from 1 in 625 to about 1 in 1300. A significant change IMO! However using the doranchak calculation of all numbers above two being divisible changes the 1 in 625 to 1 in 50. A similar ratio and reduction would apply to what I dubbed the Bentley Weekend Limitation Factor or BWLF.

Doranchak: I was curious about this question: How many ways are there to select 4 random dates that have these qualities:

* All four dates form sums that are divisible evenly by 5
* When the same number is subtracted from each sum, 4 out of the 5 clock face positions are selected. The 5 possible clock face positions are 6, 8, 9, 10, and 11.

Here's how I tried to answer this question. Warning: Lots of math and combinatorics. The executive summary: There is reason to believe the pattern is just coincidence, so we need more evidence to confirm the pattern. Here's the full analysis:

First, let's look at this date range: 1/1/68 through 12/31/70. There are 1,095 days between those dates (including 1/1/68 and 12/31/70). Think of the 1,095 days as a big deck of cards. How many ways are there to select 4 cards from the deck, where order doesn't matter? (For example, picking 7/4/68 and 8/1/70 is equivalent to picking 8/1/70 and 7/4/68)

Answer: 1,095 choose 4 = 59,574,855,795 (about 59.5 billion).

How many of those 1,095 days add up to numbers that are divisible by 5? Answer: 217, which is very close to the expected 20% of the 1,095 days. You can see the full list here.

So, we now have a smaller deck of 217 cards. How many ways can we select 4 cards from this smaller deck?

Answer: 217 choose 4 = 89,857,530 (about 90 million). Notice that this works out to about 1 in 663 of all possible date selections, which is close to the 1 in 625 that was predicted previously.

Now we want to know this: How many of these 90 million "hands" will select four out of five of the following clock positions: 6, 8, 9, 10, and 11?

First, the way we select a clock position is by looking at the sum formed by the date parts. For example, 7/4/69 = 80. And 80/5 = 16. To reach the clock position 6, we subtract 10 from 16. I'll refer to 10 as the "offset".

There is no reason to assume that 10 is the only possible offset value. We have to consider all offset values that can produce four out of five of the clock positions. Here is an example:

Consider the clock positions 6, 8, 9, and 10. Let's say we have dates that form the sums 70, 80, 85 and 90. If we divide those sums by 5, we get: 14, 16, 17, and 18. If we use an offset of 8, we get: 14-8 = 6, 16-8 = 8, 17-8 = 9, and 18-8 = 10. Therefore, we matched the clock face sequence 6, 8, 9, and 10 by using an offset of 8.

How many ways are there to do this?

Observe that the smallest sum that can be produced in the range 1/1/68 through 12/31/70 is 1+1+68 = 70. This is because there is no month smaller than 1, no day smaller than 1, and no year smaller than 68. Similarly, the largest sum that can be produced in the same range is 12+31+70 = 113. Thus, the range of sums that are divisible by 5 is: 70 through 110. Therefore, every sequence of sums will fall in this range. We can look at every possible sequence of sums, and count up how many combinations of dates produce them.

To do this, let's create a bucket for each sum. Each bucket is assigned a number from 70 through 100. So we have nine buckets labelled 70, 75, 80, 85, 90, 95, 100, 105, and 100. If we go through each date between 1/1/68 and 12/31/70, we can compute its sum and place the date in the appropriate bucket. For example, 1/1/68 will go into the "70" bucket. After we do this, we can count up how many dates are in each bucket:

Bucket 70 has 1 date in it.
Bucket 75 has 15 dates in it.
Bucket 80 has 30 dates in it.
Bucket 85 has 36 dates in it.
Bucket 90 has 36 dates in it.
Bucket 95 has 36 dates in it.
Bucket 100 has 34 dates in it.
Bucket 105 has 22 dates in it.
Bucket 110 has 7 dates in it.

In AK's example, the dates formed this sequence of sums: 80, 90, 100, and 105. How many unique combinations of dates will produce that sequence? The answer is the product of counts for each bucket:

(count of dates in bucket 80) * (count of dates in bucket 90) * (count of dates in bucket 100) * (count of dates in bucket 105) = 30 * 36 * 34 * 22 = 807,840 combinations.

Finally, we need to tally up those combinations for every possible sequence of sums:

Clock face selections: 6,8,9,10:
70,80,85,90 offset 8 1*30*36*36=38,880
75,85,90,95 offset 9 15*36*36*36=699,840
80,90,95,100 offset 10 30*36*36*34=1,321,920
85,95,100,105 offset 11 36*36*34*22=969,408
90,100,105,110 offset 12 36*34*22*7=188,496
total: 3,218,544

Clock face selections: 6,8,9,11:
70,80,85,95 offset 8 1*30*36*36=38,880
75,85,90,100 offset 9 15*36*36*34=660,960
80,90,95,105 offset 10 30*36*36*22=855,360
85,95,100,110 offset 11 36*36*34*7=308,448
total: 1,863,648

Clock face selections: 6,8,10,11:
70,80,90,95 offset 8 1*30*36*36=38,880
75,85,95,100 offset 9 15*36*36*34=660,960
80,90,100,105 offset 10 30*36*34*22=807,840
85,95,105,110 offset 11 36*36*22*7=199,584
total: 1,707,264

Clock face selections: 6,9,10,11:
70,85,90,95 offset 8 1*36*36*36=46,656
75,90,95,100 offset 9 15*36*36*34=660,960
80,95,100,105 offset 10 30*36*34*22=807,840
85,100,105,110 offset 11 36*34*22*7=188,496
total: 1,703,952

Clock face selections: 8,9,10,11:
70,75,80,85 offset 6 1*15*30*36=16,200
75,80,85,90 offset 7 15*30*36*36=583,200
80,85,90,95 offset 8 30*36*36*36=1,399,680
85,90,95,100 offset 9 36*36*36*34=1,586,304
90,95,100,105 offset 10 36*36*34*22=969,408
95,100,105,110 offset 11 36*34*22*7=188,496
total: 4,743,288

Grand total: 13,236,696. This means that of all possible selections of dates, about 13 million of them will select 4 out of 5 of the clock face positions.

Therefore, if you look at any set of 4 dates, where they all form sums divisible by 5, there is a (13,236,696/89,857,530) = 14.7% chance of stumbling upon a selection of 4 out of 5 of the clock face positions.

Clearly, if we consider other divisibility factors besides 5, and variations of the numerological scheme, the numbers will be completely different. And the chances of finding a pattern will just keep going up.

Conclusion: The analysis shows that the chance of selecting 4 out of 5 clock face positions (from among dates that are divisible by five) is high enough to be a coincidence, especially if we consider introducing variations to the divisibility factors and numerological scheme. If Zodiac can be definitively linked to more crime dates that fit the pattern, then the chances will decrease. At this point, I think there is too much doubt to know for sure if the pattern is real, without confirming it with more evidence.

(As always, I welcome questions, corrections, and criticisms of the analysis)

AK WILKS: Great work. Very detailed and excellent in conception and execution.

I see no flaws and agree with most of your conclusions even.

To be clear the odds of four random dates hitting a divisible number pattern is 2%, while the odds of Zodiac mentioning four correctly in five dates marked is almost 15%. Correct? I agree 15% is too high to rely much upon. Could just be chance.

If I may ask one thing - lets assume Johns was a victim. Zodiac does mark the four confirmed dates plus a number that matches to the Johns date. Agreed?

If possible can you calculate the odds of getting 5 out of 5 right randomly? Thanks.

Doranchak: Yes - I determined the probabilities of two events: 1) The probability of picking four dates that have any divisibility pattern, and 2) The probability of selecting four out of five clock face positions when the set of dates that are divisible by five are already selected.

In my calculation for #2, I only considered one factor (the number five), and one numerological scheme (offset subtracted from the sum divided by five). Making matters worse is the fact that there are many possible variations to both numerological schemes (the one producing the sum in the first place, and the one that produces the clock face positions). Including those variations will greatly increase the chances of stumbling across patterns.

Sure -

Calculation for 5 matches:

How many ways to select 5 dates from all 1,095 dates? Answer: (1,095 choose 5) = 12,999,233,534,469 (about 13 trillion)

How many ways to select 5 dates from the 217 dates, so all 5 dates are divisible by 5? Answer: (217 choose 5) = 3,827,930,778 (about 3.8 billion)

How many of those fit all of the clock face positions 6, 8, 9, 10 and 11?


70,80,85,90,95 offset 8 1*30*36*36*36=1,399,680
75,85,90,95,100 offset 9 15*36*36*36*34=23,794,560
80,90,95,100,105 offset 10 30*36*36*34*22=29,082,240
85,95,100,105,110 offset 11 36*36*34*22*7=6,785,856
Grand total: 61,062,336

61,062,336 / 3,827,930,778 = 1.6%.

Once 5 dates are selected that are divisible by 5, there is a 1.6% chance of accidentally selecting all clock face positions using this scheme.

AK WILKS: "Once 5 dates are selected that are divisible by 5, there is a 1.6% chance of accidentally selecting all clock face positions using this scheme."

AK Wilks: Well I must admit I do find that potentially interesting. I think a 1.6% chance is rather low. Of course I do think Johns was a Zodiac victim - if others do not, than this does not apply. I also think Lass was a probable Z victim, and the "7" spot is noticably missing here in the chain Z marked of 6, 8, 9, 10 , 11. That is debatable. I also think it is POSSIBLE that Hakari, Bennallack and Radetich were Z victims, and I think it is PROBABLE that Z hinted at credit foe Radetich, Lass and Bennallack. And ALL of those - Johns, Hakari, Radetich, Lass and Bennallack - fit into BOTH the number pattern and marks made by Zodiac on his symbol (with the exception of Lass, who is perhaps notable for the missing 7 spot).

So I still lean in favor of their being something to this number theory, though I accept that the numbers, ranging from low to not so low, will not be exceptional enough to convince most researchers or anyone in LE.

So I don't know where to go with this. Doranchak does show number patterns happen more by chance than you may think, so there are good grounds for skepticism here too.

But I am still impressed by Zodiac marking numbers that match to the four confirmed cases + Johns, and also that the next 1970 cases mentioned as possible Zodiac all fit.
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Re: Number Theory Linking Zodiac Murder Dates

Postby KITE on Tue Sep 25, 2012 7:59 pm

AK, I have the odds of Zodiac "coincidentally" committing or otherwise claiming murders that fell, not only on dates adding up to being divisible by FIVE, (and this is why my number is so high), "NOT ONLY divisible by five", I repeat, but ending up in a 80-85-90-95-100-105 pattern, at:
15,187,500 to 1.
YES THAT IS 15 MILLION and change!
THIS ISN'T ABOUT ZODIAC ACTING ON DATES THAT WERE DIVISBLE BY FIVE. IF THIS IS ALL THAT WAS ABOUT, I'D TAKE THE 625 to 1, put it in my back pocket, yawn, and move on. Those numbers are all different! (80-85-90-95-100-105)! And this is all interwoven with the SIX signs of the zodiac whose numbers-for-letters are:

Now add all those to ZODIAC=58 (numbers-for-letters)


Now when you put these numbers to the points of a MULTIPLES of FIVES Zodiac symbol, in other words subtract 60 from the 4 under 120(Leo,Libra,Aries,Gemini) and subtract 120 from the other 2(Capricorn,Aquarius). That just brings it down to the lowest rotation (a multiples of five Zodiac symbol goes to 60). That gives you 30,35,40,45,50,55 and those are POINTS 6,7,8,9,10,11 (You don't have to subtract the numbers, otherwise just go around the Zodiac symbol/circle until you reach the point of the numbers, still resulting in POINTS 6,7,8,9,10,11.
SO I ACTUALLY BELIEVE THE X's ZODIAC LEFT ON THE ZODIAC SYMBOL WERE MEANT TO REPRESENT THOSE SIX SIGN OF THE ZODIAC. In other words, I believe it's a MULTIPLES OF FIVE Zodiac symbol. But then the SIX homicide dates were meant to essentially mirror that pattern.

And then also: (numbers-for-letters)
VALLEJO(77)+ZODIAC(58)=135x2=270 (2 homicide dates for Vallejo)

Giving the 3 points in degrees for a circle: 90,180,270

But back to how I got 15 MILLION AND CHANGE.
The way I would pose the question is: WHAT ARE THE ODDS OF ZODIAC COINCIDENTALLY ACTED ON DATES THAT ENDED UP ADDING UP TO:80-85-90-95-100-105.(7/4/69)(9/6/70)(10/11/69)(6/19/70)(12/20/68)(9/27/69) So, I ask that, but I also want to factor in a SEVENTH date(3/22/70) with the idea that Zodiac intended to act on this (95) but was unsuccessful and then followed again with a (95) claiming to act on 6/19/70.
So, remember with each date locked in, the number of remaining days for the other numbers gets smaller. So, I don't see it as (1 in 5) continuously. I see it as:
1 in 5 (12/20/68)
1 in 6 (7/4/69)
1 in 7.5 (9/27/69)
1 in 10 (10/11/69)
1 in 15 (3/22/70
1 in 15 (6/19/700
1 in 30 (9/6/70)

(5x6x7.5x10x15x15x30)= 15,187,508 (15 MILLION, 187 THOUSAND, 508). !!!! IN MY OPINION.
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Re: Number Theory Linking Zodiac Murder Dates

Postby KITE on Wed Sep 26, 2012 1:00 pm

In a previous post I estimated the odds of a 80-85-90-95-100-105 pattern to the Zodiac homicide dates at around 15 MILLION TO ONE.
AK, you said things like (so I don't know where to go with this) and (so there are good grounds for skepticism here too).
Seriously? 80-85-90-95-100-105
You've looked at other serial killer cases for comparison, correct? Look at Jack the Ripper, the commonly accepted FOUR dates and I'll throw in an earlier possible date:
Not even close to the crisp 80-85-90-95-100-105. That pattern is crisper than Canadian air blowing into Montana in January, if you ask me.
Zodiac was different than most other serial killers in his prolific "game-playing" with his ciphers, and remember he uses EQUAL signs and refers to RADIANS. And so it is this "game-playing" serial killer who just happens upon a 80-85-90-95-100-105 homicide pattern?
Speaking of game-playing and odds, what are the odds of the 3 Zodiac locations for the 4 Zodiac homicide date, when adding their numbers-for-letters to the word ZODIAC numbers-for-letters(58), all equal out to 3 different points of a circle in degrees?
VALLEJO(77)+ZODIAC(58)=135X2=270 (2 Vallejo dates)

90,180,270 POINTS OF A CIRCLE. I estimate the odds of that at ONE BILLION TO ONE. Well, I did that sort of quickly but I'll stick to that estimate for now. And, it's not as if I'm pulling the ZODIAC and the CIRCLE out of thin air. Even if I had, the numbers would still be completely amazing, BUT THIS GUY CALLED HIMSELF THE ZODIAC AND SIGNED HIMSELF WITH A CIRCLE, with 12 points.
Remember that Zodiac struck on July 4th.
Remember that Zodiac possibly offered the anagram (I Robet Emet The Hipie) in the 408 cipher final 18. With the historical Robert Emmet being a revolutionary, that possibly meant that Zodiac thought it was his destiny to rile up the people of that time and place into a state of revolution.
The Unabomber wrote in The Manifesto something along the lines of how various resisters to the system put pressure on the system. Zodiac may have thought something similar, that with his murderous action and publicity seeking, he was putting pressure on the system, possibly to the point, to the brink of revolution. That might be considered the main message of the Manifesto: Revolution against technology and the system.
And let's not forget the 340 cipher, columns FIVE and TEN, no less. (KEPT BARS LEASH)(ATAK LOSE). Attack the restraints. Revolution.
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Re: Number Theory Linking Zodiac Murder Dates

Postby KITE on Tue Oct 02, 2012 1:29 pm

Don't get me wrong in any way, AK, I enjoy reading analysis of odds and that sort of thing. It's definitely very interesting and a sound approach, in my opinion. Of course, my opinion is qualified by my qualifications (or lack of), you could say. In other words, I'm not particularly educated in that area of odds analysis, probability, and the like, and so some of the calculations might be too complicated for my comprehension.
But even as I plead inferiority to the complication, I don't sigh
Because this kite don't need rocket science to fly

So, to my way of looking at things, a pattern concerning numbers or letters from the Zodiac case can be subjected to analysis and then the observation and the results can be navigated by common sense through the circumstances of the Zodiac case. For example, you have a 80-85-90-95-100-105 possible pattern that has resulted from a serial killer that seems to enjoy playing games, so I think that makes the possible number pattern all the more suspicious. In other words, is that yet ANOTHER game from Zodiac? And then with multiples of FIVE(80-85-90-95-100-105), you might further surmise a connection to a clock face with it's one thru twelve, multiples of FIVE minutes pattern. And to be continually kept in mind is that this guy has called himself the Zodiac and signed himself with a 12 pointed circle. So there may be circumstances that might seem to jump out and cry that these numbers must be intentional but analysis could throw caution on counting unhatched chickens, because patterns can result from chance. For example, that birthday problem did really surprise me to learn that only 23 were necessary to establish 50/50 odds. But, on the other hand, let's not forget the temptation that exists to too loosely associate names, numbers, and letters ETC.
For example, and I'm just making this up, but imagine a claim is made that Zodiac might actually be in his EIGHTIES, age-wise, a suggestion based on the 80 result for 7/4/69 and that the location Blue Rock Springs (BRS), minus EIGHT from each is TJK, and the real Zodiac knew of Ted Kaczynski. That along with the 3 EIGHTS in the 13-code and let's not forget the Zodiac diction use of SHALL can remind one of Jack the Ripper and he struck in 1888.
Now, at this point, my opinion and my reply would be that with each and every additional "epiphanous" eight-related observation entertained, you have gradually gravitated to grazing in the gray grass of a free and fast fanciful flitting and freewheeling association with the facts when perhaps it was preferential to refrain in the first place. I might also suggest that my good Southern sensibilities won't allow me to equationally entertain what I might regard as disingenuous numerical technicalities. So, in other words, that could be an example of observing a number pattern and then possibly fetching too far afield for it's possible meaning. There are enough EIGHTS in the Zodiac case to draw interest, in my opinion, but the link to Zodiac being 88 years old is the loose association. Of course, I used an obvious example to illustrate what might be a far finer point. But, that's just a made-up example and I'm not saying anyone is doing that with the 80-85-90-95-100-105 pattern, although I know I've been guilty of too loosely associated on previous occasions. But, perhaps I needed not write any of this and was better served and best expressed to simply repeat something I wrote in an earlier post:
80-85-90-95-100-105 SERIOUSLY?
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Re: Number Theory Linking Zodiac Murder Dates

Postby akwilks on Wed Oct 03, 2012 12:47 pm

What I meant was that in my experience most police detectives are not very interested in possible cryptic clues. Which is unfortunate as I think it is clear Zodiac used them. So I don't know where to go with this - anyone who wants to contact police or media can.

What Doranchak showed was that when you factor in ANY pattern - divisible by 3, or 4, or 5, etc. - the numbers do go up. Meaning the odds go up. Which IMO is something to consider. But even Doranchak's number of 1.6% is IMO still impressive. Things that only happen about 1% of the time by chance do NOT frequently "just happen."

When you add to that all the elements mentioned by Kite, such as two independent events, the date pattern appearing and Zodiac marking the numbers that match on the symbol, plus the bunching noted by Kite, you do get into the one in a million area, and even lower.

I think the number theory is likely valid.

Also nobody has yet come up with an alternate plausible explanation of the markings by Zodiac.
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Re: Number Theory Linking Zodiac Murder Dates

Postby KITE on Thu Oct 04, 2012 12:29 pm

The thing is, when you say ANY pattern, and I could be mistaken, but isn't an 80-85-90-95-100-105 pattern more difficult to get than an 80-82-84-86-88-90 pattern, by TWO, for example? I would certainly be intrigued by that TWO pattern nonetheless especially if there were a co-ordination and consistency with other factors as I believe exists with the FIVE pattern. But, by using numbers added from dates anchored by 68, 69, or 70, getting 80, for example, would require smaller numbers like 7-4-69 and 100, for example, needs larger like 12/20/68. In other words, there's going to be a place like, let's say, 82-96, where a lot of the numbers would fall into by chance. So, in other words, getting 85-86-87-88-89-90 would seem easier to arrive at then the more spread out 80-85-90-95-100-105, considering that particular anchoring of 69 and 70? At any rate, I think I will start a new topic (Is This The Game Zodiac Was Playing?) to put all the numbers into a nice, neat, easy-to-follow arrangement because I believe all of the numbers go thru the Zodiac symbol with the alluding of going around a circle as a promotion of revolution(since going around a circle once is called a revolution). (When I say all of the numbers, I mean also that the 0,3,6,9 and 0,10,20,30 Caesars fit on the points of the Zodiac circle, the same as 0,90,180,270).
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