The same eight people sit in a certain church pew every week, but not always in the same order. Every week, each person hugs the people immediately to his or her left and right. How many weeks does it take (at a minimum) for every pair of people to hug at least once?
+62 Let's say, that we want as many weeks to go by before person A hugs person B. Becuase the 8 people sit in a circle, we have to account for rotational symmetry. So, I'll break this up into different cases.
Case 1) A and B sit opposite to eachother.
We can pick any spot for A to sit in, and there is only 1 spot for B to sit in. Since we have 6 seats of any choosing, we have \(\frac{6!}{2} =360\)cases where A and B sit opposite to eachother
*We divide by 2 because there are 2 cases where the order is exactly the same.
The other cases are where B sits 3 and 2 spots away from A. I'll let you work them out.
