Knight Movement on Board for Questions
Assume the white Knight starts on square b1 as shown in figure (a). How many paths can the Knight take to reach the b8 square marked ”X”? Assume that the knight must make forward progress towards the 8th row with each move - it can never move backwards. One such path is shown.
How many paths can the Knight take to reach X assuming that the Knight can never land on any of the 4 center squares marked in diagram (b). Again, only forward moves are allowed.
For convenience sake, I've reversed it from going from $b8$ to $b1$ so that I can fill out my grid from top to bottom rather than from bottom to top to make typing easier. The process is otherwise the same. I've also added lines in my array to make reading it easier. They have no mathematical significance.
Here is a good approach which does involve setting up a grid but where you effectively have all of the information about all of the paths present in the same grid. Begin with the first row with a $0$ in every position except at $b8$ where you have a $1$... The other positions in other rows begin blank.
... From there, fill out the numbers in the second row (and then third row etc...) by looking at the position and asking if there are any moves from the previous row(s) who could have moved to the position in question, and adding any numbers from those previous positions together and writing the sum in the spot for the position in question. Repeat this process. The numbers you are writing are the number of distinct paths you can take to arrive at that position. Read the number in position $b1$ to complete the problem.
The beginning of the array looks like this:
$\left[\begin{array}{cc|cc|cc|cc}0&1&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\\hline1&\color{blue}{1}&1&\color{blue}{0}&0&1&0&0\\\color{blue}{1}&0&2&1&\color{blue}{2}&0&0&1\\\hline1&3&\color{red}{4}&\dots&&&\dots\\\vdots&&\vdots&&&&\vdots\end{array}\right]$
The number most recently filled in was marked in red and was found by adding together the numbers marked in blue, those being the possible positions preceding this one in the sequence of moves. I leave it to you to finish filling out the grid yourself and finding the final answer.
For the second problem, the technique also works. Just place zeroes in any illegal positions (i.e. in your positions $d4,d5,e4,e5$) ahead of time and skip the step where you would have otherwise calculated their number by adding possible preceding position's numbers together.