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‘Excuse me? Prepare?’ Kusanagi’s eyes widened.

Yukawa looked over at Utsumi. ‘You told me what For -ensics thought when they examined the filtration system – that there was only one way to get the poison in there, which was to remove the hose, insert the poison, then reconnect it. Correct? Well, Forensics was absolutely right. That’s exactly what happened … one year ago.’

‘No way,’ Kusanagi spluttered, then fell silent.

Utsumi shook her head. ‘But – if she did that, she couldn’t have used the water filter!’

‘That’s right. She didn’t use the water filter for an entire year.’

‘But that just doesn’t make sense. Wasn’t there evidence that the water filter had been used?’

‘The crud in the filter wasn’t from the past year. It was from the year before.’ Yukawa opened his desk drawer and produced a piece of paper. ‘Remember when I had you check on the filter part number? Well, I called up the maker and asked when that particular part was on the market. It turns out that number range was used two years ago. They went so far as to say it was highly unlikely a part with that number had been used to change a filter only one year ago. In other words, when they changed the water filter a year ago, our killer immediately replaced the new filter with the old one, thinking that if the filter was found to be unused after the poisoning, she’d be discovered. That’s when she put in the poison.’

‘Impossible,’ Kusanagi whispered, his voice hoarse. ‘That’s simply impossible. You’re saying she put the poison in, then didn’t use the filter for a whole year, not even once? It doesn’t make sense. What if someone else had used the water from the filter? How dangerous is that?’

‘Very,’ Yukawa agreed coolly. ‘But she pulled it off. For an entire year, whenever her husband was home, she never went outside, and she never let anyone else near the kitchen water system. Whenever they held a party at their house, she did all the cooking. And she always had bottled water in good supply, making sure they never ran out. Everything to ensure that her plan worked.’

Kusanagi shook his head several times. ‘I don’t believe it. It’s impossible, Yukawa. Show me one person capable of pulling that off.’

‘Actually, it is possible,’ Utsumi said. ‘Yukawa had me ask all kinds of questions about the couple’s life after they got married. I talked at length with Hiromi Wakayama. Even though I didn’t know why I was asking the questions at the time, I get it now. You were checking whether anyone other than Mrs Mashiba had a chance to touch that water filtration system, weren’t you?’

‘Correct. The most compelling piece of evidence by far was how the Mashibas spent their days off. Ayane would sit on the living room sofa, working on a patchwork all day long. Why did she choose that room? So she could keep an eye on the kitchen.’

‘You’re delusional,’ Kusanagi said, his voice almost a groan.

‘Logically speaking, it’s the only possible answer. Our killer was extraordinarily tenacious, and extraordinarily strong-willed, you have to admit.’

Kusanagi shook his head, muttering, ‘no way,’ over and over. But as the argument sank in, his denials gradually lost their strength.

He remembered something Ikai had said about Ayane’s peculiar devotion to her husband.

‘She was the perfect wife, you know. Utterly dedicated to him. Whenever he was home, she would sit there on the living room sofa, doing her patchwork, ready to serve if he needed anything.’

His thoughts travelled back to his visit with Ayane’s parents at their home in Hokkaido. Hadn’t they told him that she had never been a good cook, until those classes she took right before getting married?

In both cases, she was just making sure that no one else would ever enter the kitchen.

‘So when she finally wanted to kill him … she didn’t have to do anything,’ Utsumi said.

‘That’s right. Nothing at all. All she had to do was leave her husband alone in the house. Actually, that’s not quite true; there was one thing. She emptied a few of the bottles of mineral water first, leaving only one or two behind. As long as Yoshitaka was drinking those, nothing would happen. He probably used the bottled water the first time he made coffee. But when he made it for himself the second time, he was down to one bottle, and since he wanted to save that for later, he used water from the filter. After sitting in place for a whole year, the poison finally had its chance to do its job.’

Yukawa picked up his coffee mug from the table. ‘She could have killed him at any point over the last year, but she didn’t. Instead she watched carefully over her trap to make sure he never poisoned himself by accident. Most killers worry about how they’ll do the deed, but in this case, it was the complete opposite. All of her efforts went towards not killing him. It was a very unusual kind of murder, you have to admit. While it is theoretically possible, it’s entirely unrealistic. An imaginary solution, if you will.’

Utsumi took a step towards Kusanagi. ‘We have to go and confront Mrs Mashiba and get her to turn herself in.’

Kusanagi took one look at the victorious expression on the junior detective’s face before turning his gaze to Yukawa. ‘Is there proof? Is there any way to show what she did in court?’

The physicist took off his glasses, resting them on the table beside him. ‘Of course there’s no proof,’ he said.

Utsumi looked at him in surprise. ‘Really?’

‘Give it a moment’s thought and you’ll see. If she had done something, she might’ve left a trace. But she killed him by doing nothing. Look all you want for some trace of what she did, and you won’t find a thing. About the only thing you have to go on is the poison they found in the filter, but you told me yourself that evidence is inconclusive. And the detail about the part number on the filter is circumstantial at best. In other words, there’s no way at all to prove she killed her husband.’

‘I don’t …’ Utsumi’s protest trailed away into silence.

‘Didn’t I tell you before?’ Yukawa’s face was grim. ‘It’s the perfect crime.’

TWENTY-NINE

Utsumi was going through paperwork at the Meguro station when Mamiya came in from outside and shot her a look. She stood and went over to his desk.

‘I was just talking to the section chief about that case,’ Mamiya said, sitting down. He had a glum look on his face.

‘Did we get a warrant?’

Mamiya gave her a shake of his head. ‘No, we didn’t, and we won’t unless there’s some significant development. We don’t have enough material to identify our killer. I’m impressed as always with Galileo’s detective work, but without proof, we can’t go to trial.’

‘That’s what I was afraid of,’ Utsumi said, her head hanging. This was playing out just as Yukawa said it would.

‘Everyone in administration is pulling their hair out. Who the hell arranges everything for someone to poison themselves, then watches carefully for a year to keep them from doing it? No one believed me at first. Not even I believed me at first. It’s clearly the only answer here, but that doesn’t make it any easier to swallow. It just sounds so … impossible.’

‘I didn’t believe it myself when Professor Yukawa first laid it out.’

‘There’s no accounting for what some people are capable of, clearly. That goes for this woman, Ayane, as well as the professor. You have to wonder what’s going on in those heads of theirs, sometimes,’ Mamiya said, a bitter look on his face. ‘We still don’t even know if he was actually right. And as long as we can’t say for sure, we can’t touch Ayane Mashiba.’