AoPS Wiki:Problem of the Day/August 23, 2011
Twenty bored students take turns walking down a hall that contains a row of closed lockers, numbered 
to 
. The first student opens all the lockers; the second student closes all the lockers numbered 
, 
, 
, 
, 
, 
, 
, 
, 
, ; the third student operates on the lockers numbered 
, , 
, , 
, : if a locker was closed, he opens it, and if a locker was open, he closes it; and so on. For the 
student, he works on the lockers numbered by multiples of 
: if a locker was closed, he opens it, and if a locker was open, he closes it. What is the number of the lockers that remain open after all the students finish their walks?
