Help with Geocaching puzzle

[Synch]

Sorry if this is in the wrong place, I wasn't too sure where to post but I figured its sort of a forum game.

I do Geocaching in my spare time (for those unaware, its like orienteering mixed with a scavenger hunt - you follow co-ordinates on a smartphone app or GPS device and try to find hidden caches; millions of people play worldwide and there are probably some near your house; great if you love taking walks outdoors and puzzles/finding things.)
There's one in my area which has a tricky puzzle (only 35 people have ever found this one) and I've spent ages trying to solve it, but I suck at algebra, and the question mark help thing in the middle didn't help me either, I couldn't work out the matrix.

http://www.geocaching.com/geocache/GC1F ... o-auckland

I'd really appreciate any maths whizzes who could work out the answer or at least show me how to solve it.

[Davecom3]

Okay, first tip. As A connects to H, I, B, and C with a top number of 11, and B connects to I, A, C, and D with a top number of 21, we know H+I+B+C=11 and 21=I+A+C+D. Combining these, H+I+B+C+21=I+A+C+D+11 or removing I, C, and 11 from both sides, H+B+10=A+D. We then need to proceed to do this for the remaining letters.

The equations we come up with are:
H+B+10=A+D
I+C+4=B+E
A+D=C+F+9
B+E+1=D+G
C+F+4=E+H
D+G=F+I
E+H+1=G+A
F+I=H+B+8
G+A=I+C+3

Note, you can replace any side with whatever's on its opposite side in any equation, For instance, we can easily say from this:
B+E+1=F+I, or B+E+1=H+B+8, which means E=H+7, and now all Es can be replaced with H+7. Further, maybe you just want to replace a part of a side. For instance, replacing B+E, (I+C+4)+1=F+I, simplified as C+5=F, meaning any F can be replaced with C+5 or vice verse.

Solving all variables for C, just one of the 9 choices, I obtain:
A-6=C or otherwise A=C+6
B-3=C or otherwise B=C+3
D-8=C or otherwise D=C+8
E-8=C or otherwise E=C+8
F-5=C or otherwise F=C+5
G-4=C or otherwise G=C+4
H-1=C or otherwise H=C+1
I-7=C or otherwise I=C+7

From here, it's simply a matter of replacing all variables with their C equivalent.

If you went to S36 56.308 E174 38.541, you did something wrong.

► Show Spoiler

It said not to go there because the location is bogus.

[ChuckDaRighteous]

Dave beat me to it and did a lot of the work. The way to finish is to solve for c. (I haven't checked his work but I'm assuming its right)

A + B + D + E = 25
So if you replace all of those with their c equivalent:
C + 6 + C + 3 + C + 8 + C + 8 = 25
or
4C + 25 = 25
C = 0
plug that in and...
A=6
B=3
D=8
E=8
F=5
G=4
H=1
I=7

[Davecom3]

It's possible to use matrices as well, but for those not as skilled with them it's probably better to use the method I provided. And it's been a while since I did things like that with matrices.

[Synch]

Wow you guys are awesome! I've been struggling on this one for days, I had misread and didn't realise one of the letters could equal 0! And the way you guys explained the methodology made it very easy to understand. Thank you so much.

Davecom3, I think the final co-ordinates you came up with are correct because on google maps it looks like its on a walking track I know of. He mentioned the original co-ordinates he posted (S 36┬░ 56.000' E 174┬░ 39.000') are bogus. I will share a picture of my success once I find the cache

Do you guys want another to solve thats been stumping me?

[Davecom3]

I could give it a go, if you need help.

[ChuckDaRighteous]

Yeah, it was fun, though Dave doing most of the work before I got to look at it took some of the fun out of it.

[Synch]

http://www.geocaching.com/geocache/GC2M ... t-auckland

This is the other one I haven't solved in my area.
I don't really know where to begin with this one - it doesn't look like binary code or morse code, And I think there must be something within the text (1st Jan 2:35) but I don't know where that leads to.
The background is a shell too, so I'm wondering if that has some relevance.
The hint says: Possibly Dan Brown's next novel, which at that point would have been Inferno. Again, I don't know the relevance of this.

Where do I start?!

[Davecom3]

Huh, Fibbonachi. Which bring the question of how to decode a Fibbonachi code. The shell, if a hint, has to do with phi, the golden ratio.

[Synch]

Oh why didn't I think of that! 1,1,2,3,5 - Fibonacci sequence. That makes sense then, there's Fibonacci codes in Dan Brown's books. I'll do some research, thanks!

[Davecom3]

Also written there is something that reminds me of The Tale of Two Cities. Not sure if that relates.

And if you wonder what I meant by Fibbonachi, that's what I immediately thought of when I saw the date, 1, 1, 2, 3, 5...

The Fibonacci code is closely related to the Zeckendorf representation, a positional numeral system that uses Zeckendorf's theorem and has the property that no number has a representation with consecutive 1's. The Zeckendorf representation for a particular integer is exactly that of the integer's Fibonacci representation, except with the order of its digits reversed and an additional "1" appended to the end.

To encode an integer X:

To decode a token in the code, remove the last "1", assign the remaining bits the values 1,2,3,5,8,13... (the Fibonacci numbers), and add the "1" bits.

This is a Zeckendorf representation, not a Fibbonachi code. Now that you know this, it shouldn't take long to solve. Remember that the third and sixth numbers need to be in the thousandths place.

I've solved it and Geochecked it, but I'll let others work it out for themselves. All the relevant information on how to solve this is already here. Though he left out the S and E.

[Synch]

It relates because the cache is apparently hidden between two trees.

I didn't use the Golden Ratio, I just solved it by hand.

Laying out the Fibonacci Sequence, I then matched the Code to the sequence (I had to read that for some reason you ignore the 0 and the first 1 in the sequence, and you ignore the last digit in the Fibonacci code as it acts as a stop. Eg:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
- - 0 1 0 0 0 0 0 11 = 2 + 34 = 36
- - 1 0 0 0 0 0 0 0 11 = 1 + 55 = 56

And finally got S36 56.208, E174 38.042, which also seems to be on a walking track!

So I'll go and find these two tonight. Thanks for your help Dave, you should do Geocaching if you don't already, you're good at puzzles!

Sorry for not giving you a look in on this one Chuck, I'll find another puzzle for you

Edit: we posted at the same time Davecom3! Looks like we both solved it.

[Davecom3]

I don't have a GPS.

[Synch]

Oh, neither do I. I just use the Geocaching app on my iPhone. There's one on Android too, if you have a smartphone.

[Davecom3]

Nope, nor a cell-phone. But I thought everybody had a cell-phone these days. Nope.

And the reason the 0 and first 1 are skipped is rather simple. 0 adds nothing to the number, so it's an unnecessary digit. The second 1 will never be used as you'd either be using the 2 or the 3 instead, so once again, you have an unnecessary digit that can be left out because it will never be used in the representation of a number. Therefore, they simplify the code by leaving the two unnecessary digits out.

Just so you know, I'm kidding around about that above fact, though it's still true. I just know the Weather Channel here is stating everybody has a cell-phone in one of their shows.

[ChuckDaRighteous]

Sorry for not giving you a look in on this one Chuck, I'll find another puzzle for you

Its ok. And I would get into these to but I don't have a smart phone. Cell coverage is so shitty here that my dumb phone is near useless, so never felt the need to upgrade for something that would be equally useless.

[Synch]

Yeah fair enough. Its a shame smartphones and GPS devices are so expensive, its a lot of fun. Can take the family, can take a friend, exercise the dog, good for some quiet alone time etc.

[Davecom3]

And the shells make sense as a hint to Fibbonachi as well, as the farther you get in Fibbonachi numbers the closer F_n:F_n-1 gets to phi.

[Synch]

First cache found, in the dark no less!



Couldn't find the second however, looked around the area but no sign. It hasn't been found in over a year, so I might get the owner to come and check to make sure its still in the same place and hasnt been moved or destroyed.

[Synch]

Oops, never updated but I found the other one too:)

I found another extremely hard puzzle guys. If you can help me solve this one then good on ya, because no-one has been able to solve this in 3 years!

http://www.geocaching.com/geocache/GC3E ... Submit6=Go

It utilises the codes they used in WWII and the recent movie Imitation Game; I've done some research and one of the other geocachers has made a useful start in the comments but I just have no idea!

[Quarg]

To solve this you must build your own Turing machine...No really...I don't see any way to break that code without the damn machine itself...

[WearsHats]

You mean like this one?

[Quarg]

You mean like this one?

No I was referring to the actual machine logic turring himself built to test multiple possible settings...

edit: Actually there are British Bombe simulators available but most rely on having a crib or a set of plain text likely to correspond to a message to determine the wheel settings...

[Davecom3]

They'll be pointing you towards downloading codegroup.zip there. As all parts of the code are made of groups of five letters, we can assume that we have a code where five letters correspond to a single letter or digit. If you look up codes that use a group of five letters (particularly capital letters) to correspond to either a number or a letter, you'll find that the most commonly mentioned code of this type on the internet is codegroup. That'd be where they are pointing you.

[Synch]

Good man Davecom3, thanks for the point in the right direction!