How many traps are there?

Difficulty controls the trap count. Easy rounds hide one trap among eight teeth (~88% survival per tap). Harder rounds widen the mouth to as many as sixteen teeth and add more traps, so the survival odds per tap can drop below 60%.

Is there any skill, or is it pure luck?

Each individual tap is luck · the player can't see which teeth are trapped. The skill is in knowing when to commit and when to walk away · longer streaks mean longer odds, so the meta-question is "do I tap again or quit while ahead?". It's the same skill that matters in any decision-under-uncertainty game (Press Your Luck, Deal or No Deal).

Does it work in the Labyrinth and Polymath?

Yes. Crocodile Dentist ships as a labyrinth encounter and shows up in the cross-game Polymath rotations whenever its reasoning tag is in the active pool.

Why include luck games at all?

Because nerve-and-pacing is a real skill that pure reasoning games don't train. The labyrinth gauntlet bundles a few luck-and-nerve encounters (Crocodile Dentist, Scratch Card, Bubble Wrap) so a complete run trains decision-making under uncertainty alongside the deduction puzzles. Tolerance for chaos is part of polymath fitness.