My reply is

Fourth image

Pattern (on the left aspect):

Assume the dotted line is a path, and it is advisable observe it (we’re take a look at the left of the puzzle proper now). Then black dots don’t have an effect on your path. However after we attain a white dot we go round it, turning twice. Then we test to see if we’re subsequent to some other white dots, and if we’re we go round that twice once more. If not then we hold going straight. Furthermore, if we crash right into a aspect of the field we flip, and the trail by no means self-intersects. Last factor we discover is that the trail should go to all of the locations that are corners of the dots

Reasoning for the proper aspect:

We can get this:

Now if the highest proper dot is white then the paths should intersect earlier than reaching the underside. Thus it’s black.

We can get this subsequent. Note that the highest left must be black. If the underside left is white then once more they be a part of too early, thus it’s black. Lastly the underside proper have to be white as a result of if it is not then the paths do not be a part of.