About This Series
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 to find as a reward. Study the lesson,
complete the exercise, and you'll find the location of a
geocache.
Each of those caches contains a piece of information you'll need
to take the final exam (the tenth cache in the series). Bring some
way of recording those clues for later ... paper and pen/pencil
would come in handy, or perhaps a camera. (A hammer, chisel, and
very large rock would work but probably wouldn't be very
handy.)
Lesson 5: Mathematics
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.
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, alegbra, 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.
This puzzle was inspired by (but is completely unrelated to)
the
Plimpton 322 cache, one of my favorite math puzzle
caches.
