Which fields have the longest solutions to problems?

[Simfish]

Longest solutions to typical problems.

Like - if your solution is appearing to be too long in math, you clearly realize that you're doing something wrong - as the solutions that come in the solutions manual are all short and elegant (especially so for the hard problems). We're also taught that "short and elegant" is always best. And indeed, most of the solutions I've seen at all levels are all very short, including the solutions I've seen for the International Math Olympiad.

Which then begs the question - which fields have problems with solutions that MUST be pages long? Engineering?

EDIT: I should have posted this in Academic and Career Guidance. Oh well - there might be some other insights here

 

[trinitron]

Umm, http://modular.fas.harvard.edu/edu/Spring2004/129/references/flt/flt.pdf

 

[mgb_phys]

I think the solution to Fermat's last theorem was also a bit of a heavyweight, no wonder it there "wasn't space in this margin".

 

[morphism]

Another classic: http://mathworld.wolfram.com/Feit-ThompsonTheorem.html

But even at a more elementary level, solutions to math problems can be pages long. I would guess the same applies to almost all the other disciplines.

 

[Simfish]

What about solutions to say, coursework? It seems that the solutions to most coursework problems are usually short - including solutions to some graduate level textbooks - this applies to the hardest problems in them as well (which then makes me wonder how people get prepared for solutions that really are pages long when they aren't encountered in undergrad coursework).

And long solutions probably occur in all fields - I'm just wondering if there are any fields with coursework that is notorious for assignments with long solutions (though this could be dependent more on teacher/textbook).

Are solutions like http://ocw.mit.edu/NR/rdonlyres/1358DFF6-8B15-4729-8F6E-FED4F0A132C5/0/sol2.pdf and http://ocw.mit.edu/NR/rdonlyres/9BF3DE8E-AA53-4089-841A-AA1F2CCF7803/0/sol5.pdf among the longest you've ever seen for coursework? (I just looked them up - they don't look as long as some numerical computation problems - but they do seem quite long compared to what I've seen elsewhere)

 

[las3rjock]

Umm, http://modular.fas.harvard.edu/edu/Spring2004/129/references/flt/flt.pdf

And that's not even the complete solution--it turned out that his solution had a small gap that needed another 19 pages to fill in: http://www.jstor.org/view/0003486x/di976377/97p0064m/0

 

[las3rjock]

What about solutions to say, coursework? It seems that the solutions to most coursework problems are usually short - including solutions to some graduate level textbooks - this applies to the hardest problems in them as well (which then makes me wonder how people get prepared for solutions that really are pages long when they aren't encountered in undergrad coursework).

And long solutions probably occur in all fields - I'm just wondering if there are any fields with coursework that is notorious for assignments with long solutions (though this could be dependent more on teacher/textbook).

Are solutions like http://ocw.mit.edu/NR/rdonlyres/1358DFF6-8B15-4729-8F6E-FED4F0A132C5/0/sol2.pdf and http://ocw.mit.edu/NR/rdonlyres/9BF3DE8E-AA53-4089-841A-AA1F2CCF7803/0/sol5.pdf among the longest you've ever seen for coursework? (I just looked them up - they don't look as long as some numerical computation problems - but they do seem quite long compared to what I've seen elsewhere)

In my experience those solutions are pretty close to the average length for homework problems at the graduate level in mathematics, physics, or engineering.

The solutions in solution manuals tend to be extremely brief--they often skip steps and tend to be more of an outline than a pedagogical solution. The other thing is that in advanced coursework, the solutions to problems are often constructed using the solutions to previous problems. Thus the solution may seem short because it simply refers to a few previous solutions. What bridges the gap between paragraph-length textbook solutions and book-length solutions of famous problems like Fermat's Last Theorem is knowing how to decompose a big problem into smaller problems.

A good course for bridging this gap is the "advanced calculus"/"introductory analysis" course offered at the advanced undergraduate level.

 

[symbolipoint]

Some problems may be solvable with long solutions; but many such problems often are composed of simpler problems --- yes, very general, but probably applies in many situations, both academic and engineering.

 

[fourier jr]

i would say there are very long (not necessarily difficult) proofs in every subject. the proof of the existence of the haar integral is probably the longest one I've ever seen & that's not from engineering or even applied math. i guess that's not a typical problem though... as far as 'typical' problems go I remember series solutions to differential equations taking up a lot of paper. I don't remember any shortcuts either, just a lot of calculating. solutions using laplace transforms & Fourier tramsforms took up a lot of paper also.

 

[Ivan Seeking]

Which then begs the question - which fields have problems with solutions that MUST be pages long?

Pure mathematics. Here, on page 362, Whitehead and Russell finally manage to prove that 1+1=2.
http://humor.beecy.net/misc/principia/

 

[Schrodinger's Dog]

Where as most people snuggle up with a good book. Pure mathematicians it is said go all moist about proofs

The end of my course delves deep into proofs, I can't say I'm particularly looking forward to it. Let's hope all the proofs are short and elegant, I have a formal exam to take at the end of this one, might need some time to revise.

 

[Simfish]

Textbook (and math competition) problems. Most of the problems in textbook are designed to be short and elegant (you'll rarely find real world problems with long solutions in textbooks) - which then demands a conceptual leap between textbook problem solving and real world problem solving (especially when people are so overreliant on strategies that they've used specifically for textbook problems for years)

 

[Schrodinger's Dog]

Good I'm sure they save the mind melting stuff for those doing maths at degree level, but some of the proofs I've seen took a few readings to get the gist of it so I was nervous. They'd say at the end we didn't need to replicate these proofs just understand them, but even so it was a tad worrying.

 

[turbo]

If you want long, difficult, problematic solutions, you should look to real-world application of engineering principles to process systems. I spent a year doing a heat-and-mass balance around the water systems in a huge Kraft pulp mill. The result was a series of recommendations that resulting in efficiencies for the mill, but teasing insights out of the mountains of data was a bear! Luckily, one of the younger engineers was a whiz at Fortran and he helped me tremendously with the data-reduction.

 

[humanino]

I remember participating to a mathematical competition back in 1996. 4 problems, 5 hours to complete them, nobody in the country finished everything (it was designed like that). The first problem was elementary geometry. It took me slightly less than one hour, but was of course the easiest problem. Well, every step I took was following very logically from the previous one, but so much calculation was involved, to be done by hand, that I could not believe I would not make an error somewhere. After 8 pages, I ended up with "the length is 1996". I was pretty happy, having confidence that it was indeed the correct result. Now you might object that math is not about calculation. It was not just that. There was more, because no step was really trivial, it required quite some imagination. But it was not short.

That is just one exemple, but I could give many others. You know, when you do real research, you keep having tedious systematic tasks, checks, and calculations.

At the same time I took this competition, I also remember this other anectdote. I was following lectures in a prestigious school in France. Many well-known mathematicians, I mean major ones worldwide, had been through the same lectures as we did. There was this guy, he was really clever as hell, at the back of the classroom. He used to read books instead of taking notes, only checking what was going on once in a while. Everybody knew he was better than all of us together, including the teacher herself. The teacher was a very interesting person as well. She probably gave the same lectures for at least 25 years she had been here. Extremely rigorous, with every theorem written in a different color from definitions, and also different colors for examples, as well as demonstrations. Underline "theorem" or "definition" or "lemma", then put the $$\begin{array}{|c|}\hline\text{statements}\\ \hline \end{array}$$ in a box on the blackboard... This settles you the stage.

We had just finished our first lecture on linear algebra. She went on like that : "now, to see who among you who has really understood what I was talking about, here is a very important exercise. If you can solve it, you probably have understood why linear algebra is relevant, and powerful". The other guy checked what she wrote, and went back reading. Everybody worked for half an hour or so, and nobody found the solution. So she goes "very well, let me show you" and 4 blackboards of demonstration followed. Probably the same four blackboards she wrote every year for more that 20 years. When my friend looked at it, he seemed to have quite some fun. He let her finish however. Once it was done, he raised his hand, made two definitions, two lemmas (lemmata), and demonstrated the theorem in 5 lines. The teacher was stunned, she could not believe it.

He his know a very promising researcher in the field of formal languages and computer aided-proofs.

 

[Integral]

What do you mean by long?

The "longest" solution I encountered was a derivation of the Runga Kutta 4th order method for numerical solution of a DE.

Fortunately, this was in the days of fan fold paper, I used 4 connected sheets of paper to write a single line of algebra along the long side.