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The solution is correct.
- Gayle L McDowell June 03, 2013Okay, let's say C = 1. What will happen in this case? The person with the hat will see that no one else has a hat and remove it that night.
So, if C = 1, the hats are all gone after the first night. Do you agree with that? Are you sure? Okay. Great.
Suppose C = 2. Both hat-wearing people look around and see one other person with a hat. They know that C = 1 or C = 2, but they don't know which of these it is. But they *do* know that, if C = 1, then that other person would have removed their hat the first night [from the above paragraph]. So, after the first night, if both people see that the other person is still wearing a hat, then they must conclude that C != 1. Therefore, C = 2 and they have the other hat. That night, they both remove their hats.
So, if C = 2, the hats are all gone after the second night. Note that *both* hats are removed simultaneously.
Suppose C = 3. Again, the hat-wearers know that either C = 2 or C = 3. They also know that, if C = 2, then the hats will be removed on the second night. When the hats are not removed, they conclude that C = 3. They all remove their hats on the third night.
... and so on.