collection of mathematically elegant tricks

Hi all,

I am an engineering student who is interested in math. I don't remember how 
often I have been amazed by nice and elegant mathematical tricks that make 
difficult problem suddenly very easy and trivial... Many nice tricks 
frequently appear in these newsgroups... I am wondering if anybody has seen 
a collection of mathematical tricks ranging from high school math up to 
graduate school math? Any website, Internet resources, books that have these 
kind bags of tricks? If not, I may want to start collecting and compile one 
such resources.

Recently, one very striking trick is offered by "Scott Hemphill" in 
computing the expected waiting time for a certain pattern to occur in coin 
tossing... I also remember many of the other tricks that have been 
contributed by many other authors in these newgroups...

Thanks a lot 

kiki skrev:
> Hi all,
>
> I am an engineering student who is interested in math. I don't remember how
> often I have been amazed by nice and elegant mathematical tricks that make
> difficult problem suddenly very easy and trivial...

That's the "art" part in "the art of engineering"...

> Many nice tricks
> frequently appear in these newsgroups... I am wondering if anybody has seen
> a collection of mathematical tricks ranging from high school math up to
> graduate school math? Any website, Internet resources, books that have these
> kind bags of tricks? If not, I may want to start collecting and compile one
> such resources.

Haven't seen what you ask for, but your idea of starting collecting
yourself seems to be a good one. Recognizing such tricks, mathematical
or others, when they see them, contemplating and understanding them,
and remebering them, is often part of what distinguishes the very good
engineers from the merely good ones.

Rune

"kiki" <lunaliu3@yahoo.com> writes:

> I am an engineering student who is interested in math. I don't
> remember how often I have been amazed by nice and elegant
> mathematical tricks that make difficult problem suddenly very easy
> and trivial... Many nice tricks frequently appear in these
> newsgroups... I am wondering if anybody has seen a collection of
> mathematical tricks ranging from high school math up to graduate
> school math? Any website, Internet resources, books that have these
> kind bags of tricks? If not, I may want to start collecting and
> compile one such resources.

One very good one I have seen so far is about "Buffoon's needle", one
weird way of calculating pi: what is the probability that a needle of
length d cast in random orientation and position on a line grid with
distance d will cross a line?

The solution was striking, and somebody called it "Buffoon's noodle":
the expected values of infinitesimal pieces of the needle add up to
the total expected number of line crossings, since expected values
even of dependent random variables are additive.  The expected number
of line crossings thus depends only on the length l of the shape
(could be a noodle...).  To figure out the ratio, throw a circle of
diameter d, which always crosses the grid twice, so the expected
number of crossings is 2*l/(pi*d), and in particular the needle of
length d will cross a line with probability 2/pi.

This is so much more elegant than trying to integrate over all
orientations and positions; and even if you do that, you still only
get the result for the needle, not arbitrary shapes.

Here is one for the engineering guys: when doing multidimensional
Fourier transforms, the differential operator nabla transforms to 2 pi
j vec f, where vec f is something like (u,v,w) if that's what the
variables in the transform domain are called.

-- 
David Kastrup, Kriemhildstr. 15, 44793 Bochum

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message 
news:1122230044.613276.226710@g44g2000cwa.googlegroups.com...
>
>
> kiki skrev:
>> Hi all,
>>
>> I am an engineering student who is interested in math. I don't remember 
>> how
>> often I have been amazed by nice and elegant mathematical tricks that 
>> make
>> difficult problem suddenly very easy and trivial...
>
> That's the "art" part in "the art of engineering"...
>
>> Many nice tricks
>> frequently appear in these newsgroups... I am wondering if anybody has 
>> seen
>> a collection of mathematical tricks ranging from high school math up to
>> graduate school math? Any website, Internet resources, books that have 
>> these
>> kind bags of tricks? If not, I may want to start collecting and compile 
>> one
>> such resources.
>
> Haven't seen what you ask for, but your idea of starting collecting
> yourself seems to be a good one. Recognizing such tricks, mathematical
> or others, when they see them, contemplating and understanding them,
> and remebering them, is often part of what distinguishes the very good
> engineers from the merely good ones.
>
> Rune
>

Thanks... I guess I don't want to be merely an engineer... I want to be a 
(applied) mathematician or statistician ... Looks to me that "talking math" 
is quite fashionable :=) 

"David Kastrup" <dak@gnu.org> wrote in message 
news:853bq43tx5.fsf@lola.goethe.zz...
> "kiki" <lunaliu3@yahoo.com> writes:
>
>> I am an engineering student who is interested in math. I don't
>> remember how often I have been amazed by nice and elegant
>> mathematical tricks that make difficult problem suddenly very easy
>> and trivial... Many nice tricks frequently appear in these
>> newsgroups... I am wondering if anybody has seen a collection of
>> mathematical tricks ranging from high school math up to graduate
>> school math? Any website, Internet resources, books that have these
>> kind bags of tricks? If not, I may want to start collecting and
>> compile one such resources.
>
> One very good one I have seen so far is about "Buffoon's needle", one
> weird way of calculating pi: what is the probability that a needle of
> length d cast in random orientation and position on a line grid with
> distance d will cross a line?
>
> The solution was striking, and somebody called it "Buffoon's noodle":
> the expected values of infinitesimal pieces of the needle add up to
> the total expected number of line crossings, since expected values
> even of dependent random variables are additive.  The expected number
> of line crossings thus depends only on the length l of the shape
> (could be a noodle...).  To figure out the ratio, throw a circle of
> diameter d, which always crosses the grid twice, so the expected
> number of crossings is 2*l/(pi*d), and in particular the needle of
> length d will cross a line with probability 2/pi.
>
> This is so much more elegant than trying to integrate over all
> orientations and positions; and even if you do that, you still only
> get the result for the needle, not arbitrary shapes.
>
> Here is one for the engineering guys: when doing multidimensional
> Fourier transforms, the differential operator nabla transforms to 2 pi
> j vec f, where vec f is something like (u,v,w) if that's what the
> variables in the transform domain are called.
>
> -- 
> David Kastrup, Kriemhildstr. 15, 44793 Bochum

Thanks David... but that "multidimensional FT" is a very general idea... do 
you have specific/detailed tricks that make a difficult problem suddenly 
turn into easy piece? 

On Sun, 24 Jul 2005 11:14:52 -0700, "kiki" <lunaliu3@yahoo.com> wrote:

>... I am wondering if anybody has seen 
>a collection of mathematical tricks ranging from high school math up to 
>graduate school math? Any website, Internet resources, books that have these 
>kind bags of tricks? If not, I may want to start collecting and compile one 
>such resources.

Try some competition problem books. The math competitions often have
problems where the solution is simple and elegant after applying some
inspired tricks.

2 very good ones that I know of:

   The USSR Olympiad Problem Book

   500 Mathematical Challenges

quasi

"kiki" <lunaliu3@yahoo.com> writes:

> I am an engineering student who is interested in math. I don't
> remember how often I have been amazed by nice and elegant
> mathematical tricks that make difficult problem suddenly very easy
> and trivial... Many nice tricks frequently appear in these
> newsgroups... I am wondering if anybody has seen a collection of
> mathematical tricks ranging from high school math up to graduate
> school math? Any website, Internet resources, books that have these
> kind bags of tricks? If not, I may want to start collecting and
> compile one such resources.

Ah, one thing that I remember: prove that the Eigenvalues of an
Antihermitian/Hermitian matrix are purely imaginary/real.

The proof for that is just pretty.  Something like

A x = lambda x

(for lambda an Eigenvalue, and x the corresponding Eigenvector)

<Ax, x> = <lambda x, x>   (scalar product)

<x, A^{T*} x> = lambda <x,x>  (factor gets transposed and conjugated
                               when switching sides)
<x, +/- A x> = lambda <x,x>  (Hermitian/Antihermitian property)
<x, +/- lambda x> = lambda <x,x>  (Eigenvector equation)
+/- lambda^* <x,x> = lambda <x,x>
+/- lambda^* = lambda             (<x,x> is nonzero)

Im/Re lambda = 0

-- 
David Kastrup, Kriemhildstr. 15, 44793 Bochum

David Kastrup wrote:

   ...

>  "Buffoon's needle", 

   ...

That's "Buffon's Needle", named after the French mathematician who first 
described the simplest case presented in 
http://www.mste.uiuc.edu/reese/buffon/buffon.html

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

wait until you look at your paycheck!

"quasi" <quasi@null.set> wrote in message 
news:oq78e15ss5hk4pro0q4qanj32resstk7td@4ax.com...
> On Sun, 24 Jul 2005 11:14:52 -0700, "kiki" <lunaliu3@yahoo.com> wrote:
>
>>... I am wondering if anybody has seen
>>a collection of mathematical tricks ranging from high school math up to
>>graduate school math? Any website, Internet resources, books that have 
>>these
>>kind bags of tricks? If not, I may want to start collecting and compile 
>>one
>>such resources.
>
> Try some competition problem books. The math competitions often have
> problems where the solution is simple and elegant after applying some
> inspired tricks.
>
> 2 very good ones that I know of:
>
>   The USSR Olympiad Problem Book
>
>   500 Mathematical Challenges
>
> quasi

Hi Quasi,

Are these problems real problems that will appear in study and research? 
These mathematical Olympiad always gives me an impression that they are 
unreal and heavenly; that working out them does not improve one's study, 
work, and research... am I correct?