Only the lights connected to switches with a number that has an odd number of factors are left on.  These are the square numbers.  The largest square square number less than 1000 is 31^2 = 961, so there are 31 lights on connected to the switches 1^2, 2^2, … 31^2.

/*
Yep Thats Perfect!
Following is a java program to do the same thing.
*/
class k1{
public static int[] factorize (int a[], int p)
{
int count=0,no=p;
int b[]= new int[a.length];
b=a;
for(int i=2;i<=p;i++)
if(p%i==0){b[count++]=i;}
b[count++]=1;
return b;
}

public static int no_of_factors(int b[])
{
int counter=0;
for(int i=0;i<b.length-1;i++)
{
if(b[i+1]!=b[i]&&b[i]!=0) counter++;
}
return counter;
}

public static void main(String args[])
{
int on_count=0;
int a[]= new int[50];
for(int i=1;i<=1000;i++)
{
factorize(a,i);
int x = no_of_factors(a);
if(x%2!=0) on_count++;
for(int j=0;j<50;j++)a[j]=0;
}

System.out.print(on_count);

}// end of main
}// end of class