# Binary Coordinates Puzzle

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.

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.

• Maria just added a bounty. I hope somebody manages to solve this for you! :-) – Rand al'Thor Sep 8 '16 at 18:47
• Thank you so much Maria Deleva and rand al'thor, I really appreciate it! – Juan Vazquez Sep 8 '16 at 18:49
• Do you have a general idea of where the geocache is (which country even) as it could have narrow down possible solutions. – gtwebb Sep 8 '16 at 19:01
• Yea, I'll include in the question right now as well but 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. – Juan Vazquez Sep 8 '16 at 19:05
• Google picked that up fast. – John Sep 8 '16 at 19:12

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.

• I've been trying to solve this puzzle for so long and was ready to give up on it once again. Then I was just notified by email of this post. I can't believe that someone was able to finally solve this. I read everything you wrote and it makes complete sense, checked the final coordinates and they are correct. I really appreciate this, thank you for everything. I know there are many puzzles posted on this sight daily and I just want to say thank you for taking the time out to help me with this one, you made my day. – Juan Vazquez Sep 16 '16 at 0:34
• @JuanVazquez I enjoyed it. I'm a geocacher as well, so this one piqued my interest. You can also thank Maria Deleva, because without her bounty, I wouldn't have noticed this question. (Plus, she claimed that bounties were useless, because hers hadn't generated any results in almost 7 days, so I felt the need to prove her wrong...) – GentlePurpleRain Sep 16 '16 at 1:49

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.)

• I see this has been accepted -- is that because something in it turned out to lead to the right answer, or just because no one else came up with anything better? :-) – Gareth McCaughan Jun 27 '16 at 19:39
• Unfortunately I still haven't been able to solve it. But I do really appreciate your random thoughts! I was given a hint that "one line helps you figure out how to parse the other line" and "The coordinates are all in the first line. The second line will help identify where they are." But even with that I'm still pretty lost. – Juan Vazquez Sep 8 '16 at 18:22

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.

• This is really interesting! Been looking into this a little bit but not sure what to make of it as of yet. I know those coordinates probably aren't correct because they look like they're in a front yard which wouldn't be where a geocache would be hidden. But the fact that they come out to be so close to the original location seems to be more than coincidence. – Juan Vazquez Sep 8 '16 at 23:04