A sister question asks when the Collatz Conjecture will be resolved - here we ask which way it will turn out.

Again, let's say that the Collatz Program in pseudocode is:

collatz(n) = 
  if (n is 1) return 1
  else if (n is even) return collatz(n/2)
  else return collatz(3n + 1)

where n is a positive integer.

The Conjecture is that for all integer inputs the Collatz Program halts (and returns 1).

For any particular execution of the Collatz program, there are three possible outcomes:

1) It moves up and down through input arguments of different sizes, until it encounters a power of 2, and then cascades down to 1, and halts.

2) It moves up and down through numbers of different sizes until it repeats a number. From that point onward it will repeat a cycle, and never halt.

3) It moves up and down through numbers of different sizes, but keeps expanding its frontier of numerical size, without ever repeating an input or encountering a power of 2. In this case, it will never halt.

Per Wikipedia, Jeffrey Lagarias in 2010 claimed that based only on known information about this problem, "this is an extraordinarily difficult problem, completely out of reach of present day mathematics."

This question will resolve positively if there is a positive proof of the Conjecture (i.e. that the Collatz Program halts for all integer inputs) in a major Mathematics journal before June 21, 2520. It will resolve negatively if there is a publication of a disconfirmation in a major mathematics journal before that time.

If the Conjecture has neither been proven nor disproven before that time, it will resolve as ambiguous.

Other questions on the Collatz Conjecture: