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# Lesson 5: Mathematics

This is the fifth in a nine part series originally written and published by ePeterso2 as part of his Puzzle Solving 101 Cache Series. Reprinted here with permission of the author. Minor edits have been made by Skottikus to apply this lesson to the Kingston Geocaching Area.

The first nine caches in this series will help you build your puzzle-solving skills. Each one contains a lesson focusing on a specific skill, examples of how to use that skill, an exercise to test that skill, and a cache (in Florida) to find as a reward. Study the lesson, complete the exercise, and you'll find the location of a geocache. Save your answers as they can be used to solve a special remote solver TB from here in Kingston as well!

### Introduction

 Counting sheep when you're trying to sleep, Being fair when there's something to share, Being neat when you're folding a sheet -- That's mathematics! Tom Lehrer, "That's Mathematics"

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:

epi*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.

Think you paid attention in class? Try Exercise 5 found at GCYXZ5 Puzzle Solving 101 - Lesson 5: Mathematics.