The cache is not at the posted coordinates
About This Series
This series is based on the original puzzle solving 101 by ePeterso2. The first twelve 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 to find as a reward. Study the lesson, complete the exercise, and you'll find the location of a geocache.
When you check your coordinates, each cache will reveal a piece of information you'll need to take the final exam (the thirteenth cache in the series).
Lesson 10: Mathematics
Introduction
Mathematics is a topic that is hated by many. Yet we use it every day in some form or another. If you have ever added two numbers or projected a waypoint, you used mathematics. Our GPSr calculates its position every second using trigonometry. Mathematics is everywhere. Unsurprisingly there are a number of math puzzles out there for which more or less math skills are required. But the use of math is not limited to math puzzles, many other puzzle types also use math to get to the final coordinates. It is frequently used in multi caches to calculate the next waypoint.
In this lesson we hope to show you that the math used in most puzzles usually is not difficult. For the few puzzles where somewhat complicated math is used, we'll show you some great resources that will get you to the right answer, even though you might not understand how you got there!
Topics in Mathematics
Constants
A mathematical constant is a special number that is "significantly interesting in some way". What makes them "interesting" is apparently a matter of taste. Some constants are specific to a certain field, others have applications to many different fields of science. Some common constants are:
- Archimedes' constant π (3.141592653589...)
- Euler's number e (2.71828182845904...)
- The golden ratio φ (1.6180339887498...)
There are a lot of puzzles that can be created with these numbers. For instance, did you know that you can find almost any number within the sequence of digits for π? There is even a search page for that! Apparently my birth date occurs on position 75,839,200 counting from the first digit after the decimal point. Since this search can be done both ways, it is a great way to hide coordinates. For instance, the posted north coordinates of this cache can be found at position 8,043,924.
Interesting Properties of Numbers
Although some say that all numbers are interesting, some numbers have interesting properties. A prime number for instance can only be divided by itself or 1. This property has many applications in the world of cryptography and many modern (computer) ciphers are built on this principle.
Another class of number are the perfect numbers for which the sum of its divisors equals the number twice itself. The first perfect number is 6 since 2*6 = 1 + 2 + 3 + 6. The next three perfect numbers are 28, 496 and 8128.
Other properties of numbers are lucky, amicable, sociable, practical and deficient, to name just a few. For all of these numbers, they can be used with their indices (the n-th perfect number), or in other formula's (961.389193575304 is π6 which could represent the number 6 in a coordinate).
Alternate Bases
Numbers are represented with a particular base. Our normal base is 10, meaning we have numbers ranging from 0 to 9 forming a decimal system. The number 1970 for instance means 1*103+9*102+7*101+1*100.
Other well known bases are 2 (binary), 8 (octal) and 16 (hexadecimal). If we want to write 1970 in a septenary (base-7) system we get 5513 since 5*73+5*72+1*71+3*70 = 5*343 + 5*49 + 1*7 + 3*1 = 1970. Older civilizations had different bases for their numeric system as well. The Mayans for instance had a vigesimal (base-twenty) system. They represented each number with a combination of lines and dots. Converting numbers to a different numeric base is a useful technique to 'hide' the coordinates of a cache.
There is a great page on purple hell that allows you to convert any number in to any base. The posted Northing would for instance be 1GNVM0 in base 32.
Topology
The field of mathematics that studies the spatial properties of objects that are preserved under continuous deformation is called topology. Continuous deformation means that it can be stretched, bended, folded, etc. but not torn or glued. A famous joke is that people who study topology cannot see the difference between a coffee mug and a donut. This is because from a topology point of view they are the same: an object with a hole in it.
Sequences
In mathematics, a sequence is an ordered list of objects. A number of puzzles are based on sequences to represent the digits that need to be found. For instance we can write the Northing of the posted coordinates as "233 31.5897" which are the 51st prime (233), the 11th prime (31) and the 776th prime (5897).
Other popular sequences include:
- the Fibonacci sequence (0,1,1,2,3,5,8,13,21,34,...) where each number (except the first two) is the sum of the two previous numbers,
- arithmetic sequences like 1,5,9,13,17,21,... which are formed by adding a constant number to the previous number,
- geometric sequences like 3, 12, 48, 192, 768, ... where every term is a multiplication of the previous term with a constant
For all these sequences the trick is to identify which sequence you're dealing with and how that would be used to represent the digits for the cache location. Indices are a good starting point for this.
Functions
A function describes the relation between one (or more) value and an output value. For instance we can write y = a*x +b where x is the input value, a and b are constant and y is the output value. For every value of x we can now calculate y. Functions are available in many forms and complexities such as
There are many ways to use functions in puzzles, from calculating the coordinates by putting the given values in the function, to determining where the function intersects the x-axis (for which x holds that a*x + b = 0?).
Solving a function can be a daunting task for most but fortunately the internet is there to help you. To determine where the function above intersects the x-axis, we can use this link which shows that the function intersects the x axis four times.
Geometry
Of special interest to us geocachers is geometry. It is one of the oldest mathematical sciences and it deals with shape, the relative position of figures and the properties of space. Whenever we make a projection of a waypoint, this is the area of math that we use. When we intersect three circles, we can use geometric calculations.
In many of these calculations triangles and angles are used. Two useful 'rules' to remember are the law of sines and the law of cosines. Both 'laws' describe the relationship between the angles and the side lengths of a triangle. They allow you to calculate angles or side lengths given either two side lengths and an angle or two angles and a side length.
Resources
The most valuable resource I can point you to is "Wolfram|alpha". This is for math what Google is for searching the web. It will answer complex math problems for you and it provides helpful suggestions if you don't really know what you are looking for. Give it a try!
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.
Practice Puzzles
If you really want to dig into the math, try the Math 10 puzzle in Edmonton.
Puzzle 10: Grade 4 Math
Click on the image to open a higher resolution image that can be printed
If you think you have this solved you can check your solution.