I need some help with combinations existing within combinations.
I sometimes get myself in to various rabbit holes of thought, and this particular one has me banging my head against a wall. Any input as to what/where to search for further reading as well as ideas for approaching the problem would be appreciated. Hopefully this sub is the best one.
Lets say you have 6 people. You want to divide them into groups of 4 such that every possible group of 3 within those groups of 4 is represented with as few repetitions as possible. Weird problem I know, but best way I can explain it. Basically a 4 player game with 3v1, and I want every poasible group of 3 to play together once. For scheduling reasons, it makes sense for a group of 4 to play 4 games, rotating who is the 1, and then switch up groups and repeat.
So, 6c3 = 20 combos to represent. 4c3 =4 combos per group.
At first glance, it would seem 5 groups of 4 could cover all 20 combos, but in reality I believe repetition would force it to be more groups. I can not figure out how to mathematically prove this.
I was able to brute force playing with numbers to get 6 groups of 4 cover the 20 combos, with 2 of the 20 combos represented 3x and the other 18 combos represented once each. (I will not list the groups in case anyone wants to solve for it….likely multi possible solutions) would be great if you could do it with 5 groups, or even use 6 groups but better divide the repetition, but I dont see how.
If any of these numbers changed, especially if the 6 goes to 8, it becomes a mess to brute force. I would think you could prove the amount of forced repetition associated with any mix, I just can not figure it out.
Thanks in advance for any comments, even the useless ones!