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I'd like to find an algorithm as simple as possible to solve the problem below.

The same seven students will each day be divided and meet into two groups, one with four students and one with three. The group with four students will be accompanied with the teacher. Each student should meet every other student as equally often as possible. All students should meet the teacher as equally often as possible. The number of subsequent days without the teacher should be as short as possible. The number of days is unknown and big.

Background: This is a real problem. I'm a teacher and my students are divided into presentation groups each time I have them. I want them to have me as a teacher as much as possible, but I need to divide them because otherwise the presentations will take up too long time. This split up is the best solution. I just want to make sure each student gets to hear as many different presentations as possible and also gets my feedback as much as possible.

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  • $\begingroup$ Is "every time they meet in a group each student should meet as many new students as possible" equivalent to "each student should meet every other student equally often as possible"? $\endgroup$
    – Fabius Wiesner
    Commented Sep 28, 2020 at 16:01
  • $\begingroup$ Yes! I'll change it to that in order to make it even clearer. Thanks for input. $\endgroup$
    – Zoe
    Commented Sep 28, 2020 at 16:07
  • $\begingroup$ There are ${7 \choose 4} = 35$ ways of dividing the students into the two groups, so I think it would be useful to know the number of days. For example, if the number of days is a multiple of 35, it is enough to repeat cyclically all combinations. $\endgroup$
    – Fabius Wiesner
    Commented Sep 28, 2020 at 16:11
  • $\begingroup$ The number of days might be bigger or smaller than 35 depending on how well the concept works. This means that a cycle shorter than 35 would be more helpful. $\endgroup$
    – Zoe
    Commented Sep 28, 2020 at 16:19
  • $\begingroup$ And I imagine that you would like not only that each student meets the teacher as equally often as possible, but also that the number of subsequent days without the teacher be as short as possible, right? $\endgroup$
    – Fabius Wiesner
    Commented Sep 28, 2020 at 16:25

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