You and four other people (who coincendentally are all smarties) are in late testing room where you will take your test where there is a 6 by 6 grid of equally spaced desks with chairs in the same relative spot.

You go into the room after all four smarties have chosen their location. You have a test taking policy where you always want to sit at the midpoint between two smarties. The smarties in the room with you feel the exact opposite way, so their arrangement is always such that no smartie is at the midpoint of two other smarties

However, depending on where the smarties are sitting, you may not be able to sit at the midpoint since in all cases it would always be where there is no chair and desk. (There is a strict no moving desks or chairs rule too.) How many ways could the current 4 smarties sit such that you couldn't sit at the midpoint of two smarties if reflections and rotations count as well?

How many ways could you not find where you want to sit if there were 5 smarties other than you and reflections and rotations count as well?

The first person comes in and sits down at any of the 36 seats.

Interestingly, no matter where the first person sits there are 8 other seats where if the second person sat there you would have a midpoint seat. 

This gives the second person 36 - 1 - 8 = 27 choices of seat.

The second seating always creates 8 more seats where the third person would create a midpoint seat.  (There is no overlap from the first 8 seats ruled out)

The third person has 27 - 1 - 8 = 18 choices of seat.

Similarly the fourth person has 18 - 1- 8 = 9 choices of seat.

There are 4! = 24 orders in which these seats can be taken so the soluton is

36*27*18*9/24 = 6561 ways this situation could occur.

-Jer

 

Posted by Jer on 2004-11-05 13:00:36