Once upon a time, we wanted to split 21 people into reasonable sized groups over a couple of rounds such that everyone see each other exactly once. We couldn't solve it. Few years later, I looked back and realised that by using affine planes it can be easily solved.
Affine plane of order 5
An affine plane is a concept in abstract algebra / geometry, which allows us to solve this problem. Therefore, the name - Affine Break Out Room. More details in the theory part.
Round 1: [0, 1, 2, 3, 4] [5, 6, 7, 8, 9] [10, 11, 12, 13, 14] [15, 16, 17, 18, 19] [20, N, N, N, N]
Round 2: [0, 6, 12, 18, N] [5, 11, 17, N, 4] [10, 16, N, 3, 9] [15, N, 2, 8, 14] [20, 1, 7, 13, 19]
Round 3: [0, 11, N, 8, 19] [5, 16, 2, 13, N] [10, N, 7, 18, 4] [15, 1, 12, N, 9] [20, 6, 17, 3, 14]
Round 4: [0, 16, 7, N, 14] [5, N, 12, 3, 19] [10, 1, 17, 8, N] [15, 6, N, 13, 4] [20, 11, 2, 18, 9]
Round 5: [0, N, 17, 13, 9] [5, 1, N, 18, 14] [10, 6, 2, N, 19] [15, 11, 7, 3, N] [20, 16, 12, 8, 4]
Round 6: [0, 5, 10, 15, 20] [1, 6, 11, 16, N] [2, 7, 12, 17, N] [3, 8, 13, 18, N] [4, 9, 14, 19, N]
We are aware that there are other ways how to solve this problem, e.g. using Kirkman Triple System, which can split 21 people into 7 groups over 10 rounds perfectly. Other solutions are either inefficient (10 rounds per 3 people) or don't include all pairs. Therefore, affine plane sounds like reasonable solution for small group sizes. For other group sizes, I recommend to use BoRAT - a handy tool with a nice paper about it.
[1] Pascoe, Abraham, "Affine and Projective Planes" (2018). MSU Graduate Theses. 3233.
https://bearworks.missouristate.edu/theses/3233
[2] Bartlett, Padraic, "Minilecture 5: Affine Planes" (2014).
http://web.math.ucsb.edu/padraic/ucsb_2013_14/mathcs103_w2014/mathcs103_w2014_lecture5.pdf
[3] von Gagern, Martin, "Affine Plane of Order 4 Picture?" (2016). Mathematics Stack Exchange.
https://math.stackexchange.com/questions/1925479/affine-plane-of-order-4-picture
[4] Kerl, John, "Computation in finite fields" (2004)."
https://johnkerl.org/doc/ffcomp.pdf