On to chapter 4, and the investigation continues. Our police friends now know the identity of the corpse on the riverbank, and as the curtain opens, we find inspector Kusanagi and sidekick Kishitani on the way to estranged former wife (and, as we know, murderer) Yasuko. They are somewhat surprised she did not know the dead man in the news lately was her former husband, and ask her about her whereabouts on the night in question. Turns out, she and her daughter had gone to the movies, a ramen shop, and some karaoke. Perfectly normal, of course - and also, incidentally, all the kind of place where you're unlikely to actually be remembered or recognized by anybody. Mighty convenient, you might say, and indeed, the point is not lost on Kusanagi.
Anyway, they have no particular reason to think there's anything amiss. They leave, but as they do, they meet Ishigami, our Mathematical Mastermind, on the stairs. They figure that what the heck, and invite themselves in for a quick interview. Ishigami states, not surprisingly, that he never noticed anything out of the usual with his neighbor that day. Of course, what is he going to say? "Oh, I helped her clean out a strangled corpse in the hallway, but apart from that, nothing"? As they are leaving, Kusanagi notices a newsletter from Teito university. And indeed, Ishigami turns out to be a graduate of the very same university as Prof. Yugawa, our main character that we met last chapter - and of Kusanagi himself, though apparently not a student of the science faculty.
After the police leave for real this time, Ishigami produces a sweater and a telephone card, and walks to a nearby telephone booth. He is calling Yasuko to ask her about her interview with the police. Now here's my problem: why is he going to a nearby phone booth?
Obviously he's afraid that the police may find out they're talking. If they'd be tapping her phone it wouldn't matter from where he calls. Instead, he must be worried that they'll look at call logs and see that they have been calling each other. But he's never called from that phone booth before (and as we'll see, he keeps using it); it would have to be a very dense policeman indeed not to get suspicious with repeated calls from a phone booth right after the murder and the police interview. A phone booth, mind you, that's a quick walking distance from her apartment - what kind of person would call from a phone booth instead of walk over and talk in person?
Instead, why not just talk with each other in the stairwell? Nothing would be more natural than a couple of neighbors spending a few minutes in conversation whenever they bump into each other. Especially a single man and a divorced, attractive woman; if asked they could hint that he's trying to court her and she's not interested but doesn't mind. That would explain his frequent visits to her workplace as well and would have had the added bonus of being completely true.
No matter; this is a mystery novel, and if we'd demand stringent logic and consistency in everything very few of them would ever get written. We'll just note this, and let it pass.