To find this cache, you must forget what you know about Base-10 and Base-2. We are stepping into the world of Leonardo of Pisa and a theorem by Edouard Zeckendorf.
Zeckendorf's theorem states that every positive integer can be uniquely represented as the sum of non-consecutive Fibonacci numbers.
We ignore the first "1" in the traditional sequence to avoid duplicates.
Example:
If we want to encode the number 19:
Find the largest Fibonacci number less than or equal to 19: 13
Remainder: 19−13=6.
Find largest Fib number ≤ 6 : 5.
Remainder: 6−5=1.
Find largest Fib number ≤ 1 : 1.
Sum: 13+5+1.
Now, map it to the place values:
13 (1)
8 (0)
5 (1)
3 (0)
2 (0)
1 (1)
The code for 19 is: 101001
The cache is located at:
N 37° AA.BBB W 121° CC.DDD
To find AA.BBB, decode this Zeckendorf string:
10100000.100010100010
To find CC.DDD, decode this Zeckendorf string:
101000.10010101000101