
Famous Unsolved Math Problems as Homework - ColinWright
http://blogs.ams.org/matheducation/2015/05/01/famous-unsolved-math-problems-as-homework/
======
geon
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](http://en.wikipedia.org/wiki/George_Dantzig)

~~~
onion2k
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
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](http://en.wikipedia.org/wiki/Ivan_Sutherland)

~~~
a_bonobo
Or to quote Beyer's Grace Hopper and the invention of the information age:

>Common sense would dictate that the most experienced programmers should have
been assigned to these difficult tasks, but, as Hopper glibly explained, young
people did not know that they were supposed to fail.

------
mturmon
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.

~~~
yeldarb
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.

~~~
ggchappell
> 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.

~~~
emiliobumachar
Just to clarify why, if the result is ever one (the loop's exit condition),
then it would be about to get stuck repeating 1,4,2 forever if it didn't exit.
If you can prove the loop does not exit (i.e., never reaches one) for some n,
then you just proved that, for that n, it would never reach the 1,4,2
repetition pattern even without the stop condition.

It's a clever way to re-state the problem to look more computer-science-y.

------
MatthewWilkes
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.

~~~
pvg
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).

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

------
superuser2
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.

------
cschmidt
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_wot
What, you call that a waste of time?

~~~
cschmidt
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.

~~~
selimthegrim
Was it Jonathan Farley?

~~~
cschmidt
Nope. Although I'd rather not say exactly who.

------
hownottowrite
cached:
[http://webcache.googleusercontent.com/search?q=cache:Y5sD7Mx...](http://webcache.googleusercontent.com/search?q=cache:Y5sD7Mxe0BQJ:blogs.ams.org/matheducation/2015/05/01/famous-
unsolved-math-problems-as-homework/&hl=en&gl=us&strip=0)

------
pbhjpbhj
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/](http://jdh.hamkins.org/math-for-eight-year-olds/).

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

~~~
omaranto
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.

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

~~~
jordigh
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.

~~~
Houshalter
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.

~~~
jordigh
You're thinking of computer things like P vs NP. Any mathematician who
develops some sort of algorithm for factoring large numbers would announce
this algorithm and a proof of its correctness as the interesting thing to
announce, not merely that some large number was factored, without disclosing
how.

Or perhaps they would keep it secret, but it would be a huge cultural faux pas
to have a method to factor large numbers without also explaining (and thus
proving) how it works.

~~~
Houshalter
It was just an example of a problem where the hard part is finding an actual
answer, not just proving that the answer is correct.

Proving things seems really boring and uninteresting to me, and there is no
guarantee the task is even possible.

~~~
jordigh
There is no guarantee it's possible to factor large numbers quickly either.

"Proving" isn't distinct from "mathematics". That's what I'm trying to get at.
In a sense, a proof is equivalent to a computation (and it's possible to make
this precisely true in some contexts).

The public at large has gotten some impression that proofs are something only
some kind of mathematics needs, probably after being traumatised by high
school geometry classes. Proofs are all that mathematics is.

~~~
Houshalter
You can model many problems as proofs, but not all, and it's not usually a
useful observation. Most interesting math problems have nothing to do with
proving things. The problem is people have already solved all the interesting
problems. Proofs are just what's left.

One interesting math problem someone told me just the other day was to come up
with a series of forward and backward steps that would keep you from falling
off a cliff, if a malicious person only chose each 2nd, 3rd or 4th step. Or to
find the resistance between two points on an infinite grid of resistors. Or to
find a series of bridges to get between several island without ever crossing a
bridge twice. Or coloring a map with only 4 colors. Or in machine learning,
tons of unsolved problems involving approximating intractable inference
problems.

That's a pretty wide variety of problems just off the top of my head, none of
which are proving things.

Sure some of the problems might not have a solution, but at least I'm not
asking people to solve the halting problem, which is what half of those
"unsolved math proofs" require.

Even if you do solve them, the result is uninteresting. You could, at least in
theory, quickly explain fermats last theorem to almost anyone. Or the four
color theorem, or most of the problems I mentioned. But only a few
mathematicians understand the proofs, and sometimes no one understands them
(the ones proved computers.)

~~~
jordigh
Finding that resistance in the grid of resistors involves performing a
computation. That computation is a proof.

You have it in your mind that proofs are something that you don't like and
completely distinct from computations. You don't have a clear distinction in
your mind between what you like and you don't. It seems to me that some
mathematics is just unfamiliar to you, and when that happens, you call it
"proofs".

------
amelius
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
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.

------
ForFreedom
I think the Database error is HN math homework..

got any clues?

------
chris_wot
Getting a database connection error...

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

~~~
robertfw
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)

------
paulpauper
nowadays students will just google it and see that is unsolved

~~~
noja
Can you Google an equation?

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

EDIT: Also wolfram alpha

~~~
noja
So how does it look? Lots of parentheses? Are the special characters replaced
with something?

~~~
SAI_Peregrinus
Write it in TeX, then search for that.

