• The chapter is a flashback to Yukawa and Ishigami's college days. They meet one day while taking a physics class. Ishigami, sitting at the front of the classroom, aloof from the other students, who usually prefer sitting in the back, is busy scribbling a proof. To Ishigami's surpise, Yukawa is able to understand the proof and thereby understand Ishigami's passion for mathematics. They are able to relate to each other as fellow geniuses whose intellectual curiosity enables them to share a bond of friendship.

• The proof that Ishigami was working on at the time related to the four-color problem, proposed by Arthur Cayley in 1879.

• Are four colors sufficient to color the contiguous countries on any map, such that no two adjacent countries are ever colored the same? All one had to do was prove that four colors were sufficient, or present a map where such separation impossible - a process which had taken nearly one hundred years. While two mathematicians at the University of Illinois were able to use a computer to solve the problem, Ishigami is not content with this kind of solution, preferring to solve it the old-fashioned way, using paper and pencil, so that the proof will be full-proof. This leads Yukawa to compare Ishigami to Paul Erdos, the Hungarian mathematician who shared an attitude similar to that of Ishigami. Hence, their bond of friendship begins with a mutual appreciation for a complex mathematical problem.

• It is not just any old mathematical problem that brings them together, though. It is the four-color problem, which becomes allegorical for several aspects of the story. Ishigami must construct an alibi and cover up Togashi's murder in a manner that is analogous to solving the four-color problem: that is, Yusako's involvement in the case, although "contiguous" to the case in the sense that she is a suspect and also Togashi's ex-wife, must be kept separate and distinct from the actual murder so that she is not charged and convicted.

Additionally, the four-color problem comes to symbolize the interplay among the characters: can the involvement of Kusanagi be kept separate and distinct from that of Yukawa's? Can the involvement of Ishigami be kept separate and distinct from that of Yusako's? Kusanagi, while not wanting to betray his friendship with Yukawa nor violate Yukawa's sense of privacy with regard to his feelings towards Ishigami, attempts to follow Yukawa's tracks in order to gain insight in the case. He also must struggle to honor his promise to keep Yukawa's secret, a promise he swore to keep in his capacity as friend, in spite of his duties as an investigator. Yukawa, for his part, despite not wanting to incriminate Ishigami or interfere with Kusanagi's work, can't help intervening in the case out of intellectual curiosity about Ishigami's methods and also out of compassion for Ishigami as an old friend with whom he shares a common bond of being a misunderstood genius. While Ishigami attempts to keep Yasuko at a distance, he can't help avoiding the lunch box shop, nor can he help expressing jealousy over Kudo's appearance later on. Yusako, for her part, while wishing to respect Ishigami's sacrifice and not wanting to appear ungrateful, can't help but get involved with Kudo, which prompts feelings of jealousy on the part of Ishigami and also angers her daughter Misato. The ways in which these character arcs mesh and interweave with each other will seem to confound the simple assumptions made by the four-color problem, suggesting that human beings are not like contiguous shapes that can be drawn on simple planes and sphere such that they can be colored separately.

• Ishigami and Yukawa, despite being in the same class and sharing a common bond based on their love for math and science, eventually go their separate ways, one staying in the physics department, the other in the math department. They also go their separate ways in more ways than one, however, with Yukawa ending up a successful university professor and Ishigami ending up a high school in order to cope with difficult life circumstances, namely having to take care of sick paraents, which necessitates taking a steady job as high school teacher. This idea of being on separate paths that later eventually converge is somewhat evocative of the aforementioned four-color problem in that Ishigami and Yukawa's lives will overlap and become intertwined rather than remain separate (Being a friend to both Ishigami and Kusanagi, Yukawa illustrates how people's lives can mesh and overlap in unexpected ways). It is also evocative of tension between free will and fate, which is especially connoted in their nicknames, with "Buddha" (i.e. Ishigami) being more connotative of fate in constraining people's life choices (as well as the inevitability of suffering) and "Galileo" (i.e. Yukawa) perhaps being more indicatve of free will, considering that Galileo favored ideas that met with disapproval from the religious establishment of his day.

• They shared a common desire to describe the world around them with theorems, but they approached this task from opposite directions. As a result of a murder case, Yukawa and Ishigami will devote their attention to the same problem but from opposite directions. Ishigami will be on the side of the culprit and Yukawa on that of the police. The difference between them seems to correspond to that between strict empiricism, with an emphasis on observation and experimentation (i.e. a posteriori and synthetic knowledge) on the one hand, and strict philosophical idealism with an emphasis on logical proofs and formulas (i.e. what in philosophical terms is considered a priori and analytic knowledge) on the other. This difference is reflected in their nicknames as well: "Galileo" (i.e. Yukawa) was known for his experiments, whereas "Buddha" (i.e. Ishigami) was known more for his philosphical insights, obtained through contemplation and meditation rather than experiments. (One of these historical figures is from the West (i.e. Galileo), the other from the East (i.e. Buddha), another dichotomy that will mesh with and reinforce the other binary oppositions characterizing their relationship). These divergent approaches and worldviews (i.e. empirical vs. idealistic, a posteriori vs a priori, etc.) will also guide their approaches to handling the murder case.

• After the flashback, the story returns to present, with Yukawa visiting Ishigami at this apartment. Yukawa candidly explains how he heard about Ishigami from Kusanagi, further adding that Kusanagi is a friend who goes back to their university days. Yukawa mentions enough about his relationship with Kusanagi to justify his visiting Ishigami and perhaps draw a little suspicion, but not enough for Ishigami to conclude definitely that Yukawa is actually cooperating with Kusanagi and gathering data to help with the investigation. This is somewhat similar to how Yasuko makes use of the movie tickets for her alibi - they prove just enough of an alibi to raise questions about her actual whereabouts and draw the attention of the police but are not so weak of an alibi as to incriminate her.

• Yukawa brings Ishigami a mathematical challenge: a counterexample to the Riemann hypothesis. If the counterexample proves to be correct, then it means that the Riemann hypothesis is false. The task for Ishigami is to demonstrate that the counterexample fails to disprove the Riemann hypothesis. It is part of a larger problem known as the N = NP problem, which tries to determine whether it easier to solve a problem yourself or prove that another person's solution is wrong. This will become a metaphor for the invesigation itself, as the police will be attempt to disprove Yusako's alibi, which is a false lead that Ishigami has concocted to distract the police, rather than attempt to solve the murder mystery itself.

• Ishigami is delighted to be given the challenge, and he spends the whole night solving the problem while Yukawa falls asleep. This enables Yukawa to ascertain the extent to which Ishigami has retained his keen logical abilities. But it's also a pretext for Yukawa to stay the night. By staying overnight, Yukawa is able to hear Yasuko leaving for work next door. He is also able to accompany Ishigami on his morning commute to work. All of this helps Yukawa to gather data relevant to the case. Yukawa probably could foresee Ishigami being so caught up in the problem as to forget the time, given how Ishigami had a tendency to act the same way during his college days.

• Apart from the math problem, Yukawa and Ishigami share dinner and discuss their lives. We learn that Ishigami couldn't work at a university because he had to take care of his sick parents, which required that he have steady employmenet. He was also frustrated by the lack of genuine passion for mathematics research at the university, where professors seemed more concern with recognition and status than actual research. This reveals several aspects to Ishigami's character: he is capable of empathy, as demonstrated by his concern for his ailing parents, which also makes it more reasonable that he would be able to feel empathy for Yusako. However, he fails to understand certain social dimensions of human behavior, which explains his failure to compete with other professors for status and recognition at the university level. Such competition is driven by insecurity and the need for validation and praise, feelings which Ishigami, up until his encounter with Yasuko, had mostly shunned in order to dedicate himself purely to research. This failure to understand some of the primeval tendencies of human nature, both in himself and others, such as jealousy, insecurity, and the need for love, validation, and recognition, explains how competition with Kudo for Yasuko's affection will also be his undoing.

• His insecurity begins to reveal itself when, while walking to work with Yukawa, Ishigami asks,"How have you managed to stay so young, Yukawa? You still have a full head of hair. How differenet we are." Concern for such superfical matters as hair loss is out of place with Ishigami's usual obsession with mathematics, a fact that can only explained by his longing for Yusako's affection. Intuitively aware of his weakness in this regard, Ishigami is careful to not allow Yukawa to accompany him to the lunch box shop, which might cause Yusako to act untoward towards them and give evidence of the true nature of her relationship with Ishigami.