What are the total number of paths that can be taken to get from point A to point B ?
Rules :-
1)You can move up ,down ,left , right
2)You CANNOT return to a point that you have been to before ie no crossing your own path
3)You CANNOT move diagonally
(I have tried a similar problem ,wherein you could move only up or to the right , which was pretty easy) (Also if a general formula can be given for l*b grid , it will be appreciated)
This is a self avoiding walk problem. There is no known formula for a generic $m \times n$ grid. In fact it is rumored that this is a computationally hard (NP hard problem).
The particular grid of $4\times 4$ is doable. It comes out to be $184$. If it helps, I can upload a small python/matlab script.
NJA Sloane's IS has it listed for a set of grid size $n \times n$, unto $n=10$: http://oeis.org/A007764
There is a nice article on self avoiding walk (aptly titled 'How to avoid yourself') by Brian Hayes in American Scientist (I couldn't find the exact link, but here is a copy sourced in the internet. http://www.peterbeerli.com/classes/images/f/fa/AmSci1998Hayes.pdf)