C.S.U.Puzzles 301 Mystery Cache

About This Series

This series is planned to be ten caches. The first nine will provide lessons to assist you in building your puzzle-solving skills. Each will contain a lesson centering on a specific puzzle skill, examples of how to apply that skill, an exercise to test you on that skill, and a cache to find as a reward for your efforts. Study the lesson, complete the exercise, and you'll find the location of a cache.

Each of the caches will contain a piece of information you will need to help you take the final exam (which will be the tenth and final cache in the series). When you visit each cache, you will need to bring something to record those clues for later ...like paper and pen/pencil or perhaps a camera.

One final note: Each of these caches will have an unactivated geocoin in it as a First to Find prize. Crew416's rule: IF YOU HAVE BEEN FIRST TO FIND ON ANY CACHE IN THIS SERIES, YOU MAY NOT - NOT - NOT FIND OR LOG ANOTHER ONE IN THE SERIES FIRST UNTIL IT HAS ALREADY BEEN FOUND AND LOGGED BY SOMEONE ELSE!!! That means there should be 9 different First to Find cachers on the first 9 caches in this series. The final exam will be open to all, including the 9 first to finders on the first nine caches, and will also contain a nice geocoin prize for the first finder. This will give a few new puzzle cachers the opportunity to be First to Find on a puzzle cache, perhaps for the first time ever. This should give more cachers the thrill of solving and being there first and score a new geocoin prize... and one experienced puzzle solver doesn't end up with all the coins. So shall we continue to our next lesson?

Lesson 5: Mathematics

Introduction

If you've ever found a cache, you've used mathematics. There's a pretty sophisticated amount of trigonometry and more that makes it possible for you to punch the cache coordinates into your GPS so that you can follow the arrow to the container.

In your caching journeys, you may have encountered an offset cache - a cache that requires you to go to a certain location, find or deduce some set of numbers, and add those numbers to your coordinates to find the final coordinates. That's one of the most common types of mathematics seen in puzzle and multistage caches.

The purpose of this lesson is not to try to cram the vast totality of the mathematical body of knowledge into a few pages in a cache description. Its purpose is simply to give you some exposure to various math topics you'll occasionally come across that are used in puzzle caches, along with some references to understand them. As always, Google is your key to unlocking more information about all of these topics and more.

Topics in Mathematics

Constants

A constant is a number with a specific value, often given a single letter name for easy reference. Numbers such as i (the square root of -1), e (the based of the natural logarithm), and pi (the ratio of a circle's circumference to its diamater) are some of the more well-known. All of them appear in unexpected ways throughout the study of mathematics, most notably in the famous relationship discovered by the great mathematician Euler:

e^pi*i + 1 = 0

Interesting Properties of Numbers

A prime number has no factors other than 1 and itself. In other words, you cannot divide a prime number by any number and get a whole number as a result. The numbers 2, 3, 5, and 7 are prime, whereas 4, 6, 8, 9, and 10 are not. A number that isn't prime is called composite.

A perfect number is a number whose factors other than itself add up to that number. For example, the factors of 6 are 1, 2, 3, and 6; the sum of 1, 2, and 3 is 6.

Numbers can be happy, weird, frugal, extravagant, sublime, friendly, and more.

Alternate Bases

Our number system is what is called base 10 because it has ten different digits, zero through nine. (Okay, that's oversimplifying tremendously. Apologies to you math majors out there.) The number written "10" in base ten means that there is 1 ten and 0 ones in the value. The number "342" means 3 hundreds plus 4 tens plus 2 ones.

But what if we only had eight digits in our numbering system instead of ten? Instead of the tens place, we'd have the eights place. And instead of the hundreds place, we'd have the sixty-fours place. So 342 in base 10 is 342, but 342 in base 8 is (in base 10) (3*64)+(4*8)+2 or 226.

Computers operate in base 2 (binary), and you often see computer numbers represented in base 8 (octal) or base 16 (hexadecimal, with the letters A through F used to represent the values 10 through 15).

Topology

Topologists can't tell the difference between donuts and coffee mugs - they consider both equivalent, which is why you never see them at Dunkin Donuts (or if you do, why they have coffee all over their pants).

Topology is the study of shapes ... topologists deal with knots and twisted ribbons and holes and more. Two shapes are considered equivalent if you can stretch, twist, mold, and bend (but not tear or puncture) one shape to make another. Which is why the donut shape is equivalent to the coffee cup shape - both have exactly one hole (the coffee cup has an indentation, but that doesn't count as a hole).

Sequences

A sequence is an ordered list of items. The list may have a fixed number of items in it, or it may be infinitely long.

An arithmetic sequence is additive. If you begin the sequence with a particular number, you find the next number in the sequence by adding a fixed amount to it. The sequence 1, 2, 3, ... is arithmetic. So is 2, 5, 8, 11, 14, ...

A geometric sequence is similar to an arithmetic sequence, except you multiply instead of add. Here's a geometric sequence where each term is multiplied by 2 to get the next term: 1, 2, 4, 8, 16, 32, ... You can also multiply by numbers smaller than one or even negative numbers.

A Fibonacci sequence starts with two terms (such as 0 and 1), then adds the two together to get the next term. Then repeat with the last two terms in the sequence to get the next term. So 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, ... If you read The Da Vinci Code, you know all about this sequence.

History

The history of mathematics is full of fascinating stories of the origins of mathematics in ancient cultures and of famous mathematicians (Newton, Euler, Gauss, Erdos, and more).

Resources

Searching for the term recreational mathematics or mathematical puzzles will give you a wide variety of links to sample problems (with solutions), further topics, typical puzzles, and much more.

Wikipedia also has an excellent mathematics portal along with topic pages on

recreational mathematics and mathematical puzzles.

Of course, no discussion of math would be complete without a bunch of links to horrible math jokes

Exercise 5: A Rhind Is a Terrible Thing to Waste

The ancient Egyptians were highly skilled mathematicians. By 2700 BC, they had the earliest-known fully-developed base 10 numbering system. Using this system, they were able to study arithmetic, algebra, number theory, linear equations, and the beginnings of integral calculus. They used this knowledge in a variety of government, business, scientific, and engineering applications ... including the construction of the Great Pyramids.

It is a little-known fact that the ancient Egyptians also were avid geocachers. Below is an annotated photo of one of the Tombs of the Scribe Surveyors at Thebes in southern Egypt on the western side of the River Nile, almost adjacent to the famed Ramesseum, a grand temple built in honour of Ramesses the Great.

Unfortunately, Egyptian numerals were considerably unwieldy and cumbersome to use by today's standards.

For instance, the ancient Egyptian version of this cache description page might show the north minutes like this:

And the west minutes like this:

Hmmm ... I guess it's not much of a puzzle if I just go ahead and put the coordinates of the final location in the cache description. Oh well ... maybe the next cache will be tougher.