Problem #2 - How big is this Group ?

Don't worry, it's nothing about Group Theory. The following is an interesting combinatorics problem that I got few moments ago from one of my friend, Leonardo

"If every week $5$ person from a group are to meet with each other and they are assigned such that a person $x$ need to attend the meeting at most once a month and he/she will not see the same people ( either of the people) from previous meeting for at least $3$ months, what is the minimum number of the people in such group?" (1 month = 4 weeks)

Some clarification :

- If $A$ attends a meeting at week $W$, he can only attend again at least at week $W+4$ (so not only once a month)

- If both $A$ and $B$ meet at a meeting in week $W$, they can only meet again at week $W+13$

One simple fact is the group must have 20 distinct people to be assigned into 4 weeks of the first month (5 people each). The challenge is to find the minimum number of people in such group. Equivalently, if that minimum number is $N$, there will be no configurations satisfying the requirements in the problem with $N-1$ people.

See Solution