Leonard and Sheldon go to a High I.Q. sperm bank to make donations and earn some money. To get the attention of the receptionist,
Leonard very quickly solves all her crossword puzzle for her, which does not please her. She has them fill out their forms, but they back out after Sheldon informs Leonard that "there is no guarantee that our sperm is going to generate high I.Q. offspring", ergo, they are committing genetic fraud. (This scene is not usually aired on reruns.)
Upon returning to their apartment, they climb the stairs while Sheldon tells Leonard that if the stairsteps are off by two millimeters most people will trip. Doing his experiment he tripped his father. At the top of the stairs, they spot Penny, who is moving into the apartment across the hall from them. Leonard and Sheldon agree that she is an improvement over their old neighbor, Louie/Louise, who, according to Sheldon, was a 200-pound transvestite with a skin condition. After Penny spots them, they exchange two rounds of "hi". Penny learns that they are her neighbors from across the hall. She offers to have coffee with them later.
Back home, Leonard thinks they should invite her over for lunch though Sheldon tells him that, except for Howard and Raj, people don't come over. Penny is invited over to their apartment after a round-about invitation that includes mentioning the health of one's colon.
Penny enters their home and looking over Sheldon's whiteboard, she is impressed by their work. It includes some quantum mechanics, a little string theory doodling around the edges and a joke on the Born-Oppenheimer approximation. Sheldon agrees on Penny's suggestion that he is one of those "Beautiful Mind genius guys". Leonard also shows her his whiteboard which Sheldon says should not impress her. Penny is introduced to Sheldon's obsessiveness when she accidentally sits in Sheldon's spot on the couch. Sheldon gives his first rendition of how that spot is perfect based upon the temperature, air currents and television visibility. She is also told that they like to play their game of Klingon Boggle.
Penny begins her story that she is a Sagittarius that tells them a lot about her. Sheldon concludes that she participate in the mass cultural delusion that the sun's apparent position relative to arbitrarily defined constellations and the time of your birth somehow affects your personality. She is a vegetarian except for fish and an occasional steak, which she loves. Also is from Nebraska, worked at the Cheesecake Factory, and is writing a screenplay about a sensitive girl who moves from Nebraska to be an actress. She also fell in love with a jerk that she has been living with for four years. Even though he lied and cheated, she still loves him and breaks into tears which the guys don't know what to do about. Leonard describes her dilemma as a paradox using as example the physics understanding of the light simultaneously as a wave or a particle (although, as postulated that the Bohr's principle of complementarity, only one of natures can be detected each time). Penny doesn't really understand, but then she mentions that she feels so dirty and gross and her shower doesn't work. Leonard offer her to use their shower.
As she is taking a shower, Howard and Rajesh arrive with a video of Stephen Hawking from 1974, which excites them because it was before the time that he uses a computerized voice simulator. Howard says that phrase while doing his famous Stephen Hawking impersonation. Penny exits the shower only wearing a towel because she can't get the water to go from tub to shower, and she is introduced to them. Howard, thinking he is a professional ladies' man, hits on Penny while Raj is rendered silent since he can't speak in front of women due to his selective mutism, which was unknown at the time.
Leonard returns with her to the bathroom and gets the shower to work. She then steps into the shower and throws the towel over the curtain to an amazed Leonard. Penny then asks Leonard a favor, asking him to pick up her television from her ex-boyfriend, Kurt who lives across town. Due to his crush on Penny, Leonard agrees and drags Sheldon along. During their ride to Kurt's apartment, we discover that Leonard's last girlfriend was a North Korean spy named Joyce Kim.
Upon their arrival, they are denied access to the building, but by observing two clever girl scout types get into the building, they quickly slip in behind them. Penny's ex-boyfriend, who turns out to be a tall and very muscular man, refuses to give up the TV, and apparently finds Leonard and Sheldon so irritating that for that he robs them of their pants.
They return to their apartment, pants-less and TV-less. Penny feels sorry for Leonard and Sheldon after their incident and decides to make it up to them by paying for their dinner (along with Howard and Raj). While looking at Penny's apartment Sheldon tells Leonard that this thing with Penny isn't over. Leonard tells him that their children will be both smart and beautiful, to which Sheldon adds, "Not to mention imaginary."
In the car, the episode ends with Sheldon telling Leonard that despite the low chances of him starting a relationship with Penny, in comparison to the other males in the car; he is a "veritable Mac Daddy".