MIT's Baker House has a tradition of dropping an irrepairable piano six floors every Drop Day, the last day one can drop a class without penalty (the 2022 date is 19 April). This year, in order to prevent damage to the asphalt, you have been asked to erect a net to catch the piano as a hack.
Being a hack, your large net must be made of
- 23 aluminium poles for support
- 23 small nets, each consisting of five rings that can be threaded onto the poles, connected by elastic ropes in all possible ways (hence $\binom52=10$ ropes per small net)
To maximise the large net's strength everything should be used, while each small net should span five different poles and no two small nets should span two poles in common (i.e. each pair of poles is spanned by at most one small net). Baker House residents say you can do this, since the total number of ropes is $23\binom52=230$, less than the number of slots for ropes which is $\binom{23}2=253$.
- Find a solution to the task (an assignment of poles spanned to each small net).
- Is that solution unique up to permuting the poles and small nets? (no-computers does not apply for this one.)