So Builderboy presented a puzzle along these lines on IRC, earlier:

You have a flat surface (for all intents and purposes, 2D)
A lobster is at position x (Builder said frog, Qwerty said turtle)
The lobster moves 1 unit away in a random direction (any angle). It tires out after 3 moves

The question is, what is the chance that Lobztor will be within 1 unit of its starting position?

Here is my approach:
-First draw a circle with center at point x of radius 1. This is the target region as well as all the first move positions
-Choose a point on the circle as the center for a second circle of radius 1.
Note that 1/3 of the circle is in the 1 unit range.
-From here, draw infinitely many circles on this circle

You will notice that this creates a circle of its own with radius of 2 units! Using the powers of geometry, we find the ratio of the area of these two circles and get....1/4. Yay! It is pretty close to magic in my opinion!

EDIT: Just so y'all know, that is 530 circles drawn by Grammer in that screenie at 6MHz.

Ah clever work, although I believe that you are lucky that it worked because while it does make a circle, it is not a circle where every point is of an equal probability 

Ah clever work, although I believe that you are lucky that it worked because while it does make a circle, it is not a circle where every point is of an equal probability 

Actually, I was worried about that, too, but while every part isn't equal, there is a balance to it.

In this case, I am pretty sure the answer is yes simply because we are dealing with circles of the same radius. To test it, if the lobster particle could move in three dimensions, we should have a 1/8 chance of it ending up within a unit away. If it could only move in 2 dimensions but took an extra move, we would have a 1/9 chance. If number of steps is n, then in 2 dimensions we have (n-1)^-2 and three dimensions has 2^-n. Now I want to look at 4 dimensions and higher.

EDIT:... because it would necessitate another class of equation. We have the polynomial solution for 2 dimensions ((n-1)^-2), n-powered polynomial solution for three dimensions. I have studied functions and have defined various types in recursive form.

f(x)=a        f(0)=a ; f(x)=f(x-1)
f(x)=ax       f(0)=0 ; f(x)=f(x-1)+a
f(x)=ax^b      f(0)=0 ; f(x)=f(x-1)+a(x^b-(x-1)^b)    ;*Needs a better form
f(x)=ab^x      f(0)=a ; f(x)=f(x-1)b
f(x)=a(x!)    f(0)=a ; f(x)=f(x-1)x


I call this "Dimensional Mathematics" in my notes, by the way, because it allows definitions of many types of equations including equations working with time itself. The latter bit is typically called an algorithm

Ah, I should have stated n>2

Darn... I don't know then...

The solution to the first problem is 1/4