I don't really know how to prove something, but for the first problem, it seems like any odd number multiplied by any other odd number (in the example, they choose 3) will always be odd. This can/has been proven. Then, any odd number, negative or positive, with 1 added or subtracted to it, becomes even. Finally, any even number divided by two is still even and approaches two.<p>"Mathematicians have tried millions of numbers and they've never found a single one that didn't end up at 1 eventually. The thing is, they've never been able to prove that there isn't a special number out there that never leads to 1."<p>Why are they trying millions of numbers? It seems like those 3 statements are very easy to prove, and explain this "phenomenon." Also, isn't the multiplying by 3 part kind of arbitrary. It seems like the only important part to consider is that if the number is odd, add 1. The multiplying by 3 is unnecessary, and could just as easily be swapped for multiplying by any odd number.