That Pesky Blue-Eyed Islander Puzzle

You may already know the famous (infamous?) Blue-Eyed Islander Puzzle. Here’s my version.

99 perfect logicians are stranded on a remote island. As perfect logicians, each is able to immediately determine the logical outcome of any statement.

These perfect logicians all have blue eyes. Each Islander is able to observe all the others. However, they are all unable to communicate with one another. Furthermore, each is (somehow) unable to observe his/her own eye colour.

As a result of these circumstances, each logician knows the following to be true:

At least 98 people here have blue eyes.

Each day at noon, a flying saucer arrives. If any person knows his/her own eye colour, that person is transported to the flying saucer, in full view of everyone.

Each Islander is aware of all of these conditions with one restriction: no Islander knows his or her own eye colour. Each day, the flying saucer comes and goes. No one leaves the island.

One day, a voice from the flying saucer announces:

One of you has blue eyes!

The flying saucer then leaves.

How many of the Islanders eventually leave the island, and when do they leave?

Even once you know the solution, you can trace the whole air-tight argument, something feels wrong.

One of you has blue eyes!

Everyone already knows this, for crying out loud! What changes ?If you were on the island, how could you eventually figure out your own eye colour?

I presented one approach to a solution on YouTube.

Solution

I’m going to take a different approach here. You can be the judge of whether it clarifies or muddifies the solution.

If you’ve never seen this puzzle before, let me encourage you to spend some time on it before reading the solution. Bookmark this article and come back to it later.

We approach a puzzle like this hampered by our own omniscience. We know too much.

You know that feeling you get during Thanksgiving Dinner when Great Aunt Alita enthusiastically shares the details of her latest colonoscopy? What goes through your head?

To much information!

We approach a puzzle like this hampered by our own omniscience. We know too much. To show you what I mean, I shall introduce a new player, whom we shall call Diophantus. This new player is the embodiment of the Common Knowledge of all players. That which everyone knows, Diophantus knows.

More importantly, that is all that he knows.

Unlike us, Diophantus is not omniscient. The sum total each individual’s ignorance is found in him.

Consider one Islander whom we will call Alice. Alice sees 98 pairs of blue eyes. One piece of knowledge she lacks: the colour of her own eyes. Because Alice does not know her own eye colour, neither does Diophantus know Alice’s eye colour.

By extension, Diophantus cannot know anyone’s eye colour.

The 98 pairs of blue eyes Alice knows about are different from the 98 Bob (another Islander) knows about. The statement, “At least 98 Islanders have blue eyes” lacks the specificity to qualify as Common Knowledge. Alice means something different than Bob by “98 Islanders.”

We can place Diophantus on the spacecraft. He visits the island daily. One day, he sets the ship’s sensors to detect blue eyes. That day, the sensors tell Diophantus that one of the Islanders has blue eyes. This is news to Diophantus. Before now, he didn’t know if anyone had blue eyes.

He exclaims:

One of you has blue eyes!

If Diophantus has new knowledge, then everyone has new knowledge. How does “one of you” differ from “at least 98”? The difference is subtle: “One of you” is a specific person. It could be Alice. We don’t know. But unlike “98,” which refers to a different set of persons depending on who thinks it, the phrase “one of you” refers to one particular individual.

The day after the announcement, Diophantus returns. If Alice (say) is the only blue-eyed islander, she’ll have figured it out.

But Alice doesn’t board the ship. One other islander must have blue eyes. Diophantus now knows this, as does everyone else.

Suppose that one other person is Bob. Bob will see 97 pairs of non-blue eyes and Alice’s blue eyes. Bob will know he is the other blue-eyed islander. Alice’s mind follows a similar process.

But the next day, Alice and Bob do not board the ship. A third person — Charlie — must have blue eyes. Alice sees Bob and Charlie with their blue eyes. Only when the ship leaves a third time does she realize hers are the third pair of blue eyes.

And we’ll stop there.

Let me ask you now, Dear Reader: Does the introduction of Diophantus make the solution feel more intuitively correct? Or has it complicated matters? Please share your thoughts.