Last night, one of my student workers and I were talking. He is a psychology major interested in clinical practice. I graduated in English and religion. So, of course, we were talking about mathematics. Rob says to me, "Sometimes I try and figure out formulas in my head. Most of the time with squares."
Now, I suck at mental math. But I wrote down what he said and worked it into a formulaic expression. So, let me know if Rob is a genius or if he just stumbled on an obvious pattern that should have been obvious if math were our field.
x2 = ( x + y ) + y2, where y = x - 1
We did some random trials of 1 through 20. It seemed to hold true.
This is the essence of academia. Exploration beyond your specialty. Lots of professors forget that. Like PBS says, you never stop learning.