There seems to be little interest here in the above Bayesian cryptanalysis approach, but Z32 is important and appears to be solvable in this or a similar method, so I felt morally obligated to revisit it. This post is pretty long, but converges on a high-confidence solution even without, as it turns out, requiring the message to be in depth.

Z32 is important because DNA and fingerprints may be preserved in the internal parts of Zodiac’s contraption, the components of the device may possibly be traced back to Zodiac (his claim, not mine, but probably no longer relevant), and a 50-year-old dud can still be dangerous, e.g., to construction workers. The cipher might appear hopelessly difficult, but not if you agree with me that the transposition rule is known (a diagonal step; see image below; click any image for its high-resolution version) and Zodiac sent clues to facilitate the solution. We could also assume that the message is in some depth (i.e., partly uses previous keys, which turns out to be true), but this will not be essential for the derivation. More generally, none of Zodiac’s ciphers should seem particularly difficult to an experienced cryptanalyst if the exact reading rules are already known, as Zodiac made sure we have sufficient information on the content of the shorter ciphers. (It is unfortunate that Zodiac chose crazy rules for Z340 without realizing the implied exponential difficulty of decryption, and provided only a general, obscure hint on the main transposition in the ambiguous ZF symbol.) Below, I assume that my analysis of other Zodiac puzzles (see recap) is correct; in particular, the diagonal transpositions are used in the derivation; however, the resulting Z32 solution itself does not strictly depend on those puzzles and solutions.

Some necessary background. Zodiac arguably wanted his ciphers solved, and so provided hints, including several clues for Z13 and two known clues for Z32 (explicitly, when writing “radians+#inches” in a subsequent communication, and hidden, in one of the three transpositions in the ZF symbol). In particular, a solution to Z32 leading to a bomb should have interested Zodiac as demonstrating his technical prowess and intimidating his audience. Zodiac must have assumed that we could solve the three later ciphers once we understood all his clues. He was correct in this regard about Z13 (if you agree with my twice–derived solution) and partly also about Z340: if its exact reading rules were known at the time, it would have been arguably solved in a matter of days or weeks. Indeed, Zodiac deliberately made Z13 easier by introducing many multiplicities (avoiding most homophones in Z13 indicates some understanding of the cryptanalysis challenge imposed by such a short cipher), but used many homophones in Z32, so it must have looked reasonably solvable to his unprofessional eyes. Therefore, if we identify and understand all the clues concerning Z32, we should be able to solve it now.

The problem is that Zodiac’s clues are convoluted, and a few were understood only after no longer being needed. Several of them appear in the Halloween card, but are difficult to distinguish from unrelated taunts. We already found some cleverly hidden elements in the card, like "sorry no cipher", so it is worthwhile mentioning possible additional clues, even if some may sound implausible to you. The effort of deliberately adding the pumpkin to the card appears to go beyond the holiday spirit, so it looks like a clue that I don’t understand. The emphasized word “slaves” is a Z13 reference, but “paradice” (sic) seems like an unnecessary addition, and so a possible Z32 hint. Zodiac drew attention to the word “knife” by inverting the letter N (while leaving the N in the adjacent “gun” untouched), so another possible Z32 clue. The “Peek-a-boo” could be either a clue or a taunt. People have pointed out that the card appears to reference the Tim Holt #30 comics, where redmask, tied by rope to a water mill, escapes death to take his revenge with fire; however, it’s difficult to see where this leads. There might be other clues, and now that we know the purpose and clever planning of the card and understand some of its elements, a local or otherwise knowledgeable person might realize their significance in the correct context for the place and time.

According to Zodiac’s descriptions of the bomb, we should expect to find it in the periphery of a town (neither in a busy urban region nor on an isolated road), where Zodiac could work relatively safely without drawing too much suspicion, along a road sufficiently central to see school-bus traffic but not too busy or with many lanes, and with an elevated wayside at least on one side of the road (a tree on the other side would suffice for a mirror if used). I have argued that the most plausible interpretation of the emphasized phrase “by knife” is the green ridge known as ‘@37.7424284,-122.0066273,13z/data=!4m9!1m2!2m1!1sknife!3m5!1s0x808f92a650d74a99:0x859c211a8f3242c0!8m2!3d37.7424284!4d-122.0066273!15sCgVrbmlmZVoOCgVrbmlmZSIFa25pZmWSAQVyaWRnZZoBJENoZERTVWhOTUc5blMwVkpRMEZuU1VSRGN6WnBXQzFSUlJBQg”>the knife’ (orange structure in the image below; click it for high-res) in Alameda County, frequented by hunters, with clear views of Mount Diablo and (on a clear day) San-Francisco. This interpretation would suggest a bomb location south of Mount Diablo and probably south of ‘the knife’, but is not particularly helpful in pinpointing the bomb. People have tried to associate other Halloween-card clues with locations, for example, looking for a “Paradise drive”, but this is also too vague. If someone here makes a related connection, for example with some famous 1970 pumpkin, that would be great…

Focusing on the cipher. Some of the proposed constraints (P1-P11) in my earlier post are powerful enough to sufficiently narrow down the phase space of viable Z32 messages. In fact, those assumptions combined are too strong and cannot all apply, as this would over-constrain the solution. We need to choose the correct set of constraints that would dramatically but not excessively narrow down the phase space. Ideally, although usually unrealistically, the decryption would pick up one unique, viable, and sensible solution. But it is also acceptable to have multiple solutions, provided that they form a manageably small set to inspect, especially if an additional criterion – such as one of the aforementioned clues suddenly making sense – selects a unique and obvious solution from the set. And as I argued when asked about Z13: if you dislike some natural (not cherry-picked) assumption, but it nevertheless ends up producing a sufficiently high-confidence solution (quantified as a small p-value below a reasonable predetermined threshold), then standard practice is to accept the solution, and barring rare circumstances, also the assumption.

We know that Z32 contains polar coordinates (angle in radians + distance in inches, centered on Mount Diablo), so proposal P9 (spelled-out numbers) is essential; otherwise, decryption would be impossible. Zodiac must have known that and most likely prepared an alphabetical key with homophones quite similar in structure to those of his other ciphers. Requiring coordinates sufficiently precise to pinpoint a bomb with reasonable accuracy (given the tools available to Zodiac) places powerful constraints on the cipher, as I suggested at the end of my earlier post but didn’t elaborate. As a radian is ~57 degrees and an inch corresponds to ~6.4 miles, Zodiac must have provided:
P12. Two digits for the angle and two digits for the distance, each pair of digits separated by a decimal point.
These decimal points were likely spelled out as “dot”, “pnt”, or “point”; again, adding a special symbol in the key to represent a dot would be atypical. A three-digit angle or radius would be both impractical for Zodiac to measure, and too long for Z32.

Z32 must contain more than just spelled-out numbers and dots, but has no room for much more than that. A direction with respect to the (here: magnetic) north is usually necessary: east vs. west, or clockwise vs. counter-clockwise. Many experts would insist on going clockwise (as Zodiac labeled compass hours on the map) and denote the direction “east” (or “east of north” in modern publications, but no room for that in Z32) even if the location is actually to the west so the angle exceeds 180 degrees, but I think that most people would take the angle between 0 and 180 degrees and specify either east or west. Proposal P6 (“rad” mid-sentence, and not “radians”, which in any case is too long for Z32) seems inevitable given Zodiac’s explanation, as discussed in the first post. Now, a digit (zero to nine) has on average four letters, so combining the coordinates and the two additional words (“east”/”west” and “rad”) already exhausts or nearly exhausts all available 32 symbols (and there is no room to spell both dots as "point"). Space is even more restricted if you notice that the unusually low frequency of symbol multiplicities (only three doubles: Delta, C, and O symbols) suggests multiple homophonic “e”, “t” and “r” (cf. the Z340 key), so we should expect repetitions of the digit “three” with its longer-than-average five letters. Hence, we have no room for P5 (“bomb”, “blast”, “bus”, …) or P11 (“slaves slayer”), and don’t need P4 (Zodiac vocabulary). The message is short and not entirely devoid of multiplicities, so we don’t need P10 (number statistics) either. And we didn’t expect P7 (inch shorthand) anyway. Of course, a solution is possible only if we know the reading order, so we must assume P3 (diagonal transpositions) to proceed.

We may therefore look for solutions satisfying P3 (diagonal transpositions), P12 (two numbers, each being a two-digit decimal), P9 (spelled-out), P6 (separated by “rad”), and preceded by “east” or “west”. We could (but won’t have to) use P1 (message in depth) to choose the deepest of a few viable alternatives, which worked nicely in Z13 (after it was solved in an independent method) although we don’t really know why Zodiac introduced depth (probably amateur mistakes). The symbol multiplicities, although very rare, now suffice to enforce a unique or nearly unique solution if we accept the template:
<“east”/“west”> + <digit> + <dot> + <digit> + <“rad”> + <digit> + <dot> + <digit>
Such a solution would be robust to a padding of one or two symbols at the end of the cipher, if we allow it, rendering the ambiguous reading rule of the last two symbols immaterial. The solution would also be robust to letter swaps or word inversions (P8) that don’t directly affect any of the (rare!) multiplicities. If an unfortunate swap or inversion does directly affect a multiplicity, we may either fail to find a solution, or converge on a wrong, nonsensical solution. A similar problem would arise if Zodiac made a sufficiently damaging mistake. Let us ignore such unfortunate possibilities for now and revisit them at the end.

The template is analyzed further below, but we don’t really need it to see that under the above assumptions, the bomb is 3.3 inches, corresponding to 21.1 miles, from Mount Diablo (see red circle on the image above). Consider the diagonally detransposed Z32, shown in the top row of the table below. The 3.3-inch radius derives directly from P6, because placing “rad” mid-sentence, coincident with the (detransposed; henceforth) 17th symbol Delta (yellow up-triangle), forces us to identify the Delta in positions 17 and 30 as “r”, and thus require a “three” at the very end of the cipher. (Indeed, there is no digit with “a” or “d” to match the other letters in “rad”. You could identify Delta as “r” but try “zeroX” instead of “three” at the end of the cipher, with some padding letter “X”, but this is a dead-end yielding an impossibly short 0.0-inch or a too-long 7.0-inch.) Now the symbol O (green circle) in positions 21 and 29 must encode an “h”, leaving us no choice but to identify symbols 17-32 as the string “radthreedotthree”. (You can replace “dot” by “pnt” – that would make no difference – but not by “point”, as the text would then exceed the cipher). It is reassuring to see the emergence of multiple digits “three”, anticipated above from an entirely independent argument (namely, rare multiplicities). The string we just identified associates the symbol C in positions 1 and 24 with the letter “e”, nicely consistent with starting the cipher with the anticipated word “east” (and not “west” or any other word) – note that we didn’t need to give a higher weight to “east”. These conclusions are corroborated by my (Wolfram Mathematica, shareable) code analyzing the above template. Note that I used 6.4-mile/inch, as indicated on the map and as Zodiac most probably assumed, although I suspect that a 1:400,000 scale was used, actually corresponding to 6.31-mile/inch.

The radius seems to be quite robust, thanks to the multiplicities concentrated in the second half of the detransposed cipher, so we only need to find the angle. There are no remaining multiplicities to help us, but we can manage without them. Counting the remaining symbols (see table above), we see that the two digits making up the angle must together have nine characters, so one of them must be A={three, seven, or eight}, and the other B={zero, four, five, or nine}. Radians only reach 2*pi~6.3, so if A is the first digit, we have only the four options {3.0, 3.4, 3.5 or 3.9} radians, and if B is the first digit, we have 3*3=9 additional options. We can wrap up now by noting that equally allowing “east” and “west” implies that the angle can be no larger than pi~3.1 radians, so we must choose the southern angle 3.0 radians – the only other alternatives, {0.3, 0.7, or 0.8} radians, are all northern and fall in rural locations outside the map – but let me show that we can manage even without this argument. The above set of 13 candidate solutions is already quite manageable: we can simply check all corresponding locations. Most of these locations must fall outside the Zodiac map, or in water, or in the middle of nowhere, or too deep inside a city, or far from any road, etc. If our method is flawed and all 13 candidates are wrong, then the locations would be random, with a small probability for even one place satisfying all of our requirements: falling inside the map + also just outside a town + exactly along (within our 6.4/20=0.32 mile precision) a sufficiently central road + with an elevated wayside, etc. If one of these 13 options does match the full description – against all odds – then it is most likely the correct solution rather than a lucky fluke. As usual in hypothesis testing problems: the smaller the odds, the higher the confidence in an emerging solution. And if this option also suddenly explains one or more of the Halloween-card clues? You might want to call the bomb squad.

But we don’t have to bother checking all 13 locations, nor invoke the east/west symmetry to directly reach the 3.0 radians solution, as we can explore P1: message in depth. We already have evidence that Z32 is in some depth, and looking for the deepest solution already proved powerful in the retrospective analysis of Z13. Let us repeat the same strategy here: the table above includes rows for the Z408 (blue), Z340 (red), and Z13 (cyan) keys. For each cipher, it also shows a row (labeled inv) for key inversions including Greek, and a row (labeled img) for possible key associations based on visual symbol resemblance, accounting e.g. for Zodiac confusing similar symbols. (As usual, ill-defined associations such as the img rows must be completed before attempting a solution, to avoid any self-bias creeping in.) Now one can focus on the deepest solution: obtained if the second digit in the angle is zero (“zero” here has depth d=3 with respect to Z408), uniquely picking the solution 3.0 radians (note: yet another “three”) just like we already anticipated above without invoking depth. This solution is shown in the bottom row in the table; letter colors indicate depth with respect to the correspondingly colored key. This angle translates to 171.9 degrees (with a 180/pi/20=2.9 degree precision) from the magnetic north, or 187.9 (with a similar uncertainty) from true north after accounting for the ~16-degree magnetic declination in 1970 California (see magnetic declination calculator). The alternatives are far shallower; e.g., one can replace the zero (d=3) by four (d=1), five (d=0), or nine (d=0), but these are consistent with randomly occurring depths.

Our assumptions have thus led us to 13 candidate solutions, all of them in some depth, but one of them significantly deeper than the others. Therefore, this deep solution is the one we should choose under the present assumption P1, although inspecting 13 locations would not be a big deal, we didn’t really need depth because 3.0 radians could be inferred directly from east-west symmetry, and depth arising from an unclear origin is not a rigorously justified requirement. Finally, we test this solution by examining the location it encodes. Remember: if we’re wrong, this would be a random location, almost certain to make no sense in the present context. On the other hand, if we find this single location exceptionally promising as the bomb sight, and hence exceedingly unlikely to be just a random fluke, then we can confidently identify the solution as correct. The more promising the location, the more unlikely the statistical fluke, and so, the higher the statistical confidence level of our solution.

The angle given by our unique solution is shown in the image above as the green ray, directed almost exactly southward from Mount Diablo. The location given by combining the angle (green ray) and radius (red circle) is in north Fremont, just outside of town. Here, route 84 or Niles Canyon road meets the town periphery, with a northern wayside sharply rising towards the railway tracks. This region nicely matches the expectation for the bomb location, far more than what one would find at random (as I demonstrate below, and as you can confirm by picking truly random points on the map and estimating the fraction of locations that cannot be easily ruled out as the bomb site). The green ray is about 15 degrees left of ‘the knife’ when viewed from Mount Diablo. Although this reference is imprecise, “by knife” would have seemed like a rather elegant clue from Zodiac’s perspective (once again, a hint that cannot actively facilitate a solution, but can show off Zodiac’s cleverness after we already know the solution). Of course, it would have been much better for us if he wrote “by Walpert ridge” or “by Sunol peak”, but that would defeat the purpose of a subtle hint. In retrospect, perhaps we should have considered Zodiac sending the ‘citizen card’ from Fremont as a clue.

The image below zooms into the location indicated by the solution. The coordinates are again where the red circle intersects the green ray, essentially falling in … Paradise drive (blue line). Placing a bomb next to Paradise drive sounds very much like Zodiac, and was indeed anticipated by others. The location is most famous for the historic Vallejo mill, so could this be the purpose of the comics reference with its iconic water-mill scene? Perhaps, but unless I’m missing some context, I would expect a more coherent clue from Zodiac. The radial uncertainty (given by the worst-case scenario when using two digits) is illustrated by the north-south extent of the cyan polygon. The angular uncertainty is more substantial at this radius, so I bounded the angles to the polygon at route 84. Therefore, while the most probable region of interest is the yellow polygon near Paradise drive, a more extended region marked by the pink polygon cannot be ruled out. Perhaps someone with knowledge of the place and time could now interpret some other suspicious Halloween clue (peek-a-boo, skeletons, rope or fire, teen… and what about that pumpkin?!) in this context: that would be very interesting, and most helpful if it further narrows down the precise location, but otherwise, I think we already have enough.

I verified that Paradise drive was already there in 1970 (see for example this map from this selection), and that the vicinity hasn’t changed much since. The alternative best 12 candidate solutions dismissed above due to their shallower depths are shown as non-green rays in the zoomed-out image. As expected from random coordinates, they all yield nonsensical locations: outside the map (yellow rays) in rural places, sometimes in water, or inside the map (red rays) but in regions either too rural, or too urban, or right on the waterfront. The solution is cryptanalytically satisfactory, as indicated in comments above. In particular, the depth criterion did identify the anticipated and only reasonable solution among 13 candidates. The letter “e” has seven homophones; perhaps Zodiac added one after realizing that the message has many “three” digits. The multiple digits "three" may have given Zodiac a false impression that the cipher is easy so there is no need to remove homophones as in Z13. The message is in depth mainly in its first (detransposed) half, inconsistent with randomness; this can happen if the first homophones in the key are chosen negligently.

To confirm the manual results derived above, I wrote a Mathematica code to examine the aforementioned template with the constraints being gradually relaxed, and ended up with the same overall conclusions. As explained above, the solution is robust to letter swaps and word inversions, unless they adversely affect the rare multiplicities. The code found some additional candidate solutions when I allowed for the less probable, harmful letter swaps and word inversions, mid-sentence padding/s, changing “dot” to “point”, and similar relaxations, but all the resulting candidates were less deep than the above solution, and all the alternative locations I inspected were nonsensical. Further allowing sufficiently bad Zodiac mistakes introduces numerous spurious solutions, but again – their vast majority is nonsensical; note that correcting for a substantial Zodiac mistake is not only impossible, but also pointless when dealing with coordinates. As usual, all the above can be easily tested and challenged/confirmed; if you want to check/use/generalize my code, please send me a PM.

Edit: a subsequent analysis of the Halloween card revealed additional relevant details and context, discussed in a separate thread.

Consider Zodiac’s description of the bomb (click any image for high resolution):

We see that he placed two "photoelectric switches" on the wayside, one higher than the other, for his differential trigger mechanism. These were bulky devices back in the 1960s, and he probably hid them well. But after 50 years, there is a good chance that they shifted and became exposed by the elements. What if they are plainly visible today?

Consider the precise intersection between Route 84 and the line (blue in the image of the previous post) continuing Paradise Drive, derived above from Z32 and separately from the Halloween card. This is the center of our search box, but with some luck, Zodiac chose it as the actual bomb location. Indeed, Google’s street view shows two suspicious devices here:

A few meters down the road to the southwest, there is a viewing angle where the two devices align one on top of the other:

Zooming in, both devices appear to have circular lenses:

The devices seem as bulky as the Detect-O-Ray Photo-Electric Switch, which was popular till the 1950s; I don’t know what was likely to be used in the 1960s. The orientations and positions of the devices match Zodiac’s description. BTW, this region is somewhat greener than its surroundings, perhaps due to the substantial amount of fertilizer Zodiac introduced.

The location and properties of the devices suggest that they are part of Zodiac’s failed bomb, although we can’t be sure without a better picture or an identification of the sensor model. Someone should definitely take a look at this place, and in any case, the failed bomb must be along this section of Route 84, based on both Z32 and the highly-redundant Halloween communication.

Edit: This location and the Z32 solution were robustly confirmed later by the Sierra postcard, in which the two crosshairs symbols are separated precisely by 3.3 inches at east 3.0 radians. Zodiac must have grown frustrated with his bomb being unappreciated, and so provided a plain-sight Z32 solution in this postcard.