The Collatz conjecture, also known as 3n + 1 conjecture, the Ulam conjecture or

Question

The Collatz conjecture, also known as 3n + 1 conjecture, the Ulam conjecture or the Hailstone sequence or Hailstone numbers, was first stated in 1937 and concerns the following process:
1. Pick any positive counting number, n.
2. If n is even, divide by 2. If n is odd, multiply by 3 and add one.
3. If n = 1, stop; otherwise go back to step 2.

For example, if we start with 6, we get the sequence:

6, 3, 10, 5, 16, 8, 4, 2, 1.

The Collatz conjecture states that this process ALWAYS terminates at one, regardless of which positive integer for which we start the process. The conjecture has been checked by computer for all start values up to 3 x 253 (about 27,000 trillion) but no proof has been found. Paul Erdos, one of the most prolific mathematicians of recent and perhaps all time, said about the Collatz conjecture: "Mathematics is not yet ready for such problems." He offered $500 for its solution.

Investigate the efforts placed into finding a proof and why it is so elusive. Pick a few of your own starting numbers and report a Hailstone sequence of length twelve or more.

Explanation / Answer

There are some heuristic, statistical arguments supporting the conjecture: if one considers only the odd numbers in the sequence generated by the Collatz process, then one can argue that on average the next odd number should be about 3/4 of the previous one, which suggests that they eventually hit the bottom. Sometimes the problem is stated differently. The termination condition ("If n = 1, stop") is removed from the procedure, so the sequence doesn't end. If you state the problem this way, the conjecture becomes the statement that the sequence always ends up in the repeating loop 1, 4, 2, 1, 4, 2

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