I would solve it as follows. Think of each light as initially off. Then the cord is pulled once for each divisor of its
number. But a number of form p_1^{r_1}p_2^{r_2}...p_j^{r_j}
(where the p_i are distinct primes) has (r_1-1)...(r_j-1)
factors; this is easily proved. Now it is clear that the number of factors is odd exactly when the number is a perfect
square (i.e. all the r_i are even.) But pulling the switch an
odd number of times when it is off turns it on. (The problem
states they are all initially on, but think of them as
being off and turned on for the divisor 1.)
The trouble with enumerating the result - what if it had been
2,000,000,000,000,000,000,000 instead of 20,000?
chas Update Sorry, of course the number of factors is
(r_1+1)...(r_j+1), not (r_1-1)...(r_j-1). I guess I was
brain dead when I first posted. The conclusion is the same,
though; this is odd exactly when all the r_i are even.