Gaming Logic: Perception and Investigation

“How do you adjudicate passive vs. active Perception checks?”
Passive Perception is for noticing things out of place, hidden creatures, secret doors, traps, and other clues. It assumes the character is just moving along, being aware of the world at average capacity, which might be very high for high Wisdom or trained characters.
It also assumes the character isn’t rushed or distracted; if the character only has a few seconds to peer into a room or see a useful handhold before the cliff collapses, I’ll call for an active Perception check.
My threshold is about a minute: if a character has a full minute to notice something, I’ll allow Passive Perception to kick in; otherwise it’s active checks. Likewise, if the character is in combat and ends up in a new room, I’ll let them look around with an active Perception check; passive Perception doesn’t apply in that case, unless they’ve been fighting in the room for about a minute.
Perception and Investigation

Original artist unknown (Wizards of the Coast?); Meme from Hack & Slash:

“What about other passive skill checks, like passive Investigation or Insight?”
Passive Investigation isn’t a thing at my table. Though having a high Passive Perception can clue you in that an Investigation check might be useful.
Similarly, Passive Insight isn’t a thing at my table. If a player expresses some doubt that an NPC is telling the truth or is hiding something,
The only skill that can be used passively at my table is Perception. Even on the frequent occasions when my characters are being passive-aggressive, that’s still an active check. 🙂
“What does it mean to *fail* at a Perception check?”
Remember too that skill checks are rolled to determine if something happens in the game, when the DM and/or players aren’t sure how something will resolve. They are not exactly a measure of your character’s aptitude, though obviously that plays a part. Having a high modifier makes something more likely to happen when you attempt it, which is always through a combination of skill, circumstance, and luck.
When we’re talking about Perception, failure doesn’t mean you’re clueless or that you never pay attention. If you fail a Perception check, that can mean one of several things: 1) the thing you didn’t notice is hidden well enough to escape your keen eye, 2) you looked away at exactly the wrong moment, when you would have otherwise seen it, 3) you were distracted by something else, so you didn’t notice the thing, 4) you saw the thing but didn’t understand what it was or that it would be useful (i.e. you briefly glimpsed the candlestick but didn’t put together that if pulled, it would open a secret door).
In that last case, an Investigation check might give you that information. If you had succeeded on a high Perception check, you might notice that something about the candlestick seems odd, but you’re not sure what it is. Maybe it’s shinier than the others or there are small scratches near the base. This would be a situation where you’d follow up the Perception check with an Investigation check (possibly with advantage, depending on the clues you noticed) to figure out the connection between the candlestick and the secret door.
“What is Investigation?”
Investigation is a new skill in 5e, and it causes a bit of confusion, particularly for us old players. I’m not sure I understand it either, and I spent a lot of time ignoring it, since it overlaps with Perception a lot in my head. But I’ve been making an effort to call for it more in my games, with some success.
Perception picks up data. With Perception, you might notice a clue, but you don’t necessarily understand what it means. For instance, a room is unusually clean, even though you’re in the heart of a dungeon. Or a faint acrid smell, like something astringent. Or faint squelching sounds like something viscous burbling against rock. Any of these might be useful clues, but you aren’t necessarily going to be able to put them together.
Investigation takes the next step, putting together the data and formulating a conclusion. It also fulfills in 5e what the Search skill did in previous editions of the game: you spend some time searching a room or place and, if your check succeeds, you find what you’re looking for or at least something interesting. Investigation takes time–generally at least one minute at my table, sometimes upwards of ten minutes or an hour.
Perception, by contrast, determine if you happen to notice something on a brief glance; see above. Passive Perception might see a hidden item that you would have found with Investigation, but the DCs to find hidden things are usually high enough that few characters have high enough passive Perception to find them.
Insight does the same sort of thing, but usually it’s specific to interactions and RP. Figuring out if someone is lying to you or conveying information that isn’t on the surface. Perception might set up advantage on one of these checks as well, such as if you noticed signs that a person might be desperate or their heart is racing.
(You know, the way Daredevil does it.)

Passive Perception 30? Jeez! (Daredevil Copyright Marvel Comics)

“Skill Synergy: How do Investigation, Perception, and Insight interact?”
There is some overlap among these skills, and that suggests we should use them to complement and support each other. Perception can feed clues to support Insight or Investigation checks, while Insight and Investigation can feed each other in either order, For instance, if you use Investigation to turn up incriminating evidence against an NPC, that might aid your Insight check when you confront them, or if you observe something off about their clothes with Perception or get a weird vibe from their mannerisms with Insight, that might aid your later Investigation.
Now, do I insist that you use Perception or Insight to find clues, then Investigation to come to a conclusion? No. You can use Investigation by itself to search an area by itself without ever making Perception checks, and you can use Investigation based on a previous conversation to glean some information you didn’t have before.
But, and this is important: if you’ve made successful Perception or Insight checks that are relevant to your Investigation, you might gain advantage on your Investigation check (or vice versa).
Of all of these skills, Perception is the least supported by previous work. Doing Investigation or making an Insight isn’t necessarily going to help you notice something, unless that previous legwork has told you what to look for, such as a signet ring or red mud on the NPC’s boots. In that case, you might have advantage on your Perception check.
This serves two purposes: 1) to reward characters you make good rolls or are very thorough, 2) to encourage multiple characters to participate in a meaningful way, since Int/Wis builds aren’t especially common, and you’re likely to have one person who is very perceptive, one person who is very insightful, and one person who is very deductive (with a high investigation modifier).

Sherlock Holmes undoubtedly has both high INT and WIS stats. (As well as a decent STR stat; I see you, fellow book geeks!)

Happy gaming!