If anyone looked out of their window now, even beady-eyed Mrs.
Dursley, they wouldn’t be able to see anything that was happening down
on the pavement. Dumbledore slipped the Put-Outer back inside his
cloak and set off down the street toward number four, where he sat
down on the wall next to the cat. He didn’t look at it, but after a
moment he spoke to it.
“Fancy seeing you here,
He turned to smile at the
tabby, but it had gone. Instead he was smiling at a rather
severe-looking woman who was wearing square glasses exactly the shape
of the markings the cat had had around its eyes. She, too, was wearing
a cloak, an emerald one. Her black hair was drawn into a tight bun.
She looked distinctly ruffled. (Harry Potter and the Sorcerer’s
What does the word ‘fancy’ mean, used when meeting someone you know on a place you’ve not expected? I don’t find any right meaning in OALD or Merriam-Webster’s.
As commented by @Walter, in OP’s context, fancy is a synonym of imagine, believe. But the usage is becoming increasingly “dated”…
Many of the above instances will be the idiomatic exclamation “Well, fancy that”! (equivalent to “Imagine that!”, “What a surprise!”, “How strange!”, etc.). In OP’s context, Dumbledore could quite naturally have used any of those (possibly preceded or followed by something like “I didn’t expect to see you here!”).
Note that Dumbledore is an elderly British schoolmaster in a relatively “formal” situation (he’s greeting Professor McGonagall, one of his staff who apparently happens to have adopted the form of a cat).
Idiomatic expressions as used by Dumbledore won’t necessarily be so natural for younger speakers, and per @StoneyB’s comment below, even Dumbledore is speaking ironically here. Brits in particular still use fancy = like, want, be attracted to quite naturally today, but fancy = imagine, believe, think is now largely confined to ironic/sarcastic contexts.
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