The Collatz Conjecture. The addition of a +1 in the odd number stage intrinsically alters the number, hence why you cannot 'track' the conjecture backwards. 1x2 is 2, 2x2 is 4, 4x2 is 8, 8x2 is 16, that much we can track without a doubt, but once you hit 16, was it (5x3+1) that got you there, or (32÷2)? (5) Is odd, and (32) is even. Not only do all the numbers lead to 1 eventually, they all must lead to 16 first. And considering that the conjecture only uses whole numbers, all numbers must be brought to either (5) or (32) to complete the conjecture. From there, we can say that (5) was derived from (10), as there are no other whole numbers that can be used within the rules to bring the number (5). We can also say that (32) came from (64) for the same reason. (10) can come either from (3x3+1), or (20÷2), where x=3, 20. (64) Came from either (21) or (128), same reasonings.
We could follow the conjecture backwards all day, until we have all numbers under 100, 1000, 10000000 listed, and they are all possible candidates for completing the conjecture. If all whole numbers can be even or odd, and all whole numbers can be multiplied, divided, or added into other whole numbers, there is no whole number that cannot be considered complete within the Collatz Conjecture.
