Question 423190
We see that for the nth switch, all multiples of n and only all multiples of n are being switched. Conversely, a number k is switched if and only if n is a factor of k.


In addition, numbers with an even number of integer factors will be switched an even number of times (hence, these lights will be off). Numbers with an odd number of integer factors will be switched an odd number of times, so these will be on at the end of 240 switches.


It can be easily shown that all perfect squares have an odd number of factors, and non-perfect squares have an even number of factors. From this, we conclude that all non-perfect squares (i.e. all integers except 1, 4, 9, ..., 225) will be off. There are 15 perfect squares, so the other 225 numbers remain off.