analyzing residual plot by hand

Thanks in advance for looking at my query. I have a very basic question regarding regression. We have to analyze a residual Plot by hand.
We do this in three steps. The third step I don't understand.
1. Divide in half from zero across and the overall points above and below the line should be close
2. Divide into 3 vertical regions check for spreadness/outliers in each section.
3. "In each of the 3 vertical regions the number of points below and above zero are close so it is OK"--This is the answer, but I am not sure what is meant. Does anyone have any ideas?

Also, when we analyze the residual plot the best outcome is that there is No pattern correct? What is the significance of the residual plot having no pattern?
What would a pattern signify?

Thank you in advance.
I understand what the author of these rules is trying to do, but these rules are (1) not very clearly written and (2) trying too hard to formalize something that has subjective elements.

Basically, when you look at a residuals plot, you want to see that about the right number of error bars overlap the regression line (about 2/3 for 1-sigma error bars) and that there are no obvious patterns to the residuals. A pattern in the residuals (e.g. a sine wave in residuals for a linear fit) could indicate a statistically significant phenomenon that was not taken into account by your model.