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Famous Unsolved Math Problems as Homework

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Re: Famous Unsolved Math Problems as Homework

#71

Post by mturmon »

0xffff2 wrote: post #63
mturmon wrote: post #53

One of my favorite C.S. professors, in a first-year grad class, gave out 3 of these problems as homework, including the first one, without indicating whether it was easy or hard. The problem was posed as a while loop, and the question was, does the loop terminate for arbitrary n>0, so it seemed very tractable: given: n > 0. while n != 1: if n is odd: n = 3n + 1 else: n = n/2 It was a tease. I tried a couple of differ…

This is a pretty good example of how far computing hardware has come. I was able to eliminated 2 to 50k in 81ms in completely unoptimized C# just now. Makes me wonder who has taken this approach the farthest these days.

My prof was also more impressed by my implementing it than by my attempted proofs. (Not the first time this has happened!)

I just implemented it too. I notice that by 100K, you get an intermediate result of size just less than 2^31, but at 1M, you get an intermediate result of size about 2^36. Maybe I went up above 100K.

For more: http://en.wikipedia.org/wiki/Collatz_conjecture#Experimental..., which includes some very nice plots, and a fractal construction when iterating an extension of the map to the complex plane. Whew! (On the other hand, https://xkcd.com/710/)

Re: Famous Unsolved Math Problems as Homework

#73

Post by Houshalter »

kbart wrote: post #36
paulpauper wrote: post #15

nowadays students will just google it and see that is unsolved

It's still hard if you don't know/recognize the name of a problem and mathematical form is usually easy to disguise.

This is what https://oeis.org/ is for.

Re: Famous Unsolved Math Problems as Homework

#75

Post by selimthegrim »

cschmidt wrote: post #33
chris_wot wrote: post #20

Earlier quoted context omitted.

What, you call that a waste of time?

I did feel like it has been a waste of time. He just mixed it in with a bunch of textbook problems. It wasn't a particularly elegant or insightful, just something technical he wanted the answer to. I don't think anyone in the class got it right, as we didn't really have the tools to make progress on it. It was a first year grad class. It was one of the only times I had a really bad professor in grad school. He was vi…

Was it Jonathan Farley?

Re: Famous Unsolved Math Problems as Homework

#76

Post by pvg »

MatthewWilkes wrote: post #44

One of the first exercises in The Art of Computer Programming is Fermat's last theorem. Don Knuth rates it as 95/100 in terms of difficulty.

It's rated HM45 (higher mathematics required, 45 out of 50) on a 'logarithmic' difficulty scale. Previous editions used to rate it M50 (mathematics, unsolved research problem).

Re: Famous Unsolved Math Problems as Homework

#77

Post by TheLoneWolfling »

Houshalter wrote: post #72

Are there any unsolved math problems that aren't proving things?

Yes, sort of.

There are a number of problems that basically boil down to "what's the smallest value that has property ", which can be solved via brute force - or rather, could be, except that naive brute force would take too long, and no-one has figured out any tricks to make it fast enough to be practical.

For example, calculating the value of any but the first few Ramsey numbers (the minimum number of vertices v = R(m, n) such that all undirected simple graphs of order v contain a clique of order m or an independent set of order n).

There are also a lot of problems that could be proven or disproven via (counter)example - something like the Collatz conjecture, for instance.

Re: Famous Unsolved Math Problems as Homework

#78

Post by to3m »

ColinWright wrote: post #50
ghshephard wrote: post #34

Earlier quoted context omitted.

It's more like, "Student has a standard 15 credit semester, means 15 hours of classroom lecture + 45-60 hours of study/assignments outside of class. Student is micro-scheduling the number of minutes they have to answer each of the standard 20 Calculus questions (God, it was always 20 question) which they have to have in to the TAs before next lecture. 60 questions/week, every week, and you have about 12 hours allotte…

This sounds more like secondary school than university. Certainly for first year undergrad there are worksheets and a significant workload, but it sounds like you're describing what I think of as straight school-type work.

I had courses like this as part of my degree, though of course the ones I had could easily turn out to be something completely different. If you're any good at maths, it's very possible you just never noticed that these courses exist ;) They seemed to be designed for people who hadn't done (or had done poorly at, or mature students who had forgotten the material for) later secondary school maths exams.

This was university maths, in that it was maths, and you were at university, but in terms of the syllabus it was pretty much later secondary school standard, and there was very little emphasis that I can recall on preparing people for further study of maths. So no proving anything. Just telling you how stuff worked, and then a zillion exercises for you to do to try to make it stick.

Re: Famous Unsolved Math Problems as Homework

#79

Post by Nimitz14 »

ghshephard wrote: post #11
relet wrote: post #5

Earlier quoted context omitted.

The important part of this was: * Students are forced to depart from the “answer-getting” mentality of mathematics. * Students are forced to redefine success in learning as making sense and increasing depth of understanding. * Students are able to work in a context in which failure is completely normal. If your class tends toward revolt, they have been way too far strayed from the first point, and that may be the pro…

Undergraduates (at least for the first few years) have to be in the "Answer Getting" mentality. I'm fine with curve balls being thrown at graduate students, and maybe, just maybe, fourth year students, depending on the course - but can you imagine what a freshman/sophmore student would do if faced with this situation?

im guessing youve never been to college because no math/physics/eng student is going to complain about something about like that

Re: Famous Unsolved Math Problems as Homework

#80

Post by vlasev »

yeldarb wrote: post #64
mturmon wrote: post #53

One of my favorite C.S. professors, in a first-year grad class, gave out 3 of these problems as homework, including the first one, without indicating whether it was easy or hard. The problem was posed as a while loop, and the question was, does the loop terminate for arbitrary n>0, so it seemed very tractable: given: n > 0. while n != 1: if n is odd: n = 3n + 1 else: n = n/2 It was a tease. I tried a couple of differ…

Now I'm curious, is it guaranteed to terminate? Is there a name for this problem? It needs to land on a power of 2 at some point.

As some of the people have already said, this is the Collatz conjecture aka 3n+1 problem aka half-or-triple-plus-one (HOTPO)problem. The fact that we don't know if the algorithm will certainly land on a power of two shows that it's quite a difficult problem. The famous mathematician Erdos said that math is not yet ready for such problems.

There are heuristic to show that the algorithm should terminate but there are numbers for which it takes an arbitrarily long time for convergence. Furthermore, there are similar problems that have divergence or loops. What makes matters worse is that it's been proven that such problems are undecidable in general.

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