Here's a riddle I came across recently:
There’s a group of six friends who are all musicians. One day, they decide to have a few performances between themselves so that they can hear each other one play, each one with some of them playing and the rest in the audience listening. What’s the minimum number of performances they need so that each of them hears all of the others play from the audience?
I have the solution: 4 performances. They are:
1:$P123 \, L456$
2:$P561 \, L234$
3:$P345 \, L612$
4:$P624 \, L135$
We came to that answer without using maths, but to me it feels like there is a simple way to find the number of performances using maths. Is that the case? What would the equation be?