Donald Knuth's 20th Annual Christmas Tree Lecture: (3/2)-ary Trees [video]
youtube.com
youtube.com
My largest takeaway was just the joy of looking at the results of mathematical constructs for patterns. Not just symbolically manipulating expressions, but also simply listing the results and seeing what could be found.
That is, many of the techniques he showed involved simply recognizing patterns in output. Then, exploring the equations symbolically to see if he could explain these ideas.
Though, again, limited intelligence on my end. :)
I would also keep in mind that Donald Knuth probably spent more than an hour before understanding the topic.
Something that I think anyone can take away from this lecture is a story of discovering new mathematical relationships. He starts with a function that correlates to a logical problem, changes the inputs in an new (strange) way, then studies the output to form a theory that connects to the original problem.
The key was recognizing old results in a new problem.
This type of mathematical exploration seems to be the story of his life. See Quarter-imaginary base for him doing the same thing in high school.
...fifty years writing a 1comprehensive 12 chapter book on compilers. The reason: things like this are relevant to its scope
What am I going to do about it?