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?

If 4 smarties couldn't let me sit in the middle and the same idea was with other smarties, all options would be eliminated for a 5th smartie as other grids would be unoccupiable. Howver, 4 smarties can sit in a 2 by 2 grid in the middle of the 6 by 6 grid such that they are all adjoing. This gives the 5 the smartie 20 options of sitting on the peripheral grids (20 peripheral grids) without having a smartie in between two others. So the number of ways 5 smarties could sit is:

A = 36 -1-8 (one sit is lost and other unsittable sits are opened. This will exist for smarties 2 to 4. However all can sit in the middle-most 2 by 2 grid to leave 20 open grids on the periphery which when occupied by the 5th smartie don't violate the constraint on of "no middle smartie."

So answer for 5 smarties is 20*36*27*18*9 = 3149280

Posted by Osi on 2004-11-05 15:05:34