October 2006: The locker room

Lockers numbered 1 to 100 stand in a row at the school gym. When the first student arrives, she opens all the lockers. The second student then goes through and recloses all the even-numbered lockers; the third student changes the state of every locker whose number is a multiple of 3.

This continues until 100 students have passed through. Which lockers are now open?

Dickinson College students can submit answers to Barry Tesman or Dave Richeson. The list of solvers will be posted at the end of the month.