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George Polya tutored the child John von Neumann

An interesting bit from The Pentagon's Brain: The famous mathematician George Pólya, author of How to Prove It, tutored the child John von Neumann. From page 29:

"Johnny was the only student I was ever afraid of," said his childhood teacher, George Pólya, also a famous mathematician. "If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."

I'm surprised at what seems like a random connection. I don't recall anything in the book about it, but I wonder if George Pólya and von Neumann's parents shared a scene. Now that I look, it looks like they were both born in Budapest. And this is reminding me of an old post speculating about why there were so many famous scientists and mathematicians from Hungary in the 20th century.

I wonder about the logistics of this. How would they have met? Did they have a common community? Did someone introduce them, and if so, who? How did that person meet each of them? In other words, what brought them together?