Someone asked me:
"I am a teacher with 30 students. I want to arrange my students into groups that they meet with throughout the year. I'd like to have them gather into small groups (for example 5 groups of 6 or 6 groups of 5) each time we separate into groups. I want each student to have a way to meet with every single member of our class through this grouping system. How can I do this efficiently (is there a formula or tool I can use)? Is it possible to have groups created where they never see the same person more than once?"
This question really made me sweat. For a "30 choose 5" problem we have 142506 combinations, but how on earth do we form 6 groups of students, such that each group is unique? I can think of just two scenarios: {1, 2, 3, 4, 5} ... {26, 27, 28, 29, 30} and {1, 6, 11, 16, 21, 26} ... {5, 10, 15, 20, 25, 30}. Any other groups will violate the rule "...they never see the same person more than once". In order to achieve the main goal, I think you'll have to reduce the number of groups to 2. I have an intuition, that with more number of groups this is impossible, but is there a mathematical proof to this? Or maybe it is possible, and I'm hugely missing something?
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