13 people who are not superstitious wish to have dinner together at a round table for a few nights so that each person has different neighbours every night. For how many nights can they do this? Please help!!!
2026-03-26 03:00:38.1774494038
Graph theory: decompositions and Hamiltonian graph
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You can do it for $6$ nights. (After that everyone has had everyone else as a neighbour.) Identify the people with the numbers $0,1,2,\dots,12$. On the $d^\text{th}$ night (for $d=1,2,3,4,5,6$) seat $x$ next to $x\pm d\pmod{13}$. This works because $13$ is a prime number.