@shavixmir said
Surely timesing any uneven number by 3 and adding 1 will make it an even number.
And deviding an an even number by 2 will half its size.
So, ultimately the number will get smaller and smaller, and, eventually reach 1.
That does not (necessarily!) follow. You overlook that the odd case increases the number
more than the even case decreases it. To be precise, the increase is slightly over (because of the +1) 150% as the decrease: times three plus a bit vs. divided by two. This means that, if there were always one even case for every odd one, the numbers would rise indefinitely if jerkily: three steps forward, two back, as in Echternach.
However, that is not the case. An odd case must inevitably lead to an even one, but an even case may lead to an odd one
or more even ones. Therefore, there will be more even cases than odd ones. The crucial question is now: are there sufficiently
more even cases to counterbalance the
larger odd ones?
Collatz' conjecture says: yes, for every number.
Experience says: yes for every number we have tried, but for some only by a small margin, and we haven't tested them all yet so there may be a few, rare numbers for which the answer is no.
Proven mathematical reasoning says: we don't know, and we don't even really have any idea how to tackle the problem definitively.
We
do have a proof that it is true for 'almost all' numbers, meaning that it is not possible for the conjecture to be mostly true up to a large number but false above that. However, there are other conjectures we knew to be almost always true, believed to be always true, and then found a large counter-example, for instance the Pólya conjecture.