I never asked a shailah about puzzles, because I personally wouldn’t do one on Shabbos. It is mamesh creating something, and the intent is typically to keep it permanent, once completed. If it is permissible (and I would find that to be odd), I still wouldn’t do it.
Trivial Pursuit is my game of choice, and also games like Blurt, Taboo, or any game where you have to think of ways of expressing something without using specific common words. Charades is also a great choice.