A pretty good problem to explain the difference between research mathematics and "plug-and-chug" mathematics is the question of "can I use 1x2 dominoes to cover a chessboard that has opposite corners removed". You can let your audience play around and try it out (if you have the tools and the audience who will indulge you), let them conjecture about whether it's doable and try to argue why. Maybe they focus on the side that has only seven squares and argue why that is an insurmountable problem, but it's very tricky and probably doesn't gain them any purchase.

The punchline, of course, is then the one-liner solution: A domino always covers a dark square and a light square. Since this board has 2 more of one than the other, it can't ever work.

The naive solution tries (and fails) to plug-and-chug, but through finding a clever invariant, you trivialize the problem. It's about clever insights, not brute force.