Project Euler 14 Mystery Cache

Acknowledgements

This cache was inspired by Project Euler.

General notes

• To solve Project Euler puzzles, you will probably have to write a computer program.
• Many of the containers are small, and contain neither pen nor pencil.
• There are explicit finding notes in each geochecker.

The Collatz conjecture

Given the number n, consider the sequence formed from it by iterating the rule:

• if n is even, halve it;
• otherwise triple it and add one.

For example if n = 6, we get the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1.

The Collatz conjecture states that whichever number you start with, you'll eventually always get back to 1. Nobody knows if this is true or not, but it has been shown to be valid for all the numbers less than 5 ×10¹⁸.

Sequences

There is no simple relationship between the starting number and the length of the sequence it takes to reach 1, but this can be explored experimentally by computer. For example, if we consider the sequences which start with a number less than 100, the longest has 119 terms starting at 97:

97, 292, 146, 73, 220, 110, 55, 166, 83, 250, 125, 376,
188, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121,
...
488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53,
160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The Puzzle

To find this cache first find the longest Collatz sequence starting from a number less than 10,000,000. Having found that, extract terms 200, 249, 250, 360, 275, 222, 255, 203, 204, 205, 280, 225, 244 and 246 from the sequence. From these, you will be able to extract the coordinates of the cache.