A fairly interesting math problem from Project Euler.

By far, this path question is the only one I put some thought in it, other problems' answer are found in less than 5 minutes each.

Best idea when meeting problems like this is to change geometric objects into algebra objects.
so there is a function, f(x), which will produce the amount of paths for a x*x block.
f(1) = 2, I tried it by drawing it out
f(2) = 6, given
any patters? not really.
I don't want to get into programming to solve this problem because I can see this does not need brute force attack, it can be solved by a equation for f(x)
now, I have to introduce another function, g(x,y), where x is the width of the blocks and y is the length of the blocks. it's easy to see the special case for this function, when x=y, is f(x).
with a little reasoning, g(x,y) = g(y,x) can be deducted, because after a rotation, the shape is the same.
there are only 2 choices for the first step, either going toward the end point by following horizontal line or the vertical line. after that, the problem deduce to a (x-1)*y or a (y-1)*x block. A inspirational found, because following recurrence will save the day
g(x,y) = g(x-1,y) + g(x,y-1)
Find the explicit formula for this recurrence and you can solve for the answer.