This is the seventh in a series of caches that we hope will help cachers learn some of the tricks to solving puzzle caches. Although experienced puzzle solvers can jump in at any point in the series, each successive lesson is meant to build on concepts that were demonstrated in the previous caches. The first cache in the series contains some background information on puzzle caches as well as a link to the tools we use when solving puzzle caches.
This series is not meant to be an "end all" on how to solve every puzzle cache that exists. It is only a starting point on what to look for when you see a puzzle cache. If you go through this series, you should be able to solve most puzzle caches that have a difficult of 3 stars or less. If we gave away all our secrets, then we wouldn't have anything to do but put out lamp post caches.
This series of caches contains the following caches
Disclaimer:
Alert: There are downloadable files in our toolbox and printable copies (PDF) of the puzzles for the caches in this series. These files are not required to solve the puzzles, although they may be useful to you for both this cache series and other puzzle caches you solve. As the cache owner, I represent that these files are safe to download although they have not been checked by Groundspeak or by the reviewer for possible malicious content. Download these files at your own risk.
Introduction
Besides the basic "add this number to that number" type math problems (we've had two in this series so far), there are plenty of other puzzle caches out there that require a bit more knowledge of math. It is way beyond the scope of this cache to provide a refresher course in algebra and geometry. There are plenty of web sites that can help with this. Henrico County, Virginia has a few that are worth a look: Algebra 1, Geometry, and Advanced Algebra and Trigonometry.
This cache will describe some math concepts and a few of the more common math problems you may encounter in your geocaching adventures and ways to go about solving them.
Numbers
Constants
A constant number is a number that is significantly interesting in some way. Here are a few you may come across:
π (pi) - the ratio between the circumference and diameter of a circle. You may see this any time you're dealing with a cache talking about circles. Equal to about 3.14 (or to 100,000 digits)
φ (golden ratio) - interesting only in the fact that humans percieve things in this ratio to be more eye-pleasing. Equal to about 1.61.
c (the speed of light) - universal speed limit in a vacuum. Equal to 299,792,458 meters per second or about 670,616,629 mph.
Prime Numbers
A prime number is a number greater than 1 that cannot be evenly divided by any number other than 1 and itself. Some examples: 5, 7, 31, 97.
Perfect Numbers
A perfect number is a number that is equal to the sum of its divisors excluding itself. 6 is the first perfect number. Its divisors are 1, 2, and 3 and 1+2+3=6.
Alternate Bases
The numbering system we use every day is called base 10 because it has 10 different digits (0 though 9). It's because we have 10 fingers and toes.* When we write a number such as 813, it means we have 8 in the hundreds place (102) + 1 in the tens place (101) and 3 in the ones place (100). But what would have happened if humans evolved with only 8 fingers and toes? That would mean we'd only use the numbers 0 through 7. The number 813 wouldn't even exist!!! Well, it would, but it wouldn't look like that--it would be written 14558. In base 10, it would be equal to 1 * 83 + 4 * 82 + 5 * 81 + 5 * 80 = 512 + 256 + 40 + 5 = 81310. Other common numbering systems are binary and hexidecimal. You'll see these numbers often when working with puzzle caches that involve computer information.
Sequences
A sequence is a ordered list. Sequences can be arithmetic which means you add a fixed amount to the previous number to arrive at the next. The sequence 1,2,3,... is arithmetic because we are adding one to the previous number to get the next. The sequence 9,18,27,36,... is also arithmetic because we are adding 9.
Sequences can also be geometric. This means that we're muliplying instead of adding. If I gave you a penny on January 1st and then doubled the amount I gave you on the first of every succeeding month, on December first I'd be paying you $20.48 because we are multiplying the previous amount by 2 the first of every month.
If you think a list of numbers might be sequential or if I said the decimal seconds of a cache were the 11th number in this series (0, 1, 1, 2, 4, 9, 20), try plugging them into the On-line Encylopedia of Integer Sequences. It has a list of over 200,000 different sequences and the formulas required to solve them.
Shapes
Circles
Circles are the shapes that most geocachers are familiar with, especially if they've hidden a cache. Since the Groundspeak guidelines state that a cache may be no closer than 528 feet from another cache, we have to make sure that our new caches meet that guideline. There are some puzzle solvers who try to play battleship when solving a puzzle cache. They'll create unpublished cache listings to see whether or not their coordinates are with 528 feet of another cache and then use some math to figure out the final location of the puzzle.
Triangle
Everyone should know that triangles have 3 sides and most people know that the 3 angles of a triangle add up to 180°. There are three types of triangles:
Equilateral - where all three sides are of equal length and all three angles equal 60°
Isosceles - Two of the sides have equal lengths and two of the angles equal each other.
Scalene - A triangle with no equal sides or angles
Squares/Rectangles
A square or rectangle is a shape that has 4 90° angles. The adjacent sides always have equal length.
Others
There are many other shapes out there along with mathematical formulas that can descibe them. However, these are rarely seen in puzzle caches.**
Others
There are many other ways math can be worked into a puzzle without using shapes--temperatures, distance, time, pressure, X-Ray Fluorescence, etc. Since coordinates are nothing but two numbers refering to a point on a map, any way that the hider can come up with those numbers is valid. For example, the original north coordinates for this cache are 
--it's kind of goofy, but it's true.
We'd recommend finding some nice online math calculators that can perform a lot of the grunt work for you. We have one that we use in our toolbox.
Now What???
So, by now you're saying to yourself--"I hated this stuff in school and I don't like it any better now." You haven't seen any puzzles out there that require any fancy math besides simple addition, subtraction or projections. To tell the truth, we're not big fans of them either, but there are a few caches here in town that do require a bit of math skills to solve.
Here's an example similar to a cache we've seen on our journeys:
In the desert there are two waymarks, one at N 36 19.707, W 115 21.852, the other at N 36 19.707, W 115 21.608. The cache is located exactly 1.5 miles towards the north of each of these points. Go get it!
So, how do we get there?
| 1. Well, we know that the distance from both point A and point B to the cache is 1.5 miles or 7920 feet.
2. The coordinates for the midpoint (m) between A and B are N 36 19.707, W 115 21.730 (608 + (852-608)/2)
3. If we use FizzyCalc, we see the distance between either of the waymarks and point m is 599 feet.‡
4. We use the Pythagorean Theorem (a2 + b2 = c2) for solving right triangles. We know a = 599 and c = 7920, so
5992 + b2 = 79202
358801 + b2 = 62726400
b2 = 62367599
b = 7897.3159 feet
5. If we use the projection function of FizzyCalc, and project the distance (b) from our midpoint (m), the cache should be at
N 36 21.009, W 115 21.730
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This Cache
This cache was an adventure to place. We started by hopping aboard the westbound California Zephyr at 10:30am while it was at its third stop. It was a fun trip and we got to see parts of the country in ways we haven't done before. However, after a day and a quarter of listening to the clickity clack of the rails, we got off at the next stop. From this point we hired a pilot and her biplane that had an average cruising speed of about 110 mph. We flew due south for an hour and a half before being warned that we were entering restricted airspace. We banked hard to the right and continued flying for another 72 minutes before the engine began to sputter because it was running out of gas. There were only two parachutes and since the pilot seemed to be preoccupied with finding more sheep to buzz, we donned the chutes and bailed out, watching her spiral in as we floated safely to the ground. From here we had to start hitchhiking. The first car to come by was a nice Audi R8. Since it only seats two people, we decided that we'd meet at a Starbucks that was 7.91 miles south of the cache. Brenda hopped in the R8 and headed southeast. A half hour later I was picked up by some white haired guy in a DeLorean. Two hours, 29 minutes and 24 seconds later I gave Brenda a call on the phone to ask where she was. She said that she's been waiting at the Starbucks for an hour and 59 minutes. I did a quick calculation and saw that the cache location was directly to my east and I still had another 11.36 miles to go to meet her. If that white-haired guy would have gone just one more mile per hour faster, we could have time travelled and I could have beaten her to the Starbucks.
Some notes:
If you're using FizzyCalc, use a Rhumb line to keep the coordinates straight.
Due to rounding errors and trying to keep to whole numbers, the cache probably won't be where your final solution says it is. This is a problem with projecting distances on an ellipsoid (the earth).
If you correctly solve the math problem from either Bob's or Brenda's final locations, the geochecker will give you the correct answer. However, because of the way the geochecker works, the number of correct answers will probably always show as zero since you can only have 1 "correct" answer.
There are additional checkpoints in the geochecker so you can check your progress on the way.
* Ok...that really isn't the reason, but it sounds good anyway.
** Now that we've said that, we're sure someone out there will come up with one.
‡ When using FizzyCalc, if you project a distance from a point, and then compute the distance between the original and new point, it doesn't always match the distance projected.
