I would allow it, but I'd also allow an experienced player to randomly pick a spot to search. And no, it wouldn't both me in either case. A character wandering the world might check a spot on the wall on a hunch. The issue for me is if the player knows there is a secret door there.
They can't unring the bell, so there's no way they can randomly search that section of wall on a hunch.
Ok, good. Now I want to unpack that a little bit. Let's refer to the secret door as "the secret", because this applies to any of the examples we've been discussing.
First, I think we can agree that if
anybody at the table doesn't know a secret they should be given a chance to discover it themselves. Spoiling
genuine problem solving for others is not cool.
But let's take the case where
everybody else at the table knows the secret, for whatever reason. The newbie happens to stumble across the answer on his first try, without any in-game knowledge. We're fine with that. It seems improbable, but we
easily rationalize why he might have taken that course of action. It doesn't feel impossible or implausible because, well, it happened.
Now magine that a few months later you're talking to the newbie and you mention that story, and he laughs and says, "Oh, I was totally pulling your leg. I've been playing D&D forever; I knew right where that secret door was."
So he actually cheated, right? I know I would be a little bit annoyed; I won't pretend that wouldn't bother me.
But why? The fact that it didn't bother you or me when we thought he was uninformed means that his
actions didn't actually impact our enjoyment of the game. What bothers us is not the in-game actions of the character, but the internal thoughts of the player.
Sure, we can hypothesize about what else might have happened if he hadn't revealed the secret and say, "By his actions he eliminated other possible pathways that might have been enjoyable." (But remember, as specified above, everybody else knows where the secret door is, so somebody else having the joy of discovery is not one of those possible pathways.) But the reality is that when we were ignorant of his thoughts we enjoyed playing, and it was only when we learned what he was thinking that we felt "cheated".
It seems to me that somebody else's internal thoughts shouldn't have any bearing on our own enjoyment of the game. All that should matter is what they
do. And thus anything we can accept and rationalize from an uninformed newbie should be equally acceptable from a savvy veteran.
EDIT:
Ok, apparently I need to be very explicit about the point being made here. I'll try to use short words.
If everybody at the table knows the secret, and everybody knows that everybody else knows, then solving it with an invented, in-game rationale is not lying.
We've established that it's not the action of solving the puzzle that some find offensive, it's what's going on in the head of the other player.
Don't worry about what's going on in their head; care only about what the character does.