Challenge 342: Chess Companions
A school chess club has 12 members. Each week, a group of six of them visit another school to play chess. The groups of six are picked at random each week.
(i) Amina, a member of the chess club, has been included in a visiting group with every other student in the club. Find the smallest number of chess club visits that Amina could have been on.
(ii) If each pair of students in the club have been in a visiting group together at least once, find the minimum number of chess club visits to other schools that could have happened.