For this problem, which I believe is still unsolved, I was wondering what is wrong with this proof I thought of (probably is wrong somehow)
So my proof has 2 sub proofs. The first sub proof proves that all even numbers work. The second sub proof shows all odd numbers work. I am going to handwave somewhat, but it goes like this.
Sub proof 1:
The base case works for 1. Then the induction hypothesis is assume it works for all numbers going from 1 to n, where n is an odd number. Now for n+1, which is even number, the conjecture says that you have to divide by two. So (n+1)/2 is clearly within 1 to n. Therefore by the hypothesis, (n+1)/2 will work. So n+1 works. So this proves that all even numbers definitely works for the conjecture.
Sub proof 2:
The second sub proof uses the fact proved from the first sub proof. So now, in this sub proof, take any odd number $t$ > 0, by the conjecture, you have to multiply it by 3 and add 1. The result of that, call it $r$, is an even number always. So by the result of the first proof, $r$ works with the conjecture since its even. So that means $t$ will work.
So by both the sub proofs, that shows all even and odd numbers > 0 will work for the conjecture.
This seems pretty logical to me, but feels too easy to be the actual answer.
Does anyone know?