A professor has to schedule four different exams in a $31$-day period. No more than one exam can be scheduled in any one day.
In how many ways can he schedule the exams if there should be a strictly positive, even number of days off between any two successive exams? For instance, he can schedule the finals on May $2$, May $5$, $12$th, and $23$rd; that is, there are $2$ days off between the first two exams, $6$ days off between the second and the third, and $10$ days off between the last two exams. Notice that in this case, no two exams can be scheduled in consecutive days.
We know that the total number of ways he can schedule the exams without restriction, other than no more than one exam per day, is : $31\cdot 30 \cdot 29 \cdot 28 = \displaystyle\frac{31!}{(31-4)!}=\displaystyle\frac{31!}{27!} = 755,160$ ways.
I am stuck on how to attempt this part of the problem with the restrictions given.