Situation 01: Greatest possible number of teachers who teach three classes
Let all 37 teachers first be assigned 1 lecture each.
Remaining lectures = 64 - 37 = 27 lectures
In order to assign two more lectures to each teacher, divide the remaining lectures in pairs of 2 = 27/2 = 13 full pairs + 1
Therefore, 13 teachers will get 3 lectures, 1 teacher will get 2 lectures and the remaining 23 teachers will get 1 lecture each.


Situation 02: Least possible number of teachers who teach three classes
Let all 37 teachers first be assigned 1 lecture each.
Remaining lectures = 64 - 37 = 27 lectures
Assign one more lecture to each teacher. Now remaining lectures = 0
Therefore, 27 teachers will get 2 lectures and 10 teachers will get 1 lecture each while NO teacher will get 3 lectures.