While finishing a game of badminton with a student, Yukawa is visited by Kusanagi. Yukawa's love for badminton is the reason he met Kusanagi in college, as Kusanagi was also badminton enthusiast in his college days. Being passionate about science and sports enabled Yukawa to have a much wider circle of friends than Ishigami.

Together, Yukawa and Kusanagi go to a local bar to discuss the case. Kusanagi updates Yukawa on the police’s progress in solving the murder case. It turns out that several aspects of the case are causing much confusion for the police.

Prior to meeting with Yukawa, Kusanagi had watched the movie that Yusako and Misato claim to have seen on the night of murder. He did this in order to see if Yusako and Misato’s knowledge of the plot details are accurate. While Misato and Yusako can recount the details of the plot, the police are puzzled by the fact that Misato spoke about the film with a friend on May 12, two days after the alleged murder date, rather than the day right after. Kusanagi also wonders how a girl in junior high school like Misato, who is a member of the badminton team, which he deduced from noticing her racquet in Yasuko’s apartment, could still have the energy to go to a moive and then karaoke with her mother despite being exhausted from badminton practice.

Additionally, Kusanagi and the other detectives are puzzled by another issue: one particular witness claims to have seen the person represented by the police sketch of the victim but her description of the victim doesn’t fit Togashi’s appearance. Further, this witness claims to have seen Toagshi at one train stop before the one at which the murder took place.

The police also struggle to understand the fact that a stolen bicycle with fingerprints matching that of the victim was left at the crime scene even though the murderer went to all the trouble to burn the fingerpinrts, strip the victim’s clothing, smash the victim’s face, and even smash the victim’s teeth in order not to leave dental records. Nor can they understand why the murderer wouldn’t just completely remove the victim’s clothes instead of just partially burning them if they wished to obscure the victim’s identity.

Given all of these loose ends in the case, the police are at a loss in coming up with a plausible theory that accounts for all the facts. Kusanagi proposes a theory about how the murder may have taken place. Given the weakness of Yasuko’s movie theater alibi, there is sufficient time for her to commit the murder and go to Karaoke with her daughter. However, Yukawa is doubtful that Yusako, being a woman and much shorter than Togashi, could have been capable of strangling Togashi and then disposing of the body.

This leads many at the police station to think that Yusako must have had a male accomplice – specifically, someone who would’ve had access to a vehicle and would’ve been able to help dispose of the body.

Next, we see Ishigami teaching his math class. He has given failing grades to many of the students in his class, which causes them to complain. One student in particular, Morioka, who has reputation for being somewhat of a rebel and class clown, complains very vocally, noting how unnecessary it is for certain students to have to pass Ishigami’s math class since they won’t be needing it in their long-term studies. Since Morioka enjoys racing motorcycles, Ishigami explains the application of integral and differential calculus to motorbike racing, which is enough to silence further objections. It turns out, Ishigami has already anticipated such an objection, having answered different variations of it from different students over the years.

After class, Ishigami receives a phone call from Yukawa, who wishes to speak with him and is waiting at the gate of the school. When they see each other, Yukawa asks to go to the lunch box shop for dinner, which makes Ishigami nervous, since it would involve him seeing Yusako, whom he hopes to avoid. While at the lunch box shop, Kudo suddenly walks in. When Ishigami notices how Kudo gets on with Yusako on such friendly terms, he is noticeably frustrated and uncomfortable, indicative of his jealousy for Kudo.

After leaving the shop, Yukawa asks if Kudo has hobbies apart from math, something to help him let off steam. Ishigami can’t think of any but Yukawa offers driving as an example of hobby. Ishigami denies this. While Ishigami does have a driver’s license, he doesn’t have a car, so it wouldn’t make sense for him to go driving for fun.

Unsure of Yukawa's motives, Ishigami presses Yukawa for his reasons for coming to visit him. Yukawa claims that the police wish for Ishigami to spy on Yasuko. Instead of asking Ishigami directly, the police, according to Yukawa, had hoped to get a sense of how Ishigami might feel about the request. Uncomfortable with the idea, Ishigami refuses.

Before leaving, Yukawa poses another math riddle for Ishigami, a variation on the N = NP problem: which is harder, devising an unsolvable problem, or solving that problem?