Colin Singleton, a child prodigy living in Chicago, fears he will not maintain his genius as an adult. Over the span of his life, Colin has dated nineteen girls named Katherine, all spelled in that manner. After being dumped by his girlfriend, Katherine XIX, Colin is longing to feel whole, and longing to matter. He hopes to become a genius by having a "eureka" moment.
After graduating from high school, and before college, Colin's best and only friend, Hassan Harbish, convinces him to go on a road trip to take his mind off the breakup. Colin goes, hoping to find his "eureka" moment. After reaching a rural Tennessee town called Gutshot, they visit the supposed resting place of Archduke Franz Ferdinand. There, they meet Lindsey Lee Wells and her mother Hollis, whose family runs a local textile mill. Hollis allows Colin and Hassan to stay with her family and offers them a summer job interviewing the town's residents and assembling an oral history of Gutshot.
Colin begins to like Lindsey, though he is foiled by her boyfriend, Colin Lyford (he and Hassan call him TOC, "the other Colin"), whose father is employed by Lindsey's mother. Colin is still chasing his eureka moment, finally finding it in the theorem he created called the "Theorem of Underlying Katherine Predictability". It determines the curve of any relationship based on several factors of the personalities of the two people in a relationship. His theorem eventually works for all but one of his past relationships with a Katherine—which the novel explores.
While the back stories of Colin's life play out, Hassan finds a girlfriend, Katrina, a friend of Lindsey's. Their relationship is cut short when Colin and Hassan catch Katrina having sex with Lyford while on a feral-hog hunt with Lindsey, her friends and Lyford's father. A fight between Lyford and all the surrounding acquaintances begins when Lindsey finds out that he has been cheating on her. Injured in the fight, Colin anagrams the Archduke's name while in the graveyard to dull the pain, and realizes that it is actually Lindsey's great-grandfather, named Fred N. Dinzanfar, who is buried in the tomb.
Colin finds Lindsey at her secret hideout in a cave, where he tells her the story of every Katherine he has ever loved. Lindsey tells him how she does not feel sad but instead slightly relieved by Lyford's affair. They discuss what it means to them to "matter" and eventually confess their love for each other. As their relationship continues, Colin decides to use his theorem to determine whether he and Lindsey will last. The graph reveals that they will only last for four more days. Four days later he receives a note from Lindsey, saying that she cannot be his girlfriend because she is in love with Hassan; however, a P.S. at the bottom makes it clear that she is only joking. Colin realizes that his theorem cannot predict the future of a relationship; it can only shed light on why a relationship failed. Despite this, Colin is content with not "mattering". Hassan says that he is applying for two college classes, which Colin has been trying to convince him to do throughout the book. The story ends with the trio driving past the restaurant they were originally planning to go to, because Colin, Lindsey, and Hassan realize that they can just keep driving; there is nothing stopping them from continuing on.
