The above coordinates are NOT the location of the cache. You either like this kind of cache and find it interesting or you don’t … oh, well.Much computer encryption makes use of modular arithmetic, also called clock arithmetic in elementary school, and one-way functions. One-way functions are easy to perform in one direction but harder or impossible to perform in the reverse, or require a different set of rules.
In this cache I’ll tell you exactly how I encrypted the cache coordinates—your job is to reverse the process.
Instead of substituting one letter for another letter in the alphabet, I used trigraphs, three-letter groups of letters, and assigned them numbers. The number of possible trigraphs (different ways of putting three letters of the alphabet together) is 26^3 or 17,576. The trigraph ‘aaa’ is given the number 0. The trigraph ‘zzz’ is 17,575.
For example, to convert the trigraph ‘PAL’ into a number, I would use the following formula. ‘P’ is the fifteenth letter of the alphabet (remember, ‘A’ is zero), ‘A’ is zero, and ‘L’ is the eleventh letter. Therefore to encrypt them:
= (15 * 26^2) + (0 * 26^1) + (11 * 26^0)
which solves as
= 10140 + 0 + 11
= 10151
So the number corresponding to the trigraph ‘PAL’ would be 10151. I could just use trigraph numbers to hide a cache but that would have been too easy, wouldn’t it? We’ll call the number we got, in this case 10151, the ‘plaintext’ number or ‘P’.
Modular arithmetic, also called clock arithmetic, is what we use when we tell time. A clock has twelve numbers. For modular arithmetic we always start at zero. What time would it be fourteen hours from now? Two o’clock since we’d go around the clock once and get back to two. In modular arithmetic, the answer is the remainder, the amount left over after we wrap around the clock as many times as we can.
A clock would be modular 12 written ‘mod 12’.
14 mod 12 = 2 since you’d go around the clock once and have a remainder of 2.
What is 27 mod 12? Well, we’d wrap around the clock twice and have a remainder of 3. Therefore
27 mod 12 = 3
But we can also do modular arithmetic using any number as the modulus or base. Imagine a clock with 26 numbers. What would be the answer to
577 mod 26
See how many times 26 goes evenly into 577. It goes 22 times with a remainder of 5. Therefore
577 mod 26 = 5
Here are a few more. Make sure you understand how to get the correct answer before proceeding.
5 mod 12 = 5
1234 mod 891 = 343
(55 * 22) mod 91 = 27
So next in this cryptogram I encrypted the plaintext trigraph number using the following modular arithmetic formula. ‘C’ stands for ciphertext or in this case, the ciphertext number.
C = (159 * P) mod 17,576
Let’s do it with the plaintext number (P) we got for ‘PAL’.
C = (159 * 10151) mod 17,576
C = (1,614,009) mod 17,576
See how many times 17,576 goes into 1,614,009 and determine the remainder. It goes in there 91 times with a remainder of 14593. The remainder is the answer.
C = 14593
I could give you a list of all the ‘C’ numbers but I went a step farther.
Next I converted my ciphertext number (C) into a trigraph itself. In this case, we convert 14593 into its corresponding trigraph.
= (x * 26^2) + (y * 26^1) + (z * 26^0)
Calculate how many times 26^2 (or 676) goes into 14593. It goes 21 times so our first number (far left number) in our trigraph is 21.
21 * 676 = 14196
This leaves us with 397
14593 - 14196 = 397
Calculate how many times 26^1 (or 26) goes into 397. It goes 15 times so our second number is 15.
15 * 26 = 390
This leaves us with 7
397 - 390 = 7
So our final third number is 7.
= (21 * 26^2) + (15 * 26^1) + (7 * 26^0)
= 21 15 7
Converting these numbers into their respective alphabet letters— (again, remember A is 0 and Z is 25)
21 = V 15 = P 7 = H
Therefore, my original trigraph of ‘PAL’ is encrypted as ‘VPH’.
Okay, so here’s the cryptogram giving the cache location encrypted in the same way. The hard part—I guess among other things—is figuring out (researching) how to reverse the modular arithmetic formula.
05/08/2018 By JA1 I came across this puzzle yesterday when I was looking for one of this DT rating that satisfied the California well-rounded cacher requirements date wise. There are only 36 in the US. Started working on it with pencil and paper but that didn't go so well. It did let me get my thoughts in order. Then sat down and created a very lengthy spreadsheet. That was turning out to be very tedious. Tried using a basic program but that didn't pan out either. Sat down and analyzed the math and came up with a much more efficient spreadsheet. Fancied it up a bit and then started working on the rest of the string. Had all the answers within another 20 minutes. Great puzzle! Hope to come out there and find it sometime.
Thanks for the fun.
JA
10/12/2017 By jasperdakota Caching and hiking in the Mt. Herman area today with geodog Dakota. Thanks for the fun puzzle cache!
The lid was off, the cache had water in it and the log papers were scattered around. I dried it out and put the papers back in. The lid is cracked. Maybe the next finder can bring a replacement small lock-n-lock. SL.
08/01/2017 By kimbotjr We found this cache as part of some vacationing in Colorado. We often target a few caches when we travel, usually trying our luck on a few higher difficulty/terrain caches, puzzle caches, ones that add new counties, and others that well…just seem interesting. Wow, this was a challenging and fun puzzle. We had to get a little nudge to get started, but when we did we were on our way and ended up doing some extensive noodling and work to make the solve. We also enjoyed the hide, which is placed in a nice location. Leaving a favorite point for sure. Thanks for the very nice puzzle cache.
07/02/2016 By NYWanderer This morning anticipating a drive down to New Mexico to get another state souvenir, I spotted this puzzle which I needed to complete my Unknown fizzy grid. Read through it and quickly determined that I could solve it and finish it, if not on the drive down from Denver, surely by the time we're driving back. However our first stop was the GW14-er meet and greet so while my geopartner had to wait in the long preregistration line, I could take out my laptop and, with some help from trusty ol' awk, bc and some bash scripts, automated enough of the math guesswork to get the solve in about 20 minutes, before my friend even made it to the front of the line!
We made our way down and stopped for the find. Cool area, nice little hike and awesome final container. FP. Tftc.
This is the 12th day of my road trip to GeoWoodstock with NYWanderer. The plan for today is to attend the Meet & Greet event then head down to New Mexico to pick up a couple of caches. The Meet & Greet was the worst organized event I've ever been to. Rather than spending time catching up with friends I've made over the years at other events, I spent my whole time there on line waiting to pick up my registration packet and NYW spent his time solving a puzzle cache. The only plus was chatting with a lovely couple from New Hampshire who drove all the way out to Colorado and getting to meet MOPirate whose caches we visited when we were in Missouri last week. After we finally got out of there we headed south for some caches before getting to New Mexico. We also ran into some pretty bad storms on the way south. I was driving the south bound leg of the trip and a few times I completely lost sight of the lines on the road. It was a bit scary for a while, but we made it through. Managed to fill in a couple more grids on my D/T grid and maybe one or two more mo the on my Jasmer. It wasn't a high caching day, but the caches we did get were quality. Including FTF#7 for this trip!!
This was the puzzle that NYWanderer was solving at the meet & greet event. i love driving off-road (or at least off tarred road) about as much as NYW likes scrambling over rocks. This mean I was much happier than he was today because I got to do more off-roading than he got to do scrambling. This was an awesome cache hide!! It definitely gets a favorite point from me for originality. I enjoyed the walk up to the cache as well. Not too challenging except I've been battling the altitude since I got to Colorado especially. Thanks for the cache. I will definitely look at this puzzle when I get home to NYC.
01/23/2016 By k9jayhawk Picking up a few caches with OF-Erad...TFTC SL
01/23/2016 By OF-Erad Found on another super fun cache run with K9jayhawk. Loved the hide! Thanks for placing this cache for us to find today! SL, TFTC!
09/17/2015 By XTerra85 Plucking another caching off the solved list! Loved the hide. It definitely needs some TLC but you don't find too many like this anymore. Tftc!
02/13/2015 By RPieperCO TFTPC! Yeah, I had to have help on this puzzle. I got halfway through the puzzle and couldn't make the next leap. I contacted a friend THeSKuLL who wrote a handy dandy Javascript program to do the dirty work! It worked like a charm! So after several days we got the output we needed! Alas, Mt Herman Rd is closed till Feb 2015 for repairs. Today the weather was nice and I decided to head off to see if the road was open and to enjoy a nice little hike. I came at this cache in the wrong direction from another nearby cache. I fought my way through a thick of scrub oak like I've never been through! I sacrificed a lot of skin and blood to get to this cache! As always the way out was 1000 times easier! (which would reflect in the T rating, my method would have been 4 easy!) After a very lengthy search I finally sat down for a break and happened to look in the right spot. GEESH!!! 1 foot literally in front of my face! I had been within inches about a dozen times! GREAT hide! The cache is in perfect condition! Thanks for the great puzzle, fun hike and amazing hide! Giving this one a favorite!
01/10/2015 By colorado gypsies Absolutely terrific. I remember programming class in the nineties on the apple IIe coding random mod 4 + 1 to make my dot move up down left or right. I thought about finding a BASIC emulator to solve the cypher but ended up using another method. I loved the puzzle and the hide. We walked in about a mile past the gate to make the find. The camo is still effective :-) SLTNLN TFTC Fav.
Crypto Cache # 7 - Modular Arithmetic
| Difficulty: 
| Terrain: 
Good Will Hunting
9 
1