Goofs

Penny's clothes change after her shower, but she didn't have another set with her and it appears she didn't return to her apartment since Raj and Howard were there.

Leonard asks Sheldon if his experiment with stairs is why he was sent to boarding school, and Sheldon responds that it was his work with lasers. In Young Sheldon (2017), he never went to boarding school, he skipped multiple grades but always went to public school. A point was made multiple times that his family was too poor to send him to a private school.

At 15:30 - As Penny enters the shower and Leonard turns to leave you can see the blue top and jean shorts neatly stacked on the sink and the boots on the floor have the one on the left laying down with the sole close to the wall. Next Penny tosses the towel over the curtain and now the clothes are just tossed in place with entirely different black fabric laying on the blue top and the boots have moved. With neither character being near them.

The experiment discussed by Sheldon (Jim Parsons) at the very beginning of the episode is a quantum version of the well-known "experiment," but incompatible with the work presented on the whiteboard. His dialogue was meant to accompany the whiteboard seen in the unaired episode Unaired Pilot (2006).

In the first episode, Sheldon seems to use and understand sarcasm, however in future episodes, it is well established he does not understand it.

Correction: Sheldon doesn't understand sarcasm as is clear by the fact that Leonard says he has to hold up a "sarcasm" sign when he uses it, which he later does when Penny uses sarcasm.

Sheldon, for the first time, gives his speech as to why he likes the spot he sits in. When mentioning the two windows he opens to get his cross-breeze, the unseen window he points to is somewhere where, if it existed, would be looking out into the hallway or into the adjacent apartment. Also, it would later be found that Sheldon would not open a window simply for air due to his yet-to-be-known (to the audience) fear of birds.

Leonard (Johnny Galecki) mentions that his grandmother was there for Thanksgiving, had Alzheimer's and got naked. Later it is established that he was from New Jersey, making his ill grandmother traveling all the way across the country implausible.

Penny says that she lived with Kurt for 4 years. Penny's birthday is December 2nd, 1985. She did not graduate from high school early so the longest she could have lived with Kurt would be 2 1/2 years. Not necessarily so, as she would have graduated in May 2004, and this story takes place in Sept 2007. Knowing her math skills and her character, she may have regarded 2004 as the first year, 2005 the 2nd, 2006 the 3rd and 2007 the 4th. In any case the longest would have been 3 years, 4 months (not two and a half). Or she may have rounded up to 4 years.

Howard introduces himself as working for CalTech's Department of Applied Physics, but Howard works in the university's engineering department.