The switches and the million of light bulbs

1. Do not reason bulb by bulb; each pass follows a regular divisor pattern.
2. Count how many divisors each bulb index has.
3. Only bulbs toggled an odd number of times remain on.
4. That happens exactly for perfect squares.

Answer: Exactly 1000 bulbs remain lit: those that occupy perfect square positions. Explanation: The light bulb $n$ changes state once for each positive divisor of $n$, because it is activated by every person whose number divides $n$. Typically divisors come in pairs: if $d$ divides $n$, it also divides $n/d$. That produces an even number of changes and the bulb ends up going out. The only exception is perfect squares. For example, in 36 the divisor 6 appears paired with itself, and therefore the total number of divisors is odd. Bulbs with an odd number of changes stay on. Between 1 and 1,000,000 there are exactly 1000 perfect squares:

$$ 1^2,2^2,\dots,1000^2. $$