As you probably know, Richard Feynman had an open invitation to the residence houses at Caltech to hold parties at his house in the hills, as long as he had access to any "substances" that might be brought.
I was at such a party in the spring on 1973. I had no interest in substances and have no idea if any were around. I had been on Caltech's 1st place team in the William Lowell Putnam Mathematical Competitions held in Dec. 1971 and Dec. 1972. RF had done quite well in this when he was a student back in the 40s, so, despite my shyness, I was pushed forward and introduced to him.
"What kind of questions are they asking on the Putnam these days?", he asked in his familiar growl (I'm also from NYC, though my family had moved to LA, and found his manner of speaking familiar, nostalgic and a little comforting).
I thought quickly and remembered a favorite: "Suppose you are given 9 points in 3 space, all of whose coordinates are integers. Prove that there are at least 2 of the 9 points so that the open line segment connecting them contains another point, all of whose coordinates are integers."
He thought about it for no more than fifteen seconds and said "I give up."
"It's a pigeonhole principle problem. Think about whether the coordinates are even or odd. There are 8 choices: a point can be (even, odd, even), etc. Since you have 9 points, two of them must be in the same class and the midpoint of those two also has integer coordinates."
He said to me in his inimitable way of speaking: "That's beautiful."
At that moment, I knew that I would be telling this story for the rest of my life. And throughout my teaching career, I have often told my classes of the time I talked about a mathematical result that Richard Feynman called "beautiful" for the rest of my life. I didn't realize that I would also use a version of this proof in a research paper 30 years later.
And that, as Paul Harvey used to say, is the rest of the story.