There's a one-player math game called Tax Collector, and recently I was intrigued by it. Let me explain the game, and then I'll ask my question.
To start, choose a number to be your ceiling - the game will use all numbers from $1$ to $x$, where $x$ is your ceiling.
Then, claim a number to add to your score. The Tax Collector then claims all the factors of that number, and adds them to their score. Claimed numbers cannot be taken again.
Note that you cannot take a number with no unclaimed factors. This means that you can only take one prime number at most (with a factor of $1$), and all the others are guaranteed to go to the Tax Collector.
The game ends when you cannot claim any numbers, and all the unclaimed numbers go to the Tax Collector. Whoever has a greater total wins.
I started wondering if this game could ever be impossible to win. Instantly, I realized it could be - if your ceiling is $1$, then you can't take anything and the Tax Collector wins, $1$ - $0$. However, because you don't have a chance to make a move, I considered the game with a ceiling of $1$ to not count as a real game. So the question moved to $2$ and upwards.
I was stopped once again by the game with a ceiling of $3$, as it also proved impossible to win. Your two options were to pick $2$ or $3$ - picking $3$ was the better choice. Then, you get $3$ points, the Tax Collector gets $1$ point, and the Tax Collector gets two as well, since it has not factors left. Final score: $3$ - $3$.
However, after this, I tried greater numbers, and the games were always beatable, usually by a rather large margin. So here's my question:
After the games with ceilings $1$ and $3$, are there any ceilings that leave the game impossible to beat, no matter how well you play? I would assume no, since the prime numbers get scarcer as the numbers get larger, and primes are the Tax Collector's main source of income. However, I don't know how to prove that all games to infinity are beatable - and maybe there's an unbeatable one out there!
I would really appreciate help with this question. Maybe there's something obvious I'm not seeing, but I feel like this is more complicated than the game is itself.