2011-06-11, 02:23 AM
Well, I assumed that "know each other mutually" means that if Alice knows Bob, and Bob knows Carol, then that group of 3 counts as knowing each other, because Bob knows all 3 members.
As such, deductively:
- there can only be two or fewer groups that are linked in this way (if there are 3, take 1 from each and they don't know each other)
- each group can only have 2 people in it at most, otherwise the 3 people in that group are mutual acquaintances.
there's no way for 5 people to satisfy both.
If it requires a group of 3 people in a triangle, all knowing each other, then there's no limit on the size of the group.
However, you
- can't have loops of fewer than 4 people (forms a triangle)
- can't have a loop of 4 people, along with anyone else (take every 2nd member, plus one from the other group)
- can have a loop of 5, as long as there's noone else (your first example)
- can't have loops of 6 or more people (take every 2nd)
As such, deductively:
- there can only be two or fewer groups that are linked in this way (if there are 3, take 1 from each and they don't know each other)
- each group can only have 2 people in it at most, otherwise the 3 people in that group are mutual acquaintances.
there's no way for 5 people to satisfy both.
If it requires a group of 3 people in a triangle, all knowing each other, then there's no limit on the size of the group.
However, you
- can't have loops of fewer than 4 people (forms a triangle)
- can't have a loop of 4 people, along with anyone else (take every 2nd member, plus one from the other group)
- can have a loop of 5, as long as there's noone else (your first example)
- can't have loops of 6 or more people (take every 2nd)