I gravitated toward mathematics. And there was another reason. Our mathematics teacher was very… was… was a very eccentric guy, and also very hard – my second year mathematics teacher – also hard to please, and he had quite a reputation at Case, because, a couple of years earlier, he had flunked the entire class. He decided that none of them was learning anything; he gave F to everybody in the course.  So Louis Green was a legend at Case, and I… I was taking his class as a sophomore, called Basic Mathematics, which he had written the… the textbook for himself. And that was a course where you… stuff like… lot… things that are different from the continuous things that physicists study. And there must, you know, there must be something in the way I grew up that made integer numbers more appealing to me. I… I mean it's… it’s associated with… with computing, as everyone knows now. Of course computers didn't exist, or hardly existed in those days. I'll speak more about my first view of a computer later, but the… here we are, in… in Louis Green's Basic Mathematics class, and I'm getting to a different kind of mathematics than… than calculus.  And one day he gave a problem to the class. He said, ‘Here's a problem that I don't think… if anybody solves this problem, I'll give you an automatic A in the class’. And the problem, now, it turns out, well, I can state the problem: he said… if you… putting parentheses around… into… into a mathematical formula… if you have a… a formula with… with five variables, A, B, C, D, E, how many ways can you put parenthesis into that formula so that you’re combining two things at a time? So you could say, A, you could say, parenthesis AB, and then parenthesize C, and then D or E, or you could start with B and C and combine that with A, and so different ways to do this. So if you start out with n of these variables, and you put… and you combine with parenthesis, what… how many ways are there to do it? And this, by the way, is… is something very dear to the hearts of computer scientists now, because we call it the number of binary trees, with n leaves. But Louis Green gave us, as a problem, as a challenge, could we… could we determine this number? And if anybody could, he… he said he'd give an A to the class. Well. I don't know to this day whether he knew the answer to the problem or not, but I have found out subsequently that the answer was… was published in the 18th century, and… and had a long history. And so these numbers are so famous now that one of my friends, Richard Stanley at MIT, has… has found 128 different interpretations of these numbers, parentheses is just one of these 128 ways. And, I mean, he's collected that many ways. In fact, I… I had the honor of… of discovering number 128 last year, when I was with him in Sweden, but that's… anyway. My sophomore year, Louis gives us this… this problem, and we all knew Louis's reputation, so we figured, why work on the problem? He'll never give… he’ll never give out a problem that we could actually solve, why should we, you know, why should we waste time on this silly thing? But it turned out that I was on the football, I was in the band, actually, not the football team, and our band was going to play in Detroit, at the football game on Saturday, but I missed the bus. I got up too late, so… so I was… so I found out the bus had just left for Detroit, and I had a whole… and I had figured I'd spend all… you know, a wasted day, all day in Detroit.  So I figured, okay, I'll work on Louis… I'll spend this day thinking about Louis Green's impossible problem. And I got lucky, and figured, and found the answer to it, and so I wrote it up on two sheets of paper, and turned it in on Monday morning, and he looked it over, and on Tuesday, he said, okay, you get an A in this class.

So I'm still a physics major, but I took… took his math class, and so I cut class the rest of the quarter, and… and he lived up to his agreement, and… and I got an A on my… on the score. Well, I felt a little guilty afterwards, having cut class, so I served as… as his grader for his course, the following year. But…but that… what was I going to say? So… so, but that summer I… I switched into mathematics as a… as a major, because of my experience with the welding in physics, and because I found that mathematics was something that I would be able to actually prove… prove correctness, and that, this… this appealed to me. Still, I didn't study mathematics that much, because… because I already had the A, and an A was what I was looking for in my… in my college grades.