This may be really dumb, but I think I managed to talk to secret box-chan in the looong gallery with the least amount of puzzles completed.
I meandered through this game over the course of a few weeks at first and then I played again and again and again as I found it soothing. Then for fun I just decided to try and fill up all the fish powers and special items without doing any small puzzles (excepting the one 4x4 for chad's tank).
Edit: Hm actually now thinking of it I suppose the dream puzzles don't need to be done to get here so it could be 10 less, but that would have aggravated my brain to leave them out lol.