There are N closed lockers numbered 1,2,...., N, each assigned to a different person. The first of these N persons opens all lockers. The second person goes to every second locker and closes them. Then, the third person goes and alters the state of every third locker- that is, opens the locker if closed, or closes it if open. The fourth person alters the state of every fourth locker, and so on, until the last (Nth) person alters the state of just the Nth locker.

At the conclusion of this process, it is observed that the total number of closed lockers is precisely sixty-eight (68) times that of the total number of open lockers, with the total number of open lockers being a prime number.

Determine the value of N. How many lockers end up open?

(Inspired by the lockers problem submitted by Erin.)

I just solved this, but since I've had a few days to look at it in the queue, I'll hold off on posting.  Also, there appears to be a simple formula for starting with the ratio of closed to open lockers (68 in this problem), and determining N.

The solution to the previous lockers problem makes this one much easier.

Posted by Larry on 2006-06-23 15:52:21