Number of ways to divide dotted square grid in half Consider a $4\times4$ dotted grid that looks like this: 
By drawing a series of connected straight lines going from dot to dot, there are many ways to divide the square into two halves (of equal area), e.g. like this: 
or like this: 
Is there a way to work out how many such ways there are in total? Can we also generalise it from $4\times4$ to $n\times n$?
 A: only works for symmetric cuts.
First lets note that any cut we make works if we rotate it by 90 degrees. So lets solve the problem for cuts we make from bottom row to top row. Now lets number the points as follows\
1 2 3 4
5 6 7 8 
8 7 6 5
4 3 2 1 

Now we can pick only one number in the top row. Since picking a second number is same as starting the cut at that number. $$
\text{4 ways}$$
Now we have numbered the dots in such a way because on rotating 180 degrees the cut should look the same. 
Now we can take 1 number in 2nd row in $$\text{4 ways}$$. That gives $\text{16 ways}$ in which we can cut the square using one dot in each row. Like your first example. 
Now if we take more than 1 point on the second row then the third point can be chosen in 3 ways for each of the points. Now that gives us $$4 \times 4 \times 3=\text{48 ways}$$
in which a cut can be made. So there are total of $\text{64 ways}$. Since rotating by 90 degrees also cuts the square in half we get total of $$\text{128 ways}$$. 
Edit
Lets write instead as 
1  2  3  4
5  6  7  8 
8. 7. 6. 5.
4. 3. 2. 1. 

Now we need to find sequences that are symmetrical. Like $$1 \rightarrow 1. \\ 
1\rightarrow 7\rightarrow 7.\rightarrow 1.$$
So now lets consider how many sequences of each length can we form $$2: \quad 2\times 1 = 4$$
As if we join $1 \rightarrow 1. $ it is the same as $1\rightarrow 6\rightarrow 6.\rightarrow 1.$.
Similarly we can form sequences of upto lengths 8 or 10. 
