Math Brain Teaser?

Question

This one is from http://www.mindchallenger.com
The answers to most are here. Originally, this was asked to me by a friend and I actually worked this one out. It took a few minutes. How long will it take you? (some of the details are changed to make it harder for some to just look up the answer on the internet).

You have 1000 coins on HEADS. The first person changes that state of every coin: if it is HEADS he flips it to TAILS, and if it is TAILS he flips it to HEADS. The second person changes the state of every 2nd coin. The third person changes the state of every third coin. This repeats up until the 1000th person. How many coins are flipped from their initial position of HEADS?

Answer 1

Each coin is flipped a number of times equal to the number of divisors of that number. For example, each prime number is flipped twice. In general, if a number is not a perfect square, it will be flipped an even number of times, so at the end it will show HEADS. If the number is a perfect square, it will be flipped an odd number of times, so at the end it will show TAILS. So the answer is the number of perfect squares less than or equal to 1000. These are the numbers 1^2, 2^2, 3^2, ..., 31^2, so a total of 31 numbers will be flipped from their initial position of HEADS.

Answer 2

Each coin is flipped, but I say 1/3 of the coins end on tails.