D&D 5E - Eyes of Minute Seeing: Investigation vs Perception

Here's my interpretation of investigation vs perception:

Perception is nearly always passive. I rarely have players roll on it, instead relying on the passive score and taking into account if they'd have advantage or disadvantage. Someone standing watch at the cave entrance while the rest of the party sleeps would get advantage, so +5 to their passive perception to notice something. If it was pitch black and they were actively on guard, the advantage and disadvantage would cancel out leaving them with a normal passive perception. If they were in an animal form that could reasonably detect things by sound or smell (such as wolves), the disadvantage for darkness wouldn't apply. That kind of thing.

For secret doors or hidden treasure, it's mostly the same scenario unless they are on the lookout for something specific, but otherwise not searching around for clues or anything. If they are looking for something specific, or otherwise doing something out of the normal 'check the bodies and rifle through the drawers' type stuff, then they are essentially investigating.

Investigation is nearly always rolled at my table, but it makes a great candidate for passive checks as well. Basically, investigation is like "active perception" in many cases. While the group is 'searching the room for treasure' (passive perception), maybe the cleric says she wants to focus on the old statue. In that case, there'd probably be a few rolls taking place: history or religion, perception, and possibly investigation depending on the other results.

Now to your specific questions:

In other words: Perception is for the physical senses, while Investigation is purely mental. It's all logic and reasoning and the like.

To me, perception is more passive and investigation is active. A detective at a crime scene might notice (passive perception) that the bodies have been moved, but to make sense of that would require a proper investigation. He didn't walk in and decide to actively try and notice things; what he notices is simply the result of his general perception.

When would some ever make an Investigation check that relies on sight?

Basically anyone who needs to make use of libraries or hieroglyphics, or is maybe trying to match up a weapon with a wound (candlestick in the library), etc.. I would say it's actually the most common way of making an investigation check.

Aren't the physical senses Perception's domain?

Perception is generally limited to the 5 senses, but investigation uses them as well.

Let's just say you were running "Death House" and you happened to have a PC with eyes of minute seeing playing in the adventure. Would you let the item grant them advantage on their Perception checks to notice all the little details in the house's woodwork?

No, because they aren't investigating anything. They walk into a room and either notice something about the woodwork or they don't. If the player with the Eyes of Minute Seeing states that they want to go inspect the woodwork, then you'd have them roll investigation with advantage.

The player with Eyes of the Eagle, however, would get an advantage to his perception (or +5 to passive)to notice there there even are details about the woodwork as soon as she walks in. Eyes of the Eagle wouldn't do much for her when inspecting the woodwork details, but she may be the only one in the group that even noticed they existed.

One last thing: Skills do not always have to be tied to the assumed attributes. I had one player roll a Strength: Investigation check because he knew about a secret door, but no one could find it. His method basically boiled down to "I'm going to start pushing and pulling on crap until something gives" which made sense to me. He didn't find the secret door elegantly, but he did find it.

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