Puzzlers Lessoո 10: Irrational Numbers Mystery Cache

This cache is not at the posted coordinates.

Introduction

This is the tenth in my series of caches meant to teach people some ways to solve puzzle caches. Please read the information above the line in Puzzlers Lesson 1: Alphabets.

The posted coordinates are one of the locations used to access this section of the Weber River Parkway.  This trail is also part of the Golden Spoke Trail Network.  This access location is in a neighborhood so please take that into account when looking for this cache.  Dogs should be on a leash, and you should pick up after your pets.  No motorized vehicles are allowed.

Besides the log and my traditional trackable, I was able to put three duckies and seven mini-duckies in the cache container. Please trade fairly.

Rational Numbers

Rational numbers are numbers that can be represented by a ratio of two integers where the denominator is not zero.  This means a fraction where you are not dividing by zero.  For instance, every integer (whole number) can be represented as that integer divided by one.

An example of a rational number is the fraction ¼.  The decimal representation is 0.25.  This is a fraction that has a terminating decimal. That is that it has a finite number of digits that do not repeat.

Another example is the fraction ⅓.  The decimal representation is 0.(3) which means that the digit 3 repeats forever.  (Using parentheses is one of the notations used to indicate repeating decimals.)  Any decimal number which has a sequence that repeats forever can be represented as a fraction.  There are many articles to read if you want to learn more about how to do this.

Yet another example is the ratio ¹⁵/₇ gives the decimal representation of 2.(142857) meaning that the digits 142857 repeat forever.

Irrational Numbers

Irrational numbers are the opposite of rational numbers.  That is, no matter how much you try there is no way to represent an irrational number precisely using a ratio (fraction) of integers.  It is possible to get a close fraction, but not any fraction that will give an exact value of an irrational number.

By their nature there is no way to calculate all of the digits of an irrational number, because they go on forever and never repeat.

Like rational numbers, there are an infinite number of irrational numbers.  Some of the irrational numbers are useful enough in mathematics to have been calculated to a very large number of digits.  Some of these are ln 2, √2, φ or Φ, √3, e, √5, π, δ, e^π listed in order of increasing value.

Transcendental Numbers

"In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial with rational coefficients." All transcendental numbers are irrational numbers, but not all irrational numbers are transcendental numbers. Here is one video that explains the difference.

This difference isn't relevant in this puzzle, but could make a difference if you see the distinction made in another geocaching puzzle.

Geocaching

So, what makes this information useful to know for solving geocaching puzzles?

The fact that irrational numbers go on infinitely without repeating means that there is a list of nearly random digits to use.  These digits can then be used to encrypt the coordinates to a cache.

In encryption there are four terms that are important to know:

• Plain text: The message to be hidden.
• Key: A value usually known only to the two parties who want to communicate.
• Encrypted text (cipher): The message that is made unintelligible to those not knowing how to decipher it.
• Method or algorithm: The means of transforming the plain text to encrypted text using the key. The algorithm may be kept secret or well known. A well known algorithm relies on the key to keep the message hidden.

Here are some examples of possible methods to use irrational numbers to encrypt/encipher coordinates to make them unintelligible. It is easier to explain how to encrypt using these methods.  Reversing the process is given as an exercise to the reader.

All of these examples use the posted coordinates as the plain text. In these examples, "Part" is used as a partial cipher. That is, it is an illustration of part of the process of enciphering the plain text.

Most of the illustrations of a partial cipher use * to indicate a digit to fill in that is any digit except for the next digit in the partial cipher example. For instance, if the next digit is 8 then pick any digit but 8 to fill in for the *.

1. Count (forward or backward) in the constant (key) from current position to the next digit of the plain text:
   • Plain: 411183211159571
   • Key: e
   • Cipher: KQ-UU-DEB-C-UUBF-F-EC
   • Note: The cipher is using one of the techniques from the first lesson
   • Note that the counting can begin with the first digit, after the decimal point, or even include counting the decimal point.  Another way to make the encrypted text is to reverse the roles of the plain text and key in the above examples.
2. Separate next digits of plain text by number of digits in key:
   • Plain: 411183211159571
   • Key: e
   • Part: **4*******1*1********1**8********3*2********1**1********1****5*****9*********57****1
   • Cipher: 264303769516135847300177859765244392597056981321298950021624155332493937061225758731
3. Separate next digit of key by number of digits in plain text, similar to above:
   • Plain: 411183211159571
   • Key: e
   • Part: 2****7*1*8*2********8***1**8*2*8*4*****5*********9*****0*******4*5
   • Cipher: 222697512852665163448589199882485406320530443157695392708778010445
4. Adding (or subtracting) plain text to entire number of key:
   • Plain: 411183211159571
   • Key: e: 271828182845904
   • Cipher: 683011394005475
5. Adding (or subtracting) plain text to individual digits of key:
   • Plain: 411183211159571
   • Key: e: 271828182845904
   • Cipher: 682901393994435
   • Note: This means adding/subtracting individual digits without carry/borrow.
6. Spelling out the digits of the plain text or key may be used, then apply one of the above:
   • Plain: 411183211159571
   • Key: Two seven one eight etc
   • Part: 4Two1seven1one1eight8...
   • Cipher: 4582134044125419455648...
7. Use the digits of the irrational number, but leave out the next occurrence of the next digit of the plain text.
   • Plain: 411183211159571
   • Key: e: 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132
   • Part: e: 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132
   • Cipher: 2.718281828590452353602874735266249775724709369995957496696762772407663035354759457382752516642742746691930030599287436629043290033429526059563073832
8. Give some other number to search for in the irrational number, then the plain text will be immediately before or after that. The number of digits to use may also be indicated.
   • Plain: 411183211159571
   • Key: e
   • Cipher: 68477+3-70276+3-46377+3-18188+3-47594+3
9. Since any sequence of digits may occur in an irrational number then the entire plain text will occur in the irrational number.  However, even if the irrational number has been calculated to a billion digits, the entire plain text may not be present in that billion digits.  Segments of the number may be given in the cipher, either as absolute locations or relative to the previous segment given.
   • This is like the previous example except using longer pieces
10. Deltas:
    • This can be like previous example except instead of giving absolute location of segment, a delta (plus or minus relative location) can be given
    • This can be like two examples previous except that you always go forward from previous segment to next occurrence of that string of digits
11. Picture: Use any of the above methods to highlight digits in a long list to show a picture.
    • Think of the "Part" in examples 2 or 3 (above) except in multiple lines
    • Then the multiple lines together will make a picture containing the coordinates
    • Even if this coloring of the digits is meant to be a picture, they may not be broken down into lines of equal length. This may be one long line, or multiple lines that need to be made into lines of equal length. This may be indicated or left as an exercise to the reader.
12. One off:
    • Any of the above examples except that the key is consistently one digit off, +1 or -1
    • This should be accompanied by a clue (not always obvious) in the description that this is being done
13. Reverse:
    • Reverse the plain text then apply one of the methods above
14. Swapsies:
    • Any of the above examples except that the roles of the key and the plain text are swapped

For your information, here is the first 1000 digits of e: 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729003342952605956307381323286279434907632338298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537423454424371075390777449920695517027618386062613313845830007520449338265602976067371132007093287091274437470472306969772093101416928368190255151086574637721112523897844250569536967707854499699679468644549059879316368892300987931277361782154249992295763514822082698951936680331825288693984964651058209392398294887933203625094431173012381970684161403970198376793206832823764648042953118023287825098194558153017567173613320698112509961818815930416903515988885193458072738667385894228792284998920868058257492796104841984443634632449684875602336248270419786232090021609902353043699418491463140934317381436405462531520961836908887070167683964243781405927145635490613031072085103837505101157477041718986106873969655212671546889570350354

Also remember that the coordinates may be given in any of the coordinate formats discussed in any of the previous lessons.  This can include only the final portion of the north and west portions of the coordinates.  For instance, there may only be the lowest 3, 4 or 5 digits of the north and west coordinates.

The west coordinate, or partial coordinate, may also be given first.

Since the coordinates of the geocache are meant to be discovered, eventually, then enough information to discover the coordinates will also be given. The usual case is to give the cipher text. Often, the method and/or key will also be given. However, this is not always the case. This information may also not be made plain but may be hidden in the text description of the geocache, or even its name.

Puzzle

This time you will have to use one of the methods and one of the constants given above to decode the coordinates for this cache.  Discovering which ones will serve as an exercise in solving puzzles.

660810631582913534300783796656823796848304907509087556458249011031278207279

You can validate your puzzle solution with certitude.

Congratulations to runninbear1, del2u, Dix1, Little Tweety and Li'l Monkey on being first to find.