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I actually use Insight for a few things, including "is this combatant feinting?" and "which one of these hobgoblins is the leader?"
My discomfort with Investigation stems from the fact that it has two separate roles:
(1) Discovering physical objects like traps and secret doors, and
(2) Doing deduction on the players' behalf (deducing the location of a hidden object, finding the weak point in a tunnel).
#1 feels like something that should be Perception (or Perception AND Investigation) but was tacked on to Investigation to bulk it out a bit. In practice I often make these gated on both Perception and Investigation, e.g. Perception DC 11 to smell the goat's blood and Investigation DC 18 to realize these funny crenellations are actually booby traps, and that Goatherd Bryan's missing goat must be squished under one of the ones that have already been triggered.
#2 is something that I have trouble reconciling with game flow: I don't really like saying, "Here's a riddle, but if anyone has passive Investigation of 23+ you can flip over this piece of paper and read the answer; or you can give a Schrodinger answer and roll an active Investigation check to see if your answer turns out to be what's written here on this paper." It works okay but it doesn't feel quite right. Now, maybe I just need better game structures to make it flow more smoothly; but maybe I should just abolish the dice-based approach entirely and let a riddle be just a riddle.
How do YOU guys handle riddles and puzzles and traps? Do you give players a dice-based option, and if so how hard do you make it compared to solving the riddle/whatever with your brain?
I don't like providing an ability check to bypass/solve a riddle or puzzle, I prefer that the players solve it regardless of their PCs intelligence or wisdom. However, if I have a puzzle lock that the players have to solve before they can progress in an adventure, then I would make sure to leave clues around the place that they can find which will help them solve it since I also don't want them to get frustrated if they get stuck and can't progress. I guess I could also have them make ability checks similar to a 4e skill challenge to hand out clues, would have think on that though. I don't think most of my riddles/puzzles would be that difficult.
I had a friend who had an effective use of a riddle for our group which explained how to move across platforms. They all had numbers on with only pillars with prime numbers being safe, except for those with a 7 or 3 in them. I thought it was a good use of a riddle, although the riddle made it sound like 7 or 3 were safe so I ended up taking an acid bath, but otherwise, great riddle. We were a little focused though since we figured out which platforms were safe, but didn't figure out that shooting some statues with a ranged attack would let us ignore the slow time field that we were stuck in while trying to cross before a curtain of acid caught up to us. It was definitely a memorable encounter.