In fact, if all of the islanders knew that they had to have either brown *or* blue eyes, then the 50 people with brown eyes would leave on night 51, leaving the lonely chieftan behind.

Explanation

Let's consider some simpler cases with less people. Let's say you are an islander, and everyone else on the island has brown eyes. The chief announces that he sees one person with blue eyes. That must be you, so you leave that night, knowing you have blue eyes.

Next possibility. Let's say you're an islander, and you can see everyone else on the island has brown eyes except for one who has blue, we'll call him Sam. The chieftan makes his announcement, and if Sam is the only one with blue eyes, then he should leave that night.
But what if he doesn't leave? That means he doesn't (yet) know that he's the only one with blue eyes - because someone *else* has blue eyes. Since there's nobody else you can see with blue eyes that must be you.
Hence, on the second night, you leave, now knowing that you have blue eyes. Sam leaves as well, because he's in exactly the same situation as you - he was waiting for you to leave, and when you didn't, he realized that he must also have blue eyes.

Then consider the case where you see only two people with blue eyes. If they don't leave on the second night, then it's because they *also* see two other people with blue eyes, one of whom must be you. All three of you will leave on the third night.

And so forth.

Meaning that if you see 'n' people with blue eyes, and they haven't left on the n-th night, then there *must* be 'n+1' people with blue-eyes, and one of them must be you. But if 'n' people leave on the n-th night without you, then you do *not* have blue eyes.