Grade 12 Data Management question, involving Combinations

[Hipparchus]

I'm sorry, I didn't really know where to post this question, so I posted it here, because it is the closest to Data Management, I think.

There are 19 students in the classroom. The students are seated in a circle.
Each student shakes the hand of the person sitting to the left and the right of them.
You can change where the students sit.
The goal is to have each student share a handshake with every student in the classroom.

Find the LEAST number of different circles required in order for each student to have shared a handshake with every student.

I received this question on a test I took recently, so I do not know the correct answer.

I made two pathetic attempts at solving this question.

1: I calculated 19C2, in order to find how many handshakes must occur in total.
19C2 = 171.
I divided this number by 19 for every student in the classroom to get 9.
I only did this for the sake of writing something down, and because it seemed right. But, no logic.

2: I found the possible ways a circle could be formed with 19 students; 19!. And placing one student in every seat, and moving the others so that they do not come in contact with the same student more than once, but this seemed very long, and very difficult. And it doesn't seem like a logical approach to this question.

Thank you, in advance

 

[kurt.physics]

. The least number of circles required in order for each student to have shared a handshake with every student is 3. This can be achieved by having the students form three circles, with 6, 6, and 7 students respectively. Each student will then shake hands with the students to their left and right in each circle, resulting in every student having shaken hands with each other.