 Let us look now at solving a simpler analogous problem. This strategy requires that we consider a simpler problem that preserves the structure of the original problem. As most strategies discussed here, this might be a first step in a set of strategies to solve a problem. Consider this example. There are 1,000 mailboxes at a post office numbered 1 through 1,000. There are also 1,000 mailbox owners one for each mailbox. At the start of the mailbox challenge, all mailboxes are closed and the owners open and close the mailboxes according to the following rules. Owner Rule 1 opens every mailbox. Owner Rule 2 closes every second mailbox. Owner Rule 3 changes the state of every third mailbox. Closing it if it is open and opening it if it is closed. And so on, Owner Rule N changes the state of every nth mailbox, etc. When all the owners have taken their turns, how many mailboxes are open? If you have never seen this problem, I suggest that you stop for a few minutes and try to solve it. Considering 1,000 mailboxes might seem daunting, why not look at a simpler case? Let us take only 10 mailboxes and run through this process and see what happens. Number 1 opens every mailbox. Number 2 closes every even numbered mailbox. Number 3 changes the state of every third mailbox. Number 4 changes the state of all factors. Minus the number of everybody has 4, so that is 5. closed, open, closed. We see that we end up with only three open mailboxes, one, four, and nine. We might still not feel too confident about making a conductor, so why not look at 20 mailboxes? When we do that, we see that the open lockers are one, four, nine, six. This is one squared, two squared, three squared, and four squared. It might be that all the perfect squares are open, and if that is the case, since 31 squared is equal to nine, 161, 31 will be open at the end of the day. But how can we be sure? Why are perfect squares so special? What are they different? Well, consider any other number like 12, which is not a perfect square. When you look at the factors of 12, you have one, and 12, two, and six, three, and four. You have an even number of factors, but when you look at a perfect square, say like 16, you have one, 16, two, and eight, four, and four, because you have four here twice. You have an odd number of factors, since all perfect squares have an odd number of factors, you are taking an odd number of actions on them, which means those are the ones that will be open at the end of the day, and there will be 31 of them. Thank you.