The teacher in a certain class room allows you to pass a paper with an assignment around, and whomever it ends up on has to do it. The only two rules are you can't pass it to someone who already has had it and you can only pass it to the person to the left, right, forward, or backward.

In a room of 30 students arranged in a 6 by 5 grid, the teacher starts out with the assignment somewhere on the front row of 6 students. At some point someone is stuck holding the assignment because all his neighbors have had it and passed it on to someone else. If this happens after every student in the room has had it, what is the probablity, for each individual, that he or she turns out to be the lucky winner of the assignment?

(In reply to This is very difficult by Erik O.)

Yes it is very difficult.  At least it's not as difficult as teaching 30 factorial students, like Jer does!

In your 3x2 grid, the numbers seem to be wrong.  To clarify what Gamer meant, first a random person in the front row was chosen.  Each student passes the assignment in a random direction.  It just so happened that when someone got stuck with it, everyone had had the assignment passed to them.

Let's say the arrangement was this:

123
456

The paths branching from student 1 are:
123654
145236
145632
Student 2:
214563
236541
Paths from Student 3 are symmetrical to student 1.

Notice that the first three paths do not have even distribution.  1 passed it to 2 or 4 with equal probability, so the first path is twice as likely as either of the next two.  (If you interpret the puzzle differently, it is possible to say that the first three paths are equally likely though.)

Anyway, with that in mind, these are all the probabilities with a 2x3 grid:

1/6 1/6 1/6
1/4 0   1/4

I, too, would have a lot of trouble expanding that method to a 5x6 grid.

Edited on June 24, 2004, 4:42 pm

Posted by Tristan on 2004-06-24 16:41:45