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Magic Mouth is described as triggering when visual or auditory cues invoke it:

You implant a message within an object in range, a message that is uttered when a trigger condition is met. Choose an object that you can see and that isn’t being worn or carried by another creature ... Finally, determine the circumstance that will trigger the spell to deliver your message...

...The triggering circumstance can be as general or as detailed as you like, though it must be based on visual or audible conditions that occur within 30 feet of the object. For example, you could instruct the mouth to speak when any creature moves within 30 feet of the object or when a silver bell rings within 30 feet of it.

From the description, the Magic Mouth sees and hears in a 30 foot radius - but there is no mention of it acting like normal eyes or ears affected by obstructions. Does that make it super-“human” or is limited by the spell caster’s perception?

Thus, if Magic Mouth (MM) is cast in shopkeeper’s shop, could it automatically detect the following conditions - or would it need to roll to notice them? If it did need to roll - using what skills if it’s skills exceed that of typical casters?

  • Could MM prevent theft? Automatically trigger when any thief tries to steal any item: “Announcement...despicable thief at work right now”
  • Could MM prevent invisible infiltration? Automatically trigger when anyone invisible but audible enters a room: “Warning...I hear someone in here that I cannot see.”
  • Could MM track invisible, audible entities? Could you protect a throne room by tiling it with 5x5 tiles enchanted with Magic Mouths that indicate when anyone invisible (but audible) walks on them with each tile triggering an audio cue as someone walks across them? “HERE...Here....here....”

If a Magic Mouth can trigger with visual or audio cues - how perceptive is it?

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A Magic Mouth has arbitrarily high perception

Magic Mouth is not a creature, but rather magic. The text says that you can set the trigger conditions to be as "detailed as you like". Thus, all three of your scenarios are indeed possible. For example, if the mouth is set to speak if any creature moves within 30 feet of the object, you could conceivably ask it to speak if anything as much as twitches nearby. The magic mouth does not have eyes or ears--it relies on magic, after all.

Therefore, all three of your scenarios are technically feasible. Your first scenario is a classic panic-inducing trick in dungeons! Likewise, your second scenario is done better with the 1st-level Alarm spell, which also pierces invisibility. Your final scenario is also plausible, though the wording might take some refining, and it would be soundly defeated by someone invisible that's also not touching the ground.

Careful wording is key

However, you might have to be very careful about how you set your circumstances, given that you are restricted to audio and visual cues, and the Magic Mouth cannot interpret things. For example, you could set a magic mouth to go off if someone picks up an item in a store but doesn't go to the counter, but then someone can take two items but only pay for one.

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    \$\begingroup\$ Well-stated on the interpretation angle. A magic mouth doesn't know if your spouse is reaching into your pouch to get milk money, or a thief is after your +2 ring of awesomeness. "Someone steals from me" is not a visual or audible condition. It's an interpretation. \$\endgroup\$ – keithcurtis Feb 4 '18 at 8:10

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