In a quick recap, they basically asked if the game of balls will ever end such that one person is trying to keep the game going on forever, and another is trying to make it stop. (If you don’t remember just go back and reread it.) So, it turns out (like I thought, but was second guessing myself) that the game will always end. It may take an extremely long amount of time, but it will eventually end.
So then this brings me to my game that I had designed. The one where each person is trying to outdo another one. I just realized that these two problems may be similar, but they are somewhat different. Theirs asks if one person can create an infinite amount of time out of a finite amount of object, which is definitely a no. My question asks whether or not you can create a formula such that no formula can ‘outperform’ it. And I can’t believe how simple the proof is!
Theory: There does not exist a ‘greatest’ formula.
Proof: Assume that there is a greatest formula f(x). Therefore |g(x)| ≤ |f(x)| by definition of greatest. Allow g(x) = f(x) * x. Therefore |g(x)| > |f(x)|. We therefore have a contradiction, and there does not exist a greatest formula f(x). QED.
Corollary: The game of formulas would go on forever.
Proof: Assume that the game has a finite time. This implies that one formula will beat every other formula and is therefore the ‘greatest’ formula, but by our previous theorem there does not exist a ‘greatest’ formula. Therefore the game does not have a finite time and will go on forever. QED.
Yay for being able to do proofs! I feel prepared for grad school. Just kidding.
-The Cali Garmo