Famous Unsolved Math Problems as Homework

http://blogs.ams.org/matheducation/2015/05/01/famous-unsolved-math-problems-as-homework/

by ColinWright • 11 years ago

95 95 11 years ago

blogs.ams.org

ghshephard12• 11 years ago

The most important part of this was, "When I use these problems for in-class work, I will typically pose the problem to the students without telling them it is unsolved, and then reveal the full truth after they have been working for fifteen minutes or so."

Any professor who handed out that assignment to an undergraduate class without fair warning, would likely face a full scale revolt, possible threats of violence, and almost certainly a significant portion of the class immediately dropping him.

[Edit - to be clear, I'm commending the instructor for not hanging their students out to dry by giving a mission impossible homework assignment]

rgcase• 11 years ago

Seriously? You think students would revolt after being given an interesting problem to think about for 15 minutes?

ghshephard12• 11 years ago

"Without fair warning"

That's the important part of this - the instructor didn't give it as homework. And yes, I've seen students freak out on their instructor en-masse for far less than a mission impossible homework assignment.

omaranto• 11 years ago

I'm guessing you've never taught math to undergraduates. Unless they are math majors, for the most part they really don't like being given harder-than-average questions, let alone open problems.

relet• 11 years ago

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 professors fault.

ghshephard12• 11 years ago

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?

iamcurious3• 11 years ago

I have done that occasionally. They complain, but only in jest.

ghshephard12• 11 years ago

You've handed out a math assignment that was an open unsolved problem and not told your students what you had done prior to them leaving for the day?

pjc502• 11 years ago

Cambridge used to do this routinely. Your university may vary.

ghshephard12• 11 years ago

What year of student? Freshmen/Sophmore?

pbhjpbhj2• 11 years ago

I think so long as they say it must be independent work and give a time limit then it's fair.

toothbrush• 11 years ago

I can imagine the students now. All of them like, "omigod we're being forced to learn something by investigating things and being creative omigodomigod it has no known answer gasp".

Yeah, that'd be embarrassing.

ghshephard12• 11 years ago

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 allotted, which works out to no more than 12 minutes per question. Any questions you go over that 12 minutes, you need to steal from other questions and somehow solve more quickly, and remember, this is all new material for the student. If you blow your budget on math, it either comes out of other classes, or sleep.

That's the context in which being thrown a heretofore unsolved problem in calculus is not particularly amusing for an undergraduate.

geon• 11 years ago

As a student in 1939, the mathematician George Dantzig once arrived late to a lecture. He mistook the unsolved problems on the blackboard for homework, and solved two of them.

http://en.wikipedia.org/wiki/George_Dantzig

onion2k• 11 years ago

I imagine being told that a problem is unsolved changes the way you approach it. You'll assume that 'obvious' things have been tried before. Without that psychological barrier you can try out ideas that are obvious to you that might not have occurred to anyone yet. Telling someone that a problem is unsolved implies they won't be able to solve it unless they're better than all the people who've approached it before in some way. That isn't necessarily true; "better" doesn't really apply at the level of people attempting these problems.

riffraff• 11 years ago

relevant:

When asked, "How could you possibly have done the first interactive graphics program, the first non-procedural programming language, the first object oriented software system, all in one year?" Ivan [0] replied: "Well, I didn't know it was hard."

[0] http://en.wikipedia.org/wiki/Ivan_Sutherland

abecedarius• 11 years ago

There's something to that, but for an approach to be worth trying you don't have to expect it to work: just that you'll learn something about the problem.

vlasev3• 11 years ago

Confidence is quite important in mathematics. You would usually go further if you are confident than if you are not. That said, most unsolved problems in mathematics require more knowledge to state than these simple problems.

I think for most people, not knowing a problem is unsolved will lead them to try out many more approaches than if they knew. The problem is that most of the approaches will be approaches that have already been tried before and the rest will be useless. This does nothing for the solving of the problem but greatly aids your own understanding.

billmanman• 11 years ago

I think it's a nice story, but for the most part unsolved problems are solved by mathematicians who are aware of their difficulty.

andrewstuart2• 11 years ago

A year later, when I began to worry about a thesis topic, Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis.

I love fun stories like that. And I love that it shows this guy is crazy brilliant yet humble enough to shrug off that success and just assume he had to figure out something else for his thesis. At least that's my interpretation.

    faithInHumanity++;

vlasev3• 11 years ago

A good down to earth mathematician will always look ahead and try to solve more problems, do more math, etc. It's not surprising that he wanted to create new math despite already doing so.

cschmidt3• 11 years ago

In grad school, I had a graph theory professor who assigned a research problem he was stuck on as homework. He didn't tell us that until later, after I had a wasted a lot of time trying to solve it. It wasn't very cool.

chris_wot4• 11 years ago

What, you call that a waste of time?

cschmidt3• 11 years ago

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 visiting for six months, and you could really tell he didn't want to be teaching. He used an out of print book (Graph Theory With Applications by Bondy and Murty), which meant I had to go to the library to even read it. He lectured at a level that was beyond us, and he literally wouldn't tell us where his office was, so he wouldn't have to answer questions. Maybe with a different teacher I would have been less annoyed about the research problem.

amelius• 11 years ago

While these problems are interesting, it would be nice to have a little background on them. Where did they come from? What other problems would be solved by solving these problems?

privong• 11 years ago

The post links to the wikipedia pages for two of the problms, so it is pretty easy to get the background on them. At any rate, the point of the post was not those particular problems, but how the instructor uses them to challenge the students. I think digressions into why those problems are mathematically or practically interesting would distract from the main theme of the post.

paulpauper2• 11 years ago

nowadays students will just google it and see that is unsolved

noja2• 11 years ago

Can you Google an equation?

darkxanthos• 11 years ago

You can to a degree. I've done it a bunch for my math classes. :)

EDIT: Also wolfram alpha

chadzawistowski• 11 years ago

Try using this! http://saskatoon.cs.rit.edu/min/

Instructions at http://saskatoon.cs.rit.edu/min_instructions/

kbart• 11 years ago

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

Houshalter5• 11 years ago

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

chris_wot4• 11 years ago

Getting a database connection error...

hownottowrite• 11 years ago

cached: http://webcache.googleusercontent.com/search?q=cache:Y5sD7Mx...

pbhjpbhj2• 11 years ago

These are great problems the first two seem to have obvious modes of proof at first sight ...

What I particularly like is that the Erdos-Strauss problem is one that's accessible to a junior schooler - once you know fractions you can investigate. I could see it forming part of a project like the graph theory investigations posted formerly, http://jdh.hamkins.org/math-for-eight-year-olds/.

ForFreedom• 11 years ago

I think the Database error is HN math homework..

got any clues?

MatthewWilkes2• 11 years ago

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.

fapjacks• 11 years ago

95/100

Heh.

pvg• 11 years ago

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).

MatthewWilkes2• 11 years ago

I knew I should have checked my facts before I posted. Still, it gave me a good chuckle when I saw it.

superuser22• 11 years ago

This is great in class and for homework, but it can be taken too far. UChicago professors like to put open problems (without any indication that they are unsolved) on final exams. Half the battles is identifying which problems are unsolved to avoid wasting time on them.

mturmon2• 11 years ago

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 different ways to prove it, but of course nothing worked. Then I tried to find a counter-example. The numbers get pretty big, in some cases, because applying

  n = 3n + 1

several times in a row will blow "n" up, despite the intervening n = n/2. This was before arbitrary-precision arithmetic was a thing, so the only tool I knew that could do this was the Unix "dc". I think I was able to get up to n=10K or 50K on our campus VAX.

That was about 30 years ago. I still remember it fondly.

0xffff2• 11 years ago

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.

ggchappell2• 11 years ago

... who has taken this approach the farthest these days.

Tomás Oliveira e Silva of the Universidade de Aveiro, in Portugal, apparently. According to a page on his personal website [http://sweet.ua.pt/tos/3x+1.html] he has verified it up to 5 x 2^60.

mturmon2• 11 years ago

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/)

yeldarb• 11 years ago

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.

_asummers• 11 years ago

It's the Collatz Conjecture. [0]

[0] https://en.wikipedia.org/wiki/Collatz_conjecture

ggchappell2• 11 years ago

Now I'm curious, is it guaranteed to terminate?

No one knows.

Is there a name for this problem?

The statement that it always terminates is the Collatz Conjecture. It's the first problem in the posted article.

vlasev3• 11 years ago

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.

Houshalter5• 11 years ago

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

TheLoneWolfling• 11 years ago

Yes, sort of.

There are a number of problems that basically boil down to "what's the smallest value that has property <x>", 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.

zornthewise• 11 years ago

Finding [Ramsey numbers](http://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one. They are hard to calculate even for small values. Most open problems are either to prove something because we can make a good guess to what the answer should be.

jordigh4• 11 years ago

This isn't a meaningful question. There isn't a distinction between what is mathematics with a proof and what is mathematics without a proof. The examples that others have given about Ramsey numbers are, in fact, a proof. The proof could consist of a gigantic computation or it could be some other deep insight. But in the end, saying the Ramsey number R(5,5) is, say, 47, is not very meaningful without a convincing argument that establishes why it's that number. And those convincing arguments are a proof.

Houshalter5• 11 years ago

That's not true. There is a whole class of problems that are trivial to prove an answer is correct, but very difficult to find the answer. Like factoring the RSA numbers, or finding a large prime number, etc.

lfowles• 11 years ago

Sounds like homework I would ignore in favor of getting all of the more solvable homework finished :)

robertfw• 11 years ago

That's missing the point. The journey is often more important than the destination (i.e., learning how to explore a problem space, learning how to deal with failure, etc)

Famous Unsolved Math Problems as Homework

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