I assume that most people here have spent many long nights struggling to solve difficult math and science homework problems. What was one of the worst or most memorable problems, and why? The one that stands out the most for me was really not too difficult, but for the longest time I just couldn't see the answer. No matter what I did the answer kept coming out the same - a complex number. Now I knew that the answer had to be real since the value related to a physical quantity, but how to get there from here??? After stewing on this problem for many hours over a period of several nights, it finally hit me. The key was to recognize that I could ignore the imaginary part and use only the real part as the solution. From there the rest was easy. Having solved the problem on the first pass and then spending hours and hours staring and recalculating, I will NEVER make that mistake again.
One incident which sticks in my mind was when I was 15, and was doing GCSE maths. I had to conduct an investigation relating to algebraic series; I'd spent a while playing with plastic blocks building ever-more complex structures, and had to develop a series which could describe the number of blocks a particular design would require. Nowadays, it would be a 10 minute, back-of-the-envelope issue, but I didn't know what I know now. I remember sitting the mock exam; there were three strands, each with a maximum of 8 marks (we 'set one' pupils were expected to get at least 7,7,7). I just didn't understand it at all, and managed to score a rather embarassing 0,0,1. I absolutely shat myself when I discovered my mark, (we had to hand in coursework a few days later), and remember sitting down for hours on end trying to work out what the hell to do. I ended up getting 8,8,8, and put it down to shear brilliance, and nothing to do with the fact that I did three all-nighters in a row...
On an introductory thermodynamics exam, there was a problem about a Carnot cycle, which would have made sense with n pieces of information, but n+1 pieces of information were given - and they were mutually exclusive. So I went around and around the Carnot cycle for a couple of hours, trying to figure out why the laws of thermodynamics weren't working.
I once had one of those generic little "drop these 2 metals in a tank of water and find out the end temperpature". 3 hours later and after throwing various things at various other things in my room, i figured out what was wrong. I had it setup as (simplistic form here) mc delta T = mc delta T instead of mc delta T = -mc delta T at the very start adn never bothered to look to see if my original equation was right.
I have had a couple of memorable 'homework' experiences. One of them involved doing a lab (Hysteresis of Iron using a Rowland ring and Galvanometer) and realizing that with my working model of an inductor it could not possibly work. But it was working. When I took my questions and model to the prof, he had me write Maxwell's equations and questioned me about meaning of the term representing flux change until I was able to understand it. As the foundations of my prior understanding tumbled to the ground, I felt the earth move. Much later as part of a Homework assignment I needed to derive the 4th order Runga Kutta method for solving an ODE. The initial expression I had to reduce, by term collection and combination, was quite long, I found it very difficult to correctly and efficiently gather the terms I needed. The final solution was 4 sheets of fan fold (remember that?) printer paper. That first expression took the entire 4 sheet length to write out!