Hot tip for GMs when a player rolls badly on a knowledge/perception style check:
Don't: Tell them their character is going to argue that water is dry now
Maybe don't: Tell them something obviously wrong, or tell them no information at all. (No info is the fall back if I can't think of a more interesting result)
Maybe: Tell them that they're not sure if any of these things are related or how they could relate, throw in a red herring, or summarise what they already knew.
Fun: Tell them some almost entirely unrelated but mildly interesting trivia about the setting, or summarise some of the obvious facts for them in a way that's essentially a conspiracy theory.
I love doing this but it has also lead to situations where a character botches a knowledge roll, gets a somewhat absurd conspiracy theory, repeats it to another character who for some reason believes it, and suddenly the whole session is about hunting down the moon rabbit that has been rigging lunar city council elections...
Yeah no, the theory has to be plausible and related to what's going on, otherwise I'm putting it in as character-argues-water-is-dry but with someone going along with it because it's funny/it's a lead.
Like to clarify what I mean here, when you feed a player a wacky theory because of a low roll;
do not give any new information, right or wrong. High chance they assume it's bad info because they rolled bad, and/or act as if their character believes it if they think that would be funny. Players will happily and unwittingly derail the plot by doing what seems fun, I usually find the best way to solve that kind of tangent is to tell them OOC you made a wacky joke that didn't land/went too far and we should steer things back on track.
recontextualise what they already know in a way that emphasizes a gap in what they know, so they'll want to go fill the gap. You're signposting a lead by having it be the place/person/etc that would be their smoking gun/guy who's in on it/thing that'll blow this case wide open would be.
They're probably very different things, they just need to be able to be found in the same way. Something like searching a guy's office for the memo about the evil plan only to find out that he was fired a week ago, under some pretty dodgy sounding pretense by a guy who'd heard of before somewhere (and who the players were intended to be going after), and therefore couldn't have been the guy who did it.