Stage 5 - The Pi Vault

[Mr. Monteleone delivers this puzzle on a small piece of paper]

Somewhere, there is a vault protected by a lock with three buttons. Kevin, to open the vault, enters the following sequence of button pushes.

[Solution: In binary, the button presses are 3, 1, 4, which tell you that the next destination is room 314.]

Hint 1: There is some meaning behind the buttons Kevin pushes. Where might this “vault” be?

Hint 10: Huh? This isn’t the tenth hint!