The puzzle is solvable you just have to be fast enough after getting changed because you can't do anything if you're "dirty". Talk to the spider lady first to get a fresh change then rush over to the quiz room. Just multiply the numbers on the board and input the answer on the tv, then hit the button. Once you do that for two rooms there's a magic wand at the end that lets you change into baby form on demand and on the way you find the screwdriver. Inputting the wrong answer gives you a scene with the goblin. There's a keycard in the other room you can grab, use the screwdriver on the vent in the bathroom and you're out, then the keycard gets you past the laser so you can escape the stage.