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1. On the hardness of mathematics: I’m reading Stanley, Enumerative Combinatorics, Vol I. I decided to do all of the exercises but I got stuck on problem 2. It was a hell of a blow to my ego, especially as the problem was rated “3-” out of 5. 5s are unsolved problems, I guess 4 is hard and 3 is ‘average’. Anyway, eventually I gave up and looked at the answer. He ends up referring to a paper and when I track down what the contents were it’s clear that this is a non-trivial problem and I didn’t have a chance. So how would you rate this problem out of 5? (Bear in mind that this is at the beginning of a book on combinatorics so it’s not assuming you have a vast body of experience to draw on):

Let B(n,k) be the usual binomial coefficient. Prove *combinatorially* sum(B(2p,p)*B(2q,q),p+q=n)=4^n. (It’s easy to prove by playing with generating functions but that’s not combinatorial.)

I could probably have spent forever finding a solution even though, with hindsight, a combinatorial proof is easy to understand.

To me this epitomises why mathematics is simultaneously easy and hard. There are countless combinatorial objects out there and it takes a creative leap to come up with the right one to solve this problem. You can’t simply sit down and apply what you have been taught methodically to find a solution. And yet once you’ve seen it, the solution is essentially just elementary counting.

(I managed the next 3- problem fine, in fact I prefer my approach to Stanley’s…)

2. why isn’t coddington & levinson in print?

3. But you should read what Stanley has to say about problem difficulty before feeling bad. He said that a 2+ is the hardest problem that can reasonably be assigned in a graduate course, if I recall correctly.

4. I checked Stanley’s introduction. He says 1- to 3- are the “easier” problems and are suitable for people using the book as a text. So I guess 3- is borderline.

Despite calibration issues, I really like the book so far. Almost half the book is exercises and solutions. It seems to me that this is the way all text books should be.

5. Sigfpe: Your post reminds me of my experience in the National Mathematics Competitions I took in high-school. The first competition I entered when I was 15, and the questions were impossible; I had no clue what to do and scored nowhere. By the time I took my second competition, a year later, I had had a year of calclulus, and now the problems were straightforward. They weren’t all calculus problems, but the way of thinking that learning calculus had taught me was useful in solving the problems. By the third competition, even without any special preparation, I had no difficulty at all and won a prize.

All problems are hard, until they are solved by somebody; then, they become easy. So, mankind struggled for several millenia to invent calculus, and it took two of the world’s greatest geniuses to do it (Newton and/or Leibniz); now we teach it to 16-year-olds.

6. I used Stanley in a graduate course and on the first day the professor said, “this is not the kind of book you would read while soaking in the bathtub.” As far as I am concerned, she is correct. I wouldn’t worry about a 3- problem stumping you. If I remember correctly, there were some 3s that my class found easy and some 3- that we found nearly impossible. I also vaguely remember thinking that Stanley’s conclusion on the difficulty of a problem was slightly better then ours when judging what a calc student would think about certain test problems. At least in the beginning of the book.