The Thousand Doors

The problem

A house has 1000 doors, all closed, numbered 1 to 1000. One thousand persons walk past them in turn. Person 1 flips every door. Person 2 flips every door whose number is a multiple of 2. Person 3 flips every door whose number is a multiple of 3, and so on: person $k$ flips every door whose number is a multiple of $k$. To flip a door means to open it if it is closed, and to close it if it is open.

After all thousand persons have passed, which doors are open?

Before you begin

Don't hunt for a formula. There isn't one to apply, and that is exactly why we love this problem. Try small cases first: what happens with 10 doors and 10 persons?

1. Follow door number 12 around. Which persons touch it? How many times does it get flipped?
2. Now follow door number 16. Which persons touch it? What feels different?
3. When does a door end up open: after an even number of flips, or an odd number?

Dissect the task

1. What makes this problem right for an open circle? Where is its floor, and where is its ceiling?
2. While a class works on it, which behaviours would you watch for: who experiments, who organises cases, who asks why?