Did I hear correctly in the second video? They both has a goldfish named "George"? No one else finds that worth remarking on? Surely P(both have goldfish named George) << P(She will guess his pet's name|their pets have the same name). So why are we focusing on the latter?
More relevantly, P(both have a goldfish named George) << P(they don't, and there's some trickery going on)
It's actually quite a useful clue - obviously no psychological "mind-reading" technique could force it, so it must be some other trick, most likely the one garethrees suggests.
I just want to burn him at a stake and watch his witch's heart bubble. It’s extraordinary. Great trick. - Stephen Fry
Derren Brown does many amazing tricks - I want to focus here on his "mind reading". This is way beyond any cold reading I've seen, but he insists that he uses no actors or stooges. He's also a skeptic, very clear about not being psychic. He does reveal some of his tricks, but maintains a lot of mystery.
Reading David Frost's mind - unusually, he struggles and gets the first one wrong, and seems to reveal tiny glimpses of his technique. Then at the end he gives more hints about his technique than usual.
Pet name - getting someone on the street to read another person's mind. In the full version (from the DVD of Trick of the Mind, series one) the segment starts with Derren telling the guy (the pet owner) that sorry, it won't work on you, then later changing his mind and bringing him in.
Creepy clown - the detail here is extraordinary.
Watch the videos then scroll down, if you want to watch it without being influenced by me... I have a few thoughts, but they don't go very far in explaining it...
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Whatever he's doing, he's extraordinarily good at it. Some speculations: