There are 12 people. These people start in 4 groups - each with 3 people. They swap groups 4 times, so they are in a total of 5 groups. Is it possible for each person to be in a group with every other person at least once?
The answer is
If each person has only been in 5 groups, they've only been in a group with at most 10 people (out of 11 others).