In finding the angle for the mth dark fringe of single slit diffraction using Huygen's principle, they usually split the slit into equal portions. For example, to find the first dark fringe the slit is split into two equal spacings and if the path difference between the edge and the middle point is half wavelength then the angle corresponds to the first dark fringe. An example is shown in p.4 of

http://www.csun.edu/~rd436460/100B/lectures/chapter28-4-5.pdf
To find higher order such as 2nd order they split it into four equal spacings, as shown in p.5.

My question is, in the 2nd order case if we compare the path difference between the point at the edge and the point at the center, the path difference would be one whole wavelength (because the path difference is half wavelength in a quarter position). Thus construction interference occur. If we compare a point just below the edge and another point just below the center the path difference is still one wavelength, and so on. Then we will get a contradictory conclusion that the pattern is a bright fringe at this angle. Why is it so? I guess there is something wrong there but I cannot figure it out. Also I notice that for other optical phenomena such as multiple slit they always try to find the bright fringe first. However, in single slit or aperture diffraction they always aim at finding dark fringe. If they try to find the bright fringe using similar argument they would get the same problem as above.

I know alternatively one can locate the bright and dark fringe by finding out the intensity of light at difference position of the screen. I have no problem understanding that approach. I just want to know how to understand this in terms of Huygen's principle. Thank you.