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Content Warning: This section of the guide includes discussion of suicidal ideation and self-harm, attempted death by suicide, graphic violence, and death.
In the morning, Ishigami takes his usual route to school, passing the tent city of unhoused people. The person he has mentally dubbed the Can Man is out collecting cans along the riverbank. When Ishigami sees an empty bench, he looks away. Suddenly, Yukawa steps up alongside him. Yukawa remarks on the unhoused people, comparing them to cogs in a clock. He says, “[W]e are all cogs in the clockworks. […] the world needs its cogs, all of them.” He also asserts that “only a cog may determine its eventual meaning in the system” (219). Ishigami asks what he means.
Slowly, Yukawa dissects each piece of evidence in the murder case. He says that the police are obsessed with taking apart Yasuko’s alibi, but Yukawa has now realized that the alibi is unimportant. It is bait that the police fell for. He then says that he tested burning clothes in an oil drum to see how such a process would work. What he found was that the clothes burned very quickly and left almost no trace, which means the murderer left the clothes mostly unburned on purpose. He explains his theory that the bicycle was stolen in order to establish a timeline and to make sure that the police could identify the victim. Ishigami pretends not to understand and asks why Yukawa has not taken his concerns to the police. Yukawa says that he will, but he wanted to talk to Ishigami first because they are friends.
Ishigami recalls Yukawa’s question about formulating or solving an unsolvable problem. Yukawa suggests that the answer is: “It’s more difficult to create the problem than to solve it. All the person trying to solve the problem has to do is always respect the problem’s creator” (226). Ishigami says that this might be Yukawa’s answer, but he will shortly give him a different solution.
That evening, Yasuko and Misato have dinner with Kudo, who tries hard to make Misato like him. Misato remains sullen and quiet. At one point, Kudo confides to Yasuko that he has been receiving strange phone calls and anonymous, threatening letters. Later, Misato asks if Yasuko plans to marry Kudo. She does not think that it is a “good idea to betray that other guy” (229). Yasuko understands her daughter’s implication and wonders if Ishigami was the one who sent the threatening letters to Kudo.
Later, Yasuko receives her nightly call from Ishigami, who gives her careful instructions. He has left her a collection of letters with instructions on how to use them. She will understand when she sees them. This is the last time they will talk. He tells her that no matter what happens next, she is not to interfere, and he assures her that she and Misato will be safe soon.
That night, the police chief calls Kusanagi with important news: Ishigami has turned himself in and has confessed to the murder of Togashi.
Kusanagi sits in an interrogation room with Ishigami. In a clear, calm manner, Ishigami weaves a story, claiming that he is Yasuko’s secret bodyguard.
Ishigami says that Yasuko moved next door to him so that she could be under his protection; they communicate in codes. He claims to use a listening device to hear her messages to him through the walls. Ishigami says that one day, he caught Togashi snooping and looking for Yasuko. He knew who Togashi was because he “know[s] everything there is to know about Yasuko Hanaoka” (234). He says that he lied to Togashi and claimed that Yasuko had recently moved again; he reports that he gave Togashi a false address for a location near the Old Edogawa River, where there would be no witnesses. He then met Togashi there late at night, bringing an electrical cord with him. He snuck up on Togashi and strangled him, then smashed his face and burned his fingers out of fear that his identity might implicate Yasuko. He tried to burn the clothes, but the fire leapt too high, so he ran. He did not realize that Togashi had brought a bicycle, which is why the police found it and were able to identify Togashi despite his efforts. Ishigami adds that he turned himself in because Yasuko betrayed him by dating Kudo, and he now wants to punish her.
His language and behavior lead the police to determine that he is a delusional stalker who killed Togashi alone, without the knowledge or assistance of Yasuko. Kusanagi is unconvinced by Ishigami’s story, but he cannot find any holes in it. Kishitani and the police chief are satisfied with the story, especially when they search Ishigami’s apartment and find not only the murder weapon, but also a hole in his wall with a listening device to spy on Yasuko. They also find copies of the threatening letters on his computer. They speak to Yasuko, who admits that she has received several such letters; however, she claims that she did not think they were connected to Togashi’s death, so she never mentioned them.
Yukawa listens while Kusanagi tells him Ishigami’s story. Kusanagi admits that he does not believe it but cannot disprove it. Yukawa describes a famous math problem that asks whether it is more difficult to find a possible solution to a problem or to verify if someone else’s problem is correct. He then says that Ishigami has provided one possible solution to the problem of the murder, which fits all the variables. However, the question that remains is whether there are any other possible solutions that also fit the variables. He admits that he does not want it to be true because Ishigami is his friend. Looking sad and tired, he asks Kusanagi to leave him alone to think.
Later that day, Kusanagi follows Yukawa when he leaves the university. At first, he thinks that Yukawa is headed for Ishigami’s apartment, but then Yukawa ends up at the Shin-Ohashi Bridge on the Sumida River. Kusanagi joins him, and they walk along the riverbank, stopping to look at the unhoused people in the tent city. Yukawa tells Kusanagi about his last conversation with Ishigami, when he made the comment about cogs. He says that Ishigami’s decision to turn himself in was his answer. Yukawa does not want his friend to be accused of murder, but he does not think he can avoid it.
Kusanagi asks Yukawa again why he first suspected Ishigami. Yukawa describes the night he first visited Ishigami. Ishigami had never cared about appearances before, having always believed that the mind was all that mattered, but that night, he lamented that Yukawa still looked good while he had grown old and lost his hair. That was Yukawa’s first clue that Ishigami had changed and had developed romantic feelings for someone. Then Yukawa joined Ishigami at Benten-tei and saw his friend’s angry, jealous reaction to Kudo’s arrival. That confirmed Yukawa’s suspicions.
Finally, Yukawa says that he wants to tell Kusanagi something, but Kusanagi must promise never to share it with anyone else: not his friends, boss, or family. Kusanagi pauses, understanding that if he refuses, he and Yukawa will never be friends again. He makes the promise.
Yasuko is wracked with guilt over Ishigami’s sacrifice. She walks home from work and is surprised when Yukawa and Kusanagi stop her on the street. Yukawa asks to speak with her privately, and they sit on a bench. Kusanagi stands apart to give them space. Yukawa says that Ishigami has confessed to something he did not do. However, Yukawa also realizes that even Yasuko does not entirely understand what Ishigami has done for her. He knows that Ishigami never wants her to know, but he feels it is necessary to tell her.
She pretends not to understand. Yasuko says that she has not lied about her alibi, and he agrees that she has not. Then Yukawa asks if she ever wondered why she did not need to lie. He asks if she ever thought it was strange that the police kept asking for her alibi on the night of March 10, when she killed Togashi on March 9. Yasuko freezes. Yukawa explains.
To make certain that Yasuko would be safe, and to ensure that Ishigami himself would not lose his nerve and back out of his plan to protect her, he did something that no one could imagine. The reason the police believe that Togashi was killed on March 10 is because Ishigami killed someone else entirely on that day. This new crime both obfuscated the first crime and made Ishigami equally guilty, so that it would not benefit him to later back out even if he wanted to.
Kusanagi watches from a distance, knowing what Yasuko is listening to because Yukawa just finished explaining it to him a few moments ago. As he waits, he recalls the details of that conversation. Yukawa pointed to the tent city of unhoused people and explained that he had spoken to several men, including the Can Man, learning that a recent addition to the tent city had disappeared unexpectedly around March 10. No one had reported him missing, and Yukawa explained to Kusanagi that Ishigami had killed this unhoused man. The more Kusanagi uncovered about that death, the farther it took him from Togashi’s killer, Yasuko. Yukawa deduced Ishigami’s entire plan, including the theft of the bicycle, which ensured that the police would identify the time of death as the night of March 10; this detail cemented Yasuko’s alibi at the movie theater. Yukawa then explained that Kusanagi had helped him to realize Ishigami’s plan by mentioning Ishigami’s comment about disguising a math equation. That made him realize that Ishigami had “made a trick body look like a trick alibi” (278). The only thing Yukawa did not yet know was where or why Ishigami had disposed of Togashi’s body, but he imagined that it would appear eventually.
Now, Yukawa finishes his conversation with Yasuko and returns to Kusanagi. The two men watch as she sits on the bench, looking stunned. Kusanagi says he will need to arrest her if she does not turn herself in. Yukawa warns that if Kusanagi uses the information he got from Yukawa after promising not to, they will no longer be friends: merely a detective and his informant. Yukawa adds that it would be better if Yasuko were allowed to go free, given everything that Ishigami sacrificed for her.
Yasuko sits on the bench for a long time. Eventually, Kudo appears and gives her a ride home. He gives her a ring, saying that he is serious about her but understands if she needs time to think about things. Yasuko nods and walks to her apartment. She recalls the letters from Ishigami, which were designed to make him look like a stalker. The letters included a message in which Ishigami said that Kudo was a good man and that she should be happy with him. He asked her not to feel guilty. If she did not live a happy life, then his sacrifice was for nothing. Now, she begins to cry. Then a phone call startles her. Misato’s teacher tells her that Misato was found in the gym, having cut her own wrists in an attempted suicide. Misato has been taken to the hospital and will recover.
Ishigami sits in his cell, recalling the first time he met Yasuko and Misato. A year ago, he was dissatisfied with his life and felt that everything was a waste. He had thought about death many times, and that day, he decided to do something about it. However, as he stood on a chair, preparing to hang himself, the doorbell rang. Yasuko and her daughter had just moved in next door and came to introduce themselves. He had never seen a woman more beautiful. After that, his life changed, becoming less tedious and meaningless. He was happy just to be near them, with no pretensions that they would ever know about his feelings. He felt that it was “like his relationship with mathematics: it was enough merely to be associated with something so sublime” (291).
When he had the opportunity to help them with Togashi, he did not hesitate. He felt that he was merely repaying them for the life they had given him. He knew it would be impossible to completely hide Yasuko’s connection to Togashi, so he devised his plan. He approached the Engineer at the tent city and pretended to hire him for a job. He then sent the man to Togashi’s rented room to make sure that his fingerprints would be all over the room; this tricked the police into identifying the man’s body as Togashi’s. Ishigami then stole the bicycle and then gave it to the Engineer so that the Engineer’s fingerprints would be on that as well. He set everything up just as Yukawa deduced. He hid Togashi’s body in his own apartment bathroom, dismembered it, placed it in weighted trash bags, and sank it in the river. He knows that someone will find it eventually, but it will be too late by then, as he will already be tried and convicted as Togashi’s murderer. He is filled with darkness as he thinks about the matter, but he refuses to regret his actions.
Later, Yukawa visits Ishigami in an interrogation room. Yukawa reveals that he told Yasuko the truth. Ishigami freezes but tries to continue his act as a delusional stalker. Yukawa says that he is saddened to lose both a friend and a genius “like no other in this world” (296). Ishigami does not comment. As the police move him back to his cell, he sees Yasuko enter the police station. She runs to him, crying, and begs for forgiveness, saying that she cannot undo what has been done but can face her punishment with him. Realizing that everything he did has been for nothing, Ishigami screams. The police try to restrain him, but Yukawa stops them, saying that they should let him cry. As Yukawa stands by his side, Ishigami continues to scream.
In the last five chapters, the focus of the novel shifts away from the police investigation to a subtle but emotionally fraught intellectual showdown between Ishigami and Yukawa. This showdown is the culmination of the cat-and-mouse game that the two have played over the course of the novel. At no point do the two characters explicitly acknowledge that they are competing against each other. Instead, they speak in vague terms, circling around the issue to conceal their actual meaning. Yukawa has done this with Kusanagi throughout the narrative, motivated by a desire not to reveal his suspicions about Ishigami. A prime example of this dynamic occurs in Chapter 15, where Yukawa and Ishigami resort to the same roundabout speech; Yukawa dissects each clue in Ishigami’s carefully constructed puzzle without explicitly accusing him, while Ishigami pretends not to understand. However, both men acknowledge their competition by referencing the symbol of the math equation, which simultaneously represents the case and the novel. Specifically, they both consider the question of whether it is more difficult to create an unsolvable problem or to solve that problem. Yukawa’s dissection of the crime is his answer: He has solved Ishigami’s presumably “unsolvable” crime by “always respect[ing] the problem’s creator” (226). Ishigami, on the other hand, suggests that the competition is not yet over. Yukawa has not yet heard his solution, which is to turn himself in under the guise of a delusional stalker.
Only after Ishigami turns himself does the true complexity of the puzzle become apparent. In this light, previous red herrings presented by the narrative, such as Ishigami’s threatening letters to Yasuko, now carry a much different meaning. Yukawa also realizes that despite his best efforts to maintain a logical stance, he also failed to see the full picture, particularly when he expected Ishigami to turn himself in but did not imagine the method he would use. Yukawa’s unwillingness to explain his deductions to Kusanagi indicate his secret hope that Ishigami would confess the full truth, thereby sparing Yukawa himself from taking on the uncomfortable role of being his friend’s accuser. Though the novel never reveals Yukawa’s inner thoughts and feelings, Kusanagi’s observations about his pained and worried expressions demonstrate the depth of Yukawa’s friendship for Ishigami and the grief that he feels over Ishigami’s guilt. Thus, although Ishigami’s devotion to Yasuko is the core component of the novel’s focus on the Morality of Love and Self-Sacrifice, Yukawa’s care for his friend is also a contributing factor to this theme. On a smaller scale, he also wished to protect someone he cared about from the consequences of their own actions. Unlike Ishigami, however, his willingness to sacrifice his moral principles to that endeavor only goes so far.
Specifically, Yukawa draws the line at the death of an innocent man, a concept explored through the symbolism of cogs. In Chapter 15, Yukawa introduces the late-arriving extended metaphor of cogs in a clock, arguing that each person (or cog) has value in the world (the clock)—no matter how superfluous they may appear to others. Moreover, each person gets to choose for themselves what their role and value in the system will be. This is the core message of the novel’s stance on The Inherent Worth of Individuals. The true significance of this cogs symbol is not immediately clear, but it is later revealed in Chapter 18, when Yukawa finally explains his deductions to Kusanagi. This scene portrays the conventional Big Reveal moment in the mystery novel structure, in which the detective character lays out the truth of the crime and explains how all the clues tie together. It is also at this moment when Yukawa presents the novel’s final moral argument.
Having examined Ishigami’s complex plan, Yukawa reveals that Ishigami did not merely cover up the Togashi’s murder but actively chose to murder the unhoused man whom he dubbed the Engineer, simply because he believed that this particular “cog” was unimportant and was therefore unlikely to be missed. Just as Yukawa first suggested in Chapter 13, Ishigami’s logical approach to life allowed him to effectively transform the Engineer from a human being with inherent value into a faceless, meaningless variable in his equation; he then manipulated and eliminated this “variable” without remorse. This issue presents a new angle on the ambiguity of ethical dilemmas, for Yukawa is deeply pained by this conclusion. Although he does not say so outright, he implies that Togashi’s death, though legally wrong, might be justified, and that Ishigami’s motivation to save the woman he loves at the cost of his own freedom is inherently noble. However, he explicitly tells Kusanagi that Ishigami’s decision to murder an innocent man—to systematically dehumanize him and reduce him to a puzzle piece—is “unforgivable” (279). Thus, the novel’s emphasis on The Inherent Worth of Individuals provides one possible framework for Navigating Imperfect Justice and Moral Ambiguity. Although some things in life may be morally complicated and the criminal justice system is imperfect at best, both Yukawa and the novel argue that the value of life is a stable foundation on which to judge whether an action is right or wrong.
The final moments of the novel then return to The Morality of Love and Self-Sacrifice. On one level, the narrative suggests that Ishigami’s attempt to sacrifice himself for Yasuko’s sake is a sign of pure love and devotion. Yukawa suggests this when he tells Kusanagi that he does not believe Yasuko should turn herself in, saying, “When I think of what Ishigami would want, I’m inclined to say she should go free” (280). In this instance, he believes that the motivation of love and self-sacrifice overrides the duty of the legal system to bring Yasuko to justice for Togashi’s death. Ishigami himself believes that his motives are pure, and his attitude stems from a sense that he is merely repaying Yasuko for saving him from a death by suicide. For Ishigami, it is a simple one-to-one equation. However, as the narrative and Yukawa assert, Ishigami’s desire to protect Yasuko motivates him to commit an act judged by the author to be unforgivable: the murder of an innocent man. This conclusion suggests that even genuine, selfless love can become morally wrong in certain circumstances.
Moreover, the narrative also questions the burden that such love and self-sacrifice place on the recipients. Ishigami’s sacrifices are far beyond the bounds of what Yasuko can carry. She cannot comprehend such “deep devotion,” and she does not believe that “her heart [is] strong enough to accept such a sacrifice” (283). The narrative thus suggests that some expressions of love are harmful to the recipient. This idea is made doubly apparent in Chapter 19, when Yasuko learns of Misato’s act of self-harm. The brief moment is covered in a mere handful of sentences, but Misato’s apparent attempted suicide highlights the damaging effects of guilt and the burden of carrying the sacrifices made by others. In response, Yasuko concludes that the only way to lessen this burden is to turn herself in, thus negating the sacrifice that Ishigami made on her behalf.
Yet even here, the narrative retains its ambivalence rather than committing to an uncompromising interpretation of events. The final scene does not celebrate the rightness of Yasuko’s decision or suggest that her confession serves the interest of justice. Instead, the final scene zeroes in on Ishigami’s despair. In his view, Yasuko’s attempt to do the right thing has merely rendered his sacrifice meaningless, destroying not only her life but also Misato’s (as the girl will now live in shame, branded as the daughter of a murderer). Yasuko’s decision to turn herself in has therefore erased the single good thing that might have come out of the many injustices in the narrative. There are no winners in the end, and no positive outcome is possible.



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