
The answer is π^2/6: What’s the question?  - wglb
http://www.blog.republicofmath.com/archives/4801
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sigil
If you click through to the mathworld page [1], there's a pretty awesome
geometric interpretation of the Riemann zeta function:

"If a lattice point is selected at random in two dimensions, the probability
that it is visible from the origin is 6/π^2. This is also the probability that
two integers picked at random are relatively prime. If a lattice point is
picked at random in n dimensions, the probability that it is visible from the
origin is 1/ζ(n), where ζ(n) is the Riemann zeta function."

[1] <http://mathworld.wolfram.com/VisiblePoint.html>

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xyzzyz
That's not really the geometric interpretation of the Riemann zeta function,
but rather the series \sum_{k=1}^\infty 1/k^n. Riemann zeta function makes
sense not only in positive integers greater than 1, but also on the almost
whole complex plane, and identifying complex zeros of zeta function is of main
interest.

~~~
sigil
Okay, so it's an interpretation of Riemann zeta on positive real integers.
It's still neat. ;)

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Natsu
From the article: "Here’s a simple yet revealing question to ask people at all
levels of mathematical attainment: 'The answer is 10. What is the question?'"

I know I'd love to hear HN's answers.

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magicseth
Here's one: How do you express the number of digits that exist in any given
base, in that base?

~~~
NickPollard
If you're using Arabic numerals, that is.

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lurker19
Theorem without proof is unsatisfying.

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BrentRitterbeck
This includes a brief sketch of Euler's argument as well as a several formal
methods of proof:

<http://en.wikipedia.org/wiki/Basel_problem>

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tzs
A little harder: the answer is pi^2/(12 log 2). What's the question?

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temphn
Standard way to do this is with Fourier series on the sawtooth wave.

<http://math.mit.edu/classes/18.03/spr10/lec_22.txt>

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m314
For all values of n=pi?

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vog
The title says "π" (pi), not "n" (en).

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psawaya
I like this a lot, it seems like a classic example of divergent thinking, in a
subject (math) that doesn't traditionally involve a lot of it.

<http://en.wikipedia.org/wiki/Divergent_thinking>

~~~
vog
_> divergent thinking, in a subject (math) that doesn't traditionally involve
a lot of it._

I strongly disagree with that.

Divergent thinking as well as creativity is very important in mathematics and
has always been. Solving problems - this is what mathematics is all about. So
creativity is vital, and more important in math than in many other professions
that call themselves "creative".

It is a common misconception that math is about following calculation steps
and simple algorithms. This is "math" as taught in school, and it is okay
because those are the very basics. However, as soon as you join mathmatical
competitions, or even start studying math, things are different: It is no
longer about _using_ math but about _understanding_ and maybe even _extending_
math.

That means: defintions, theorems and proofs. And, of course, always keeping
some examples, visualizations, simplified views, and heuristics in mind. Those
help you to stay on top of things, so you don't get lost in this forest.

To find proofs, or even just good definitions (which is sometimes as
important), you have to be _very_ creative. Learning other proofs means
getting lots of ideas about what could work, but as soon as all well-known
tricks are exhausted, there are still lots of open questions whose definite
answer (i.e. proof or counter-proof) will need yet another genius. Or just
another student whose thinking diverges into directions that others haven
expoited yet. Or maybe they also thought about that, but didn't take this
seriously or didn't try hard enough.

Dedication to an concrete problem at hand is in no way a contradiction to
divergent thinking. It is more about the combination of convergent and
divergent thinking. Sticking to only one of those, and you either won't get
far enough, or be lost into totally secluded corners without even noticing.

Also don't be fooled by math competitions (which are very different from plain
calculation competitions!). Students may get excercises that are fairly well-
known to mathematicians. However, in the view of the students those are new,
unknown problems they almost certainly haven't heard of up to that point. They
might be well-prepared with lots of standard tricks and maybe know about
similar-looking issues. But in the end of the day, they have to be creative -
very creative - to find their proof.

It is not uncommon for the jury to receive a solution from a student the
original creators haven't thought of. Usually, such an exceptionally creative
solution leads to a special award on the competition.

Also, usually the students get multiple tasks, so when they get stuck with one
excersice, they can skip to the next one. When they later come back to the
first one, it usually makes a lot more sense and have some more ideas to
explore.

~~~
psawaya
I think the author of the post was talking about middle and high school
students, who don't really see proofs. (I don't think triangle proofs in
Geometry class count) My high school math felt mostly mechanical, and involved
nothing like the kind of activity the author wrote about.

~~~
vog
I agree that math in high-school should emphasize more on the creative part.
It is harder, but for those students interested in it, showing to them what
math is _really_ all about seems to be the only way to keep them motivated.
And math competitions help a lot in that regard, at least in my country
(Germany).

