Binary Coordinates Puzzle

Question

^{I was told instead of re-asking a question to rather edit my old post to bring the question back into subject. So I'm going to try to reformat it entirely.}

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The following puzzle is a geocache puzzle that should give coordinates when solved.

Puzzle:

Line 1: ZzZZzzzzzzzzzZzzZzzzzZZZzz

Line 2: ZZzZZzzZzzZzZZZzzzZzZZzzZZ

Hints:

1. z = 0, Z= 1, zz = 2, Zzz $\neq$ 6

2. "binary"

3. "The coordinates are all in the first line. The second line will help identify where they are."

4. "one line helps you figure out how to parse the other line"

The initial coordinates given for this one are N 41° 43.007 W 088° 00.410, so it should be within a 5-mile radius of that.

Answer 1

Considering the hint that one line tells you how to parse the other, I noticed that

the pattern of Z/z in the second line is much more regular than the first (no run of more than 3 the same).

That made me think that the second line might be

indicating how to separate out individual "digits" from the first line.

If we align the two lines, and then put a space at every place where the second line changes from uppercase to lowercase or vice-versa, we get the following:

Line 1: Zz Z Zz zz z zz z z zZz zZz z z zZ ZZ zz
Line 2: ZZ z ZZ zz Z zz Z z ZZZ zzz Z z ZZ zz ZZ

This parses line one nicely into

15 distinct groups, which matches up with the 15 digits in the known coordinates.

We've been given some of the values already: z = 0, Z = 1, zz = 2. If we fill those in, we get:

 Zz Z Zz zz z zz z z zZz zZz z z zZ ZZ zz
    1     2 0  2 0 0         0 0        2

Based on what the OP said about the solution's coordinates being close to the original coordinates, we can confidently assume that the north coordinate begins with 41 and the west coordinate begins with 088:

 Zz Z Zz zz z zz z z zZz zZz z z zZ ZZ zz
  4 1  4  2 0  2 0 0   8   8 0 0        2

We now have the following values:

0 = z 5 =
1 = Z 6 =
2 = zz 7 =
3 = 8 = zZz
4 = Zz 9 =

From this, we can extrapolate a pattern:

To increase your number by one,

Find the rightmost z and change it to a Z. If there are any digits to the right of the z you changed, change them all to z.

OR, if there are no z in your number, increase the total number of digits by one, and make them all z.

By following these rules, we can determine the digits we haven't yet discovered:

0 = z 5 = ZZ
1 = Z 6 = zzz
2 = zz 7 = zzZ
3 = zZ 8 = zZz
4 = Zz 9 = zZZ

This lets us fill in the gaps in the coordinates:

 Zz Z Zz zz z zz z z zZz zZz z z zZ ZZ zz
  4 1  4  2 0  2 0 0   8   8 0 0  3  5  2

Which gives us coordinates of

N 41°42.020' W 088°00.352' (Google Maps)

And is at:

The opposite side of the road of a Speedway convience store, near Historic US 66 highway, close to Chicago.

Answer 2

Random thoughts, not close to being a solution; most likely nothing here is correct. I'm going to assume you're right about the roles of the two lines, though I've no idea whether you actually have enough evidence for that to be beyond reasonable doubt.

The second line has 14 Zs in it. If you don't bother with the leading zero for the westings, that just happens to be the number of digits you need for all the coordinates. So perhaps for each digit you start reading the first line at the location of one of the Zs in the second line?

An argument against that: if the first seven Zs are for the northings and the remaining seven for the westings, it seems that the first starts 1011 and the second starts 1001, and those seem too similar for one to be encoding a 4 and the other to be encoding an 8.

If z=0 then zz=2 isn't (whatever else it may be) binary: 00 in binary is 0, not 2. So there's something distinctly weirder going on, and I bet it doesn't involve just replacing every instance of "zz" with "10" and hoping it works out :-).

Each line has 26 "bits" in it, which is a funny sort of number. It doesn't divide up nicely into anything that seems like it would make 14 or 15 decimal digits. So, if (whether by the sort of mechanism you've looked at, or the sort I've looked at, or something else) we're starting at various different points in the first line -- how do we know where to stop?

One obvious guess is that you just look at the run of contiguous "Z"s or contiguous "z"s starting there and interpret them somehow. I don't buy that because the hint seems to suggest that "Zzz" is a thing you might want to use. I wonder about this sort of scheme: you take a single "Z" if there is one, followed by whatever contiguous run of "z"s you've got. Then you count 1 for the "Z" if present and 2 for each "z" after the first. That would fit everything in the hint; unfortunately many of the available starting positions then yield "digits" bigger than 9, which seems unlikely to be what we need. (And if my guess about how the second line works happens to be right, this doesn't give us the right starting digits.)

Answer 3

Hmmm... If you call z = 1 and Z = 2, then interpret as ternary, it breaks hint 1, but some interesting coincidences arise.

Splitting line 1 in halves (and reversing the second half, since it is west) we get this (plus their decimal conversions):

2122111111111 = 1407334
1122211112112 = 882482

Those numbers bear some resemblance to the original coordinates. If we swap the first two digits, then reverse the rest of the string (padding the second with a 0 since it's a digit short), we get the coordinates:

N 41° 43.370'
W 88° 02.842'

Which is strangely close to the starting area: ~2.1 miles.

Since I blatantly ignored the first hint, this seems like it has to be a coincidence, but this is a pretty big coincidence.

No clue what to do with line 2.