First of all, let me ask “how is it possible that the cipher can have the first two layers of interpretation at the same time and these not work against each other?” Basically, how is it that we can have a homophonic substitution solution giving the coordinates of the town of Hercules alongside a numerical solution giving the name of the town, and both deriving from one and the same cipher text?

Well, when we try and answer this question we can first of all note the following key points.

Firstly, the homophonic substitution solution requires only that we meet the three constraints that we see in the three pairs of common letters or symbols. Beyond this, however, this first solution layer imposes no constraints on precisely what letters or symbols the author uses in any one instance.

Secondly then, in constrast to this, the numerical solution does depend very specifically upon which letters/symbols the author chooses, and in what order, whilst being subject to only a minor (and easy-to-overcome) limitation from the pairing constraints.

Consequently, there is only a very minimal overlap of constraints overall between the two, and so the author has almost complete freedom in adapting the two methods to whatever solutions he may wish to present in both cases.

And, we would need for this degree of freedom between the two to be present, otherwise we would hardly be able to suppose that The Zodiac could have fashioned both solutions into a single cipher in the way proposed. 

When it comes to factoring in any similar numerical method in trying to decipher a pair of initials from the self-same cipher, however, what we encounter then is the exact opposite situation.

At this point, we have almost precisely no freedom to impose a specific pair of initials on the cipher and the author can then only hope that his initials just happen to appear there, or else they don’t. And this will be the case, no matter what order we create the signature solution in.

Essentially, the problem is this:

The signature solution method that you propose is not in any significant degree independent of the solution method you have given previously and which produces ‘HERCULE/S’.

In short, I would probably wish to conclude at this point that the attempt to add a third layer on top of the two solution methods already considered amounts to simply trying to push the layering too far.

The second (numerical) solution leading to ‘HERCULES’ does produce for us the markers ‘X’ and ‘Y’, possibly indicating the author’s initials. However, beyond this, I cannot conceive of any way in which the method of decoding in this case is also internal to this one cipher also. If we are even on the right track, the solution to this puzzle must lie elsewhere.

Let me know if you can see a way in which we could infer sufficient freedoms for that to be possible.

Just to finish this idea, it can be argued that the only slight freedom that the author would have had remaining, in order to insert his initials into the cipher, would be through the placing of the ‘X’ and ‘Y’ sections, as below:

And, as the actual placing stands, it is likely that the only viable possibility would be to suggest that The Zodiac noticed that his initials were present within ‘HERCULES’ and placed the ‘X’ and ‘Y’ sections accordingly to show:

Hi @shaqmeister! I’d like to clarify one aspect of your proposal.

In this approach, you are essentially excluding the entire first row and relying solely on the second row to derive Zodiac’s initials. Moreover, you are not really following a counterclockwise reading of the letters, but simply reading the second line. This is not necessarily wrong, it could still be a valid interpretation, but it does change the reading criterion from what I understand you were proposing.

If instead we try to follow the order of the letters in HERCULES, what do we obtain?

1) Direct sum

H + E + R + C + X = 7 + 4 + 17 + 2 + 23 = 53 → 53 − 26 − 26 = 1 → B
U + Y = 20 + 24 = 44 → 44 − 26 = 18 → S

Result: BS

If we instead assume that Zodiac intended to follow the path while also introducing a specific reading direction, we can consider a variant with aligned signs.

2) Variant with signs aligned to the reading direction

H − E − R − C + X = 7 − 4 − 17 − 2 + 23 = 7 → H
U + Y = 20 + 24 = 44 → 44 − 26 = 18 → S

Result: HS

If, among these possibilities, we opt for an incremental (progressive) calculation, we obtain:

1.1) Incremental calculation (progressive sum)

H + E + R + C + X = 53 → 53 − 52 = 1 → B
H + E + R + C + X + U + Y = 97 → 97 − 78 = 19 → T

Result: BT

2.2) Incremental with signs aligned to the reading direction

H − E − R − C + X = 7 → H
H − E − R − C + X + U + Y = 51 → 51 − 26 = 25 → Z

Result: HZ

Buongiorno, @lendor-77!

Thank you for posting these four variants and showing how you have reached them. It may help me to try and explain a little more clearly the main point that I was wanting to make last night.

So, you are able to show in this way that you can generate example sums which produce 4 candidate initials: BS, HS, BT and BZ. If, then, any one of these four match the author’s actual initials, then all is well and we could say we can demonstrate how to find them there.

That said, however, it would be the same for the author as it is for us, in this situation.

Once the author has settled on the structure that gives him ‘HERCULES’, he is only able to then see if there is a way to find his initials in there also. He has essentialy no freedoms left to be able to put his initials anywhere.

Imagine for a moment that you are the author, and think about giving yourself any pair of initials whatsoever, randomly. Try, for example, MK, or AA, or QD. Does your proposal here permit you the freedom to change anything so as to be able to introduce any such initials that the author might have?

I was arguing that I cannot see that there is any such freedom left to us. And this fact, I would say, is problematic when seeking to defend your reasoning at this stage. 

Does that make sense? I don’t know if I am explaining my thinking clearly enough. Let me know if I am not.

Posted by: @lendor-77

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In this approach, you are essentially excluding the entire first row and relying solely on the second row to derive Zodiac’s initials. Moreover, you are not really following a counterclockwise reading of the letters, but simply reading the second line. This is not necessarily wrong, it could still be a valid interpretation, but it does change the reading criterion from what I understand you were proposing.

As you say, in this idea we would effectively be reading only the second row to derive the initials. That said, it is still consistent with the overall counterclockwise reading we have suggested, isn’t it. I mean, it does follow the ordering of ‘UL’ as in ‘HERCULES’, no?

Posted by: @shaqmeister

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Posted by: @lendor-77

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In this approach, you are essentially excluding the entire first row and relying solely on the second row to derive Zodiac’s initials. Moreover, you are not really following a counterclockwise reading of the letters, but simply reading the second line. This is not necessarily wrong, it could still be a valid interpretation, but it does change the reading criterion from what I understand you were proposing.

As you say, in this idea we would effectively be reading only the second row to derive the initials. That said, it is still consistent with the overall counterclockwise reading we have suggested, isn’t it. I mean, it does follow the ordering of ‘UL’ as in ‘HERCULES’, no?

I think I understand your point: once a solid method has been established to derive Hercules, it’s not easy to also “fit in” the author’s initials within the same structure. In your proposal, for example, it’s possible to read the initials UL, treating X and Y simply as markers. However, this only works if the author’s actual initials are UL and if the sequences were arranged in such a way as to leave UL in the second row. That said, we should also consider another possibility: that X and Y are not just markers, but could instead represent values to be combined with UL, acting almost like a cryptographic key. In that case, the calculations would be:

X + U = 23 + 20 = 43 → 43 − 26 = 17 → R

Y + L = 24 + 11 = 35 → 35 − 26 = 9 → J

Result: RJ

From here, one could also extend the reasoning to more complex approaches, such as incremental calculations based on the second row. In other words, it’s possible that the author saw in XYUL a coherent numerical structure, sufficient to justify encoding his initials.

Posted by: @lendor-77

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I think I understand your point: once a solid method has been established to derive Hercules, it’s not easy to also “fit in” the author’s initials within the same structure. In your proposal, for example, it’s possible to read the initials UL, treating X and Y simply as markers. However, this only works if the author’s actual initials are UL and if the sequences were arranged in such a way as to leave UL in the second row.

That is correct. And because this one idea needs ‘UL’ to be already present, it can be argued that it is likewise similarly problematic alongside any more complicated method of finding his initials already there.

All the same, I could easily imagine the following scenario.

Let’s imagine that The Zodiac wants to encode ‘HERCULES’ using the method you have suggested for this part and, just for the sake of argument, let us suppose that his initials are ‘UL’. It would not be much of a reach in reasoning to then suggest that he had noticed that his initials are present in ‘HERCULES’.

Having already decided on his method for encoding the whole word, he then realises that he could draw attention to his initials simply by introducing two extra blocks that evaluate to ‘X’ and ‘Y’, setting these in the appropriate location in the cipher to achieve this, picturing:

‘HERCXUYLES’

At this stage, of course, he does have the freedom to do this without interfering with the method that produces just ‘HERCULES’.

Suppose that he next thinks of the counterclockwise reading direction idea. With this, he would then notice that he has the further freedom to shift the starting point on the two paired symbols precisely to where it would need to be to enable him to have the ‘XUYL’ sequence land solely on the second line.

(If, however, his initials are not UL, then I would simply imagine that the possibility of finding a way to introduce his actual initials into the cipher would not have even occurred to him as something he could otherwise achieve.)

So, we would still have coincidence in this way of thinking. But it would be a coincidence that it is possible merely to see, simply upon visually considering the word ‘HERCULES’ and proceeding accordingly.

If I were the author and my initials were UL, I’d probably make that explicit by using two identical markers (something like XX) ending up with a structure like XUXL. That way, the role of the symbols would clearly be to frame or highlight the letters.

Using XY, on the other hand, suggests something different. It doesn’t really look like simple markers, but rather two distinct elements, each with its own value. That makes it more natural to interpret them as operational components, for example, values to combine with U and L, or even as a kind of cipher key applied to the initials.

In that sense, XY feels less like a purely structural feature and more like something actively involved in the calculation.

Posted by: @lendor-77

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In that sense, XY feels less like a purely structural feature and more like something actively involved in the calculation.

The issue here, though, is—as it has been my aim to show—we don’t have sufficient freedoms required for constructing a calculation giving random initials in the manner suggested. Two of the three proposed layers of decoding (‘HERCULES’ and ‘XY’) are, in this method, wholly tied into each other and cannot be separated.

We do, however, have near complete freedom to add in ‘X’ and ‘Y’ as mere markers and thus, in this last sense, the three layers are able to remain largely and sufficiently independent of each other.

As to your point regarding ‘XX’, we have to remember that the ordering of the initials, if present, is equally as important. Otherwise, we wouldn’t be able to rule out that they might be ‘LU’.

It’s all very interesting though, isn’t it?

I’m not entirely convinced that the order of the initials is that important. If, in the end, we had to choose between UL or LU, I don’t think it would make a significant difference from an interpretative standpoint.

That said, there’s one part of your reasoning I’m struggling to follow, the idea of “freedom” within the system. I’m not quite sure I understand what you mean when you say that some elements are “constrained” while others are “free.”

Could you clarify what, in your view, defines something as free versus constrained within the method?

Posted by: @lendor-77

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That said, there’s one part of your reasoning I’m struggling to follow, the idea of “freedom” within the system. I’m not quite sure I understand what you mean when you say that some elements are “constrained” while others are “free.”

Could you clarify what, in your view, defines something as free versus constrained within the method?

Ciao, @lendor-77.

I would be happy to try and explain a little more about what I mean by the difference between ‘free’ and ‘constrained’ in this context.

Suppose, then, we start with the Z32 cipher as we have it, and further suppose it to be an example of a homophonic substitution cipher with the solution as you have offered, so:

Here there are some contraints and many freedoms that must be taken into consideration when composing the cipher itself.

As to constraints, the author has required us to retain that three pairs of elements must be distinguished as having the same letter/symbol—(1, 26), (2, 32) and (6, 14). But these are the only constraints for this layer. We are not in anyway required to keep the first pair as a ‘C’, the second as a ‘Δ’ or the third as an ‘O’. As for all of the remaining letters/symbols, as the author of the ciphertext we would then have total freedom as to which we choose. Consequently we—as was the case for the author himself—have almost complete freedom to select what letters/symbols we use in encoding the ciphertext to satisfy this ‘homophonic layer’. (If, that is, we are envisaging only this one single homophonic layer of encoding.)

To illustrate the actual degrees of freedom that we have here, compare the following which, as a ciphertext, is precisely the same as the one that we were actually given and leads equally to the same solution:

In fact, we have so great a freedom at this point that, instead of letters and symbols we could substitute for these 29 pictures of different species of bee and, as long as we keep to the colour-coded pairings, we again have an unchanged cipher.

This, then, illustrates the degrees of freedom that The Zodiac would have been faced with when he came (as in your proposal) to add a numerical layer, one leading to the solution ‘HERCULES’.

Note that, with this additional layer, we no longer have anywhere near the same freedoms that we had composing the first layer.

Firstly, we are going to be using symbols as separators and letters for our actual code blocks. More than this, we will be wanting to apply the A=0 system to deduce the single results for each code block; so not even just any letters can be used here, but only certain ones. In each group, they must be such as will sum to the required total for the result letter we want to encode.

So now, having worked out a specific encoding that would achieve this, we can see that this can be done with the cipher text as we are given it, but that the second example above would absolutely not suffice. This is because the constraints have now become extremely ‘tight’, and the remaining freedoms are practically none. We can move the letters about in each block, and we can change the actual symbols we use for each separator. But, beyond this, there is very little left to change.