The ubiquitous Geocaching.com GC codes are what inspired this puzzle.
For example, this geocache's GC code is GC5KW17. The 'GC' is simply an identifying prefix, no doubt formed from GeoCache. And the '5KW17' is a serial number which, on first sight, appears to be in hexatridecimal (or more accurately but less commonly, sexatrigesimal or base 36).
The base 36 number system system uses the number 0 to 9 followed by the letter A to Z. The letters are not case sensitive and so sometimes a lower case "o" and "i" is used to differentiate them from a zero and a one. If all lower case letters were used it will have a similar problem with the number 1 and the letter L. A hand-written "V" could also be confused with the letter "U", and an "S" could be confused with the number "5" It is for this reason that the designers of geocaching GC numbers decided to omit the 4 troublesome characters and instead use a base 31 number system with the digits 0123456789ABCDEFGHJKMNPQRTVWXYZ.
The geocaching.com web site automatically translates the letters "s" and "o" to five and zero respectively so as to be user friendly for those who confuse those digits. It may not work that way in other geocaching applications - so do try to use the real numbers. Remembering that there will never be the letters I, L, O, S and U in a GC code should make that easy.
The reason for using the base 31 number system rather than the conventional base 10 (decimal) system is in order to keep the unique GC code short. Back in the early days of geocaching available GPS receivers could only accept a reference code of up to 6 characters. The number was forced to increase to 5 digits after GCZZZZ in December 2006, and GC10000 (A New Beginning) is a memorial cache to the rollover.
But, to keep this puzzle simpler, the following numbers and calculations use true hexatridecimal.
To convert a hexatridecimal number to decimal manually, or with the help of a normal decimal calculator, first manually convert each digit to its decimal form. Number 0-9 are literal and A to Z is the position of the letter in the alphabet added to 9, Thus A=10, B=11 and so on until Z=35. In that way 5KW17 becomes:
5 20 32 1 7
Then multiply each digit by 36^n, where n is the position to the left of the least significant digit. So that is 5*36^4+20*36^3+32*36^2+1*36^1+7*36^0 which gives the result of 9372715 decimal. (In base 31 this number is only about 5 million).
A floating point hexatridecimal number can be converted in a similar fashion. A.BC is 10 . 11 12 which is 10*36^0 + 11*36^-1 + 12*36^-2, or 10.3148148148 decimal.
Using the above algorithm, you can find the final coordinates for this cache from the following data:
South Y 1.o7 East P Z.KH
Round your minutes to 3 decimal places.
