The WMG page for The Disastrous Life of Saiki K.. Beware of unmarked spoilers.
Saiki got his powers from the hole to the Demon Realm in YuYu Hakusho
The way how Saiki Kusuo got his powers is unknown, but I have a theory. Kusuo got his powers from the tunnel to the Demon Realm in
YuYu Hakusho. His 200-meter radius of telepathy is like a Territory, right? And his parents lived in a different house while Kusuo was growing up, right? It could be that his parents lived in Mushiyori City and then Kurumi got pregnant with the then-female Kusuo and then (s)he was affected by the sinkhole, giving him his powers. Then, after becoming successful in the UK, Kusuke gave them the house in Hidariwakibara-cho. The thing is, it's from Mushiyori —> Other city —> Hidariwakibara-cho. Because Saiki said he'd done some newsworthy things in kindergarten. Kusuke left to go to the UK at fourteen. That would mean that Saiki was twelve at the time. That's middle school. So after incidents, they moved to another city, and then to Hidariwakibara-cho. For some reason, he can't shut off his Territory, and the limiter Kusuke gave him may have provided a way he can shut off his Territory. Then, somehow, it turned back on in Reawakened, giving his powers back. So that's how Saiki got his powers. Still doesn't explain how Kusuke has 218 IQ, though.
Nendou is a psychic.
Nendou is a psychic, he can block Saiki's ability to read his mind, he can defeat Hairo (who is always training) in any aspect, he also can read minds like when he encountered Saiki's clone, Nendou is just a psychic who likes to play dumb.
Teruhashi has mind or luck powers.
She can control whoever she wants and luck will be always on her side, maybe she just think is her beauty, but high intellectual individuals are immune to it, like Kusuo or Kusuke, it may be a small mind control that she allegates only to her beauty.
