Math Is Fun Forum

For the second question, direct computation gives 7:

F[] := (r = n = 0; 
  Do[n += 2 Random[Integer, {0, 1}] - 1; 
   If[n == 0, r++];, {i, 1, 100}]; Return[r];)
Mean[Table[F[],{i,1,10000}]]

OOh! MAthematica code!
Thats fine!
You are the first person I see writing Mathematica code except me
The problem is interesting, but I thing something simular had been posted before...
mathsy is on the right track, but he must have mistake somewhere...
[EDIT]Sorry mathsy, you're right, I'm wrong! (again)
I got answer 6 in interesting way, I'll post it.
to Ricky - what a mistake! It sertiantly is not easy-recognisible, but here is it:
----
While[previous ≠ 1 &&
    current ≠ 1,
-----
This should not be AND, this should be OR!
Now:

CoinToss[] := (r = 2; previous = Random[Integer, {0, 1}]; current = \
Random[Integer, {0, 1}];  While[previous ≠ 1 ||
    current ≠ 1, r++; previous = current; current = Random[Integer, {0, 1}]]; \
Return[r])

sum = 0; Do[sum += CoinToss[], {i, 0, 100000}]; sum/100000.0

which works just fine. And it gives 6!

Great! Fascinating!
It's really good, but the implict surface is another thing. When you have the equation of the curve - that's the way. And - something little - you said if you click below the curve, the point goes down. So you need to determine in which direction will you go. Because in general the curve may be pretty complex- with twists ect.
And last advice - you can change the step during evaluation. For example, if your point is far enough (f(x,y,z) much greater than 0) then you could use bigger step.

This is something like the Mihailescu's theorem: http://en.wikipedia.org/wiki/Mih%C4%83i … 7s_theorem

Good job!
Very interesting interpretation!

PS: "curve" in 3D is "surface".

Or we may use some non-trivial way-using generating functions instead.
First we find the generating function of

Why should you differentiate? If you have the distance function, just make two "for"-s to estimate the minimum:

double min = VERY_BIG_NUM
for(i=0;i<li;i+=step)
for(j=0;j<lj;j+=step){
d_i_j=d(i,j);
if(d_i_j<min) min=d_i_j;
}

theoretically-experimental - agrees with the theory AND with the experiments.

"Is the Gamma function something we just accept (like Taylor series) because it's too difficult to understand..."

Noo!
In mathematics we don't "just accept" things! Somethimes the things can become complicated at first look, but they are done with concrete axiomatic mathematical logic. The Taylor series may look weird, but they are something very foundamental - the complex-valued function extensions are defined mostly by them!

The Gamma function isn't an exception.  It's just confusing in the beginning because there are new terms. And I thing it will be better if you start exploring the Gamma function with some more user-friendly text. I'm not saying that the Wolfram's page is not good- no, but it's mainly for reference and it's highly formalized.

mikau, here's my point of view:
In an old clculus book the complex numbers were explained not as numbers from the form a+bi, but as doubles (a,b) with some basic definitions.
For example (a,b) + (c,d) = (a+c,b+d) ect.
With the complex numbers we want to "extend" the reals, but if we have a real function, we may extend it in many ways to complex.
So we may want to preserve some of the properties of the function.
It appears that the taylor expansion is pretty important characteristic of a function (differentiable). Many of the complex-valued functions are defined as a taylor series. We use the same series for the real and the complex function, because it preserves its properties.
But there are of course many functions which are defined over the complex plane by other ways with interesting properties - the zeta function and else.

What fixed point theory?
Do you mean fixed point theorem for functions in some weird mathemacal spaces (as  Banach fixed point theorem)