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After Yasuko Hanaoka and her daughter strangle her abusive ex-husband Togashi on March 9, their neighbour, the mathematics teacher Ishigami, hides that body. Because the police date the death to the 10th, Yasuko's alibi for that night (the cinema, a karaoke bar) is genuinely true, and every hour detectives spend attacking it only confirms it.
A fabricated alibi fails under scrutiny because each false detail is a point where evidence can contradict it. Ishigami instead makes the suspects' movements real and rehearses them in reporting those facts, so interrogation, receipts and witnesses all corroborate. The deception is moved to a premise the investigators never question, the identity and date of the victim, so their rigour works for the deceiver.
Ishigami's teaching maxim is to set a question that looks like one kind of problem, such as geometry, when it is really another kind. He builds the case the same way: police treat it as an alibi problem when it is an identification problem. The physicist Yukawa breaks it by asking what category of problem he is facing, not by answering the posed question better. Whoever controls how a problem is framed controls where solvers spend their effort.
Ishigami commits murder and then confesses to crimes he did not commit, so that Yasuko can live free. She cannot accept a freedom bought by an innocent man's death and another man's ruin, so she turns herself in and makes his sacrifice worthless. Devotion that decides for the beloved what her happiness requires removes her moral agency, and she may reject the gift in order to take that agency back.
Yasuko Hanaoka and her teenage daughter strangle her abusive ex-husband Togashi on March 9. Her neighbour Ishigami, a maths teacher, disposes of the real body and kills an anonymous homeless man on March 10, dressing him as Togashi and mutilating the face and fingers so police identify him as Togashi and date the death to the 10th.
Ishigami designs exam questions that look like one kind of problem, such as geometry, but are actually another, such as algebra, so solvers fail because of what they assume rather than any lack of skill. He builds the alibi the same way: detectives stress-test the alibi as a question of 'where was she on the 10th?' when the real question is 'whose body is this, and when did he die?' A deception is strongest when it answers the wrong question honestly, because every check confirms it.
Ishigami and Yukawa frame their contest as the question of which is harder: devising a problem no one can solve, or solving one someone else has devised. Ishigami's scheme depends on the asymmetry that a verifier checks the answers they are given, while the designer controls which questions are ever asked.
Ishigami's plan is logically flawless, but it treats Yasuko as a variable to protect rather than a person with a conscience. His extra killing and false confession turn her freedom into a debt she cannot live with. She confesses and destroys the solution. A sacrifice made on someone's behalf without their consent can become a burden on them rather than a gift, and plans that leave out the beneficiary's moral agency fail at the human step.