69,105 Keys

In this game, there are 69,105 keys, only one of which will open the door. The key you need is the only unique key.

There are many categories of keys, and you can count each category. The number 69105 is I believe a riff off of Zork I.

As a mathematician, I hoped that the puzzle would involve some kind of bizarre combinatorial computation; instead, it's mostly just trying every category until you find a pattern.