A lot of this episode felt like something out of a fever dream. Anyway, there are two things that beg mentioning. First is about houtou, a form of udon. You can look it up for yourself in that link there, but the reason that Takeda Shingen keeps appearing in the background is because legend says that he invented it. Second is a weird thing that Rin keeps saying, the wasei-eigo “match pump”. Basically it means “to stir up trouble to get credit for the solution”, and comes from “lighting a match and then putting it out with a water pump“. I have no idea where they possibly got this from.