It's almost time for another year of school! But before school starts, Principal Little needs to form classes. Because there are so many people in a class, the parents are always complaining, asking why the Principal could not just split the classes up to include another class. Being ignorant, the Principal does not pay attention. Every year, as the Principal is the one who makes the classes, it seems that there is not hope. However, this year, the Principal has made a change: Each student is allowed to request 3 friends to be with, and at least one will be in the same class as them. You, as the students' favorite teacher, have been begged by the parents to think of a plan to guarantee that there will be 5 classes instead of the usual 4. Here are the rules:
There are always 65 students in a grade
There can be at most 18 students in a class
If there is a way to have 4 classes, the Principal will have 4 classes
There are 4 pairs of siblings. Siblings can not be in the same class, and can not request the other sibling.
5 people left the school last year, and can not request or be requested
5 new people will join this year, and they can also not request or be requested
The kids will be bribed by their parents to listen to you
Is there any way for the students to work together and force the Principal to put them in 5 classes?
Can you do it if 5 very smart unknown students are trying to sabotage you?
Is there a way to have 6 classes?
Can you do it if you request 4 friends instead of 3?
What about if siblings can't request each other?